I've figured out the niches of sudoku I like to set: arithmetic, sets, and geometry. (Though I still don't want to touch anti-knight with a 10-foot long pole.) Since the role of these goals was to find my niche, in some sense, I've succeeded in my meta-goal, even if I'm technically giving up on my resolution.

How did we get here?

I boarded a plane from PIT to DFW which took off about 7:15 PM, Pittsburgh time. (It's an hour ahead of Dallas.) Everything is all hunky-dory, and we land at about 10 PM Dallas time. We bust our asses off and get on a plane from DFW to SFO which took off about 11 PM. So far so good.

But then a gas leak(!!) of all things occurs in the plane, and so we have to turn around. Initially I was a bit annoyed that there was a problem with our flight, but it quickly turns into a horrified sort of gratitude that the plane didn't blow up in the sky instead. Anyway, we hadn't been flying long, so we were able to land safely.

The plan originally was to get everyone out of the plane, including the pilots, and then all hop onto a different plane. However, the pilots reported feeling sick(?)¹ because of the gas fumes and thus were unable to fly. So we are now waiting for another set of pilots, and who knows how long that'll take because (expectedly) pilots are sleeping at 1:30 AM.

The next scheduled flight is at 5 AM, and if there is no replacement flight, I fear the passengers may need to fight for the seats on the 5 AM flight. (The plane with the gas leak was a full flight.)

So yeah, this has been a great start to my fall break. But genuinely, I am so glad to be alive right now.

Update: They found a flight at 7 AM and gave us meal/Uber/hotel vouchers. Unfortunately, the only hotels around were about 20 minutes away, but at least the hotel room was very nice. I slept for about 3 hours until I woke up at 5 to go to the airports and got McDonald's.

The 7 AM flight got to San Francisco without a hitch. I am still tired as hell, but all is alright, I suppose.

"My parents don't want to pay" or "I ran out of allowance" is not going to get your tuition waived. If you have genuine financial need, then of course that is a different story.

In some alternate universe I would've started charging for classes while saying this. In that alternate universe, someone who needs financial aid might not have applied for my class, or worse, they would have paid for it. And this fall, I am charging for classes.¹ So let me clear this up for the record.

You know, I don't like charging students extortionary fees.² In fact, I charge a nominal fee for two reasons: so everyone takes the class seriously, and to raise funds for Math Advance.

I have bad memories of debating whether to pay 1500 a semester for some AIME classes and regretting it dearly, and I don't want to put anyone else in a similar situation. Most of my students are upper-middle class Americans who can see a price tag of $100 and say, "wow that's pretty cheap let's go for it" without a second thought.

But I recognize this is not everyone.

There are countries where 100 dollars goes a lot further than it does here. People have individual circumstances. The particulars are none of my business.

In any case, the message I was inadvertently sending was that I was concerned about abuse. And well-intentioned students might think, "wow there is a problem, and I should avoid trying to be part of it".

But there is no such problem. I can think of only one instance where someone acted in bad faith in my classes (a kid posted a handout on CourseHero, hardly what I'd call a systematic pattern of abuse), and zero from my work at Math Advance.

I don't mean to brag, but I will anyway: our program attracts good students. Really good students. Many of them qualify for MOP. It wouldn't surprise me too much if we start seeing IMO qualifications soon. And all of them are hardworking, honest students who respect the program and leave it better than they found it. If you are a prospective student reading this, then this description probably applies to you too. I will say it again: we attract good students, and you hopefully will be among that number. Not a single one of our students has turned out to be a bad actor, and if you are worrying that you might become the first, you most probably are not.

I am rather more concerned that there will be a lack of use than a pattern of abuse. So if you think you need financial aid, you need financial aid. And you will get it, no questions asked. Very rarely will a program accept "trust me bro" as a reason for financial aid, and for good reason. But for better reason, we do.

To make a long story short: if you think you maybe might kinda probably sort of need a little financial aid, then you should ask for financial aid. We need to be more proactive in making sure not all our students are upper middle class Americans, and this post is the first step.

It was a success in that I further developed my range with standard variants, like killer, diagonal, thermo, etc. It also was a success in that I wrote using 4 variants I never used before: diagonal, quad, X-lines, and starry sums. Two of these are new variants invented by me: X-Lines and Starry Sums, and I particularly think the second has a lot of potential.

It was a failure in the sense that I didn't use four new standard variants, so my range of setting does not even encompass many of the common variants. There's a lot I'm missing: sandwiches, between lines, etc, that just feel out of my alleyway right now (which I suppose is why I never set them). I am not too concerned about this fact though.

So supposedly I'm slated to write a classic sudoku this month, but a couple of reasons this might not happen:

• I'm interested in writing a Monster Cages puzzle, where the clue indicates the product of the sum of shaded cells and unshaded cells. (Or I can make it a Chaos Construction and make clues indicate the product of the region sums... that would be kind of cool.)
• Also my latest puzzle, "Intersections", has an obvious dual: "Mind the Gap", where X-Sum clues in the same row/col add to less than 45. I don't know if including diagonals would be very thematic so I won't use them.
• I also want to start learning more math/CS in preparation for college next year.
• I'd like to reignite some activity and come back to teaching in Math Advance.

So who knows if it'll happen in time. And as for the April goal, probably I will not end up writing a crossword. I don't really get how they work as a solver so my debut as a setter still seems pretty far.

Given the name for this puzzle, I think it'd be very easy to use different cage totals and make this a recurring series. For example, if I wanted to use 10 and 15 cages, I could call it 10:15, etc. Kind of like a knock-off Killer Cages version of Florian Wortmann's Colorado series.

And the best part is, it only took somewhere around 2 hours. Given how long it's been since I set my last puzzle, I'm really proud of that.

More life updates, including the recounting of a track meet, my struggles in AP Chemistry, and roommate rizz, hopefully coming tomorrow :)

But for college apps I'm supposed to focus on myself, using anecdotes about myself and making observations where I can.

So the difference is, rather than writing "this is my takeaway, here are some stories" it has to be "this is my story, here are some takeaways". It's a little difficult for me to write like that since I do it so rarely.

1. Blame Everything on Consensus

Rejecting people is hard. So is doing things that aren't popular. But even harder is explaining why. Fortunately, effective leaders never have to explain anything.

"Why'd you ban me from the server?" Sorry, pal, it wasn't me --- it was the group that banned you. You can ask me to explain and I'll "consult the rest of the team" until you stop.

2. Ask People to Do Work For You

As a leader, doing work is beneath you. That's why you have to get your subordinates to do it for you. But be careful, don't fall into the trap that many others have fallen into before. Don't work to make other people do work. This includes doing things like the dreadful "communication". Your time is better spent writing angry blog posts anyway.

3. Be Pissy When Things Don't Go Your Way

Sometimes, just asking people to do your work doesn't work. Fortunately, there is a simple solution for this.

Decision theory states that all animals weigh the potential gains and losses of an action before doing it. Humans fall into the animal kingdom, last time I checked my National Geographic. If you can make the potential losses (i.e. time and sanity) greater, maybe people will do what you want next time. It's not like they can kick you out, you're the leader for crying out loud.

4. Take Second Chances

Your mama always told you everyone deserves a second chance. And as a leader, a brave pioneer, a daring venturer whose risk of failure is naturally higher, you deserve a lot more second chances than anyone else. So be brave and take them, even when no one else is brave enough to admit that you deserve it.

5. Take Breaks

Leading is hard work. So is writing this article, so I'll be taking a break from explaining this point...

6. Waste Time

Not yours, because it's too valuable to waste, but everyone else's. Find people who would be willing to do anything for you even if you have no clue what you need them to do. To prevent anyone else from figuring out you have no clue what you're doing, waste their time to distract them. They won't notice.

7. Hold Meetings

Everyone knows that important people meet often, and you're important, so meet often. As a bonus, you get to hear yourself talk. How cool is that!

As a double-bonus, you also get to fulfill the sixth point at the same time. Incredible!

8. Take Responsibility

Make sure you take responsibility for all of your successes. Don't worry about mistakes, those are what lesser leaders make.

9. Shift the Goalposts

After getting such good nuggets of wisdom it wouldn't be fair to expect yet another one from me. After all, this post was so good it might as well be worth ten of someone else's tips. Or even a hundred, pick whatever arbitrarily large number you like.

Well, that is kind of the point. There's probably a selection bias happening here: because there's an application process and I'm very vocal about only accepting very self-motivated students who take things into their own hands, the only kind of people who would apply are the ones who would be accepted anyway.

But this selection bias, while a nice side effect of the application process, is not the main point. The point is to get people used to "shooting their shot", so to say. Instead of having your parents fill out a class registration form for you, and getting driven there, students now have to reach out to me, put in their own effort to get in the class, etc. In fact, it's because enough people have taken this initiative over the last year that I decided to teach a class again to begin with.

Also, I will be a lot more expository than the lecture was. I feel like you had to connect a lot of dots in your head in the lecture, whereas the point here is to at least provide a reference for myself to turn my brain off.

No guarantees that the details are entirely right since this is off memory and Google is really, really bad when it comes to formal logic stuff. Also I will be vague since this is a general overview.

Anyway, it's math time!

First we establish the following:

(Finitary Function) A finitary function is just a function with a finite number of inputs.

An example of a finitary function is addition on the reals: it takes ℝ × ℝ → ℝ.

An example of an infinitary function (i.e. not finitary) is one that takes an infinite number of reals as input and outputs their sum; e.g. $f(1, \frac{1}{2}, \frac{1}{4}, \cdots) = 2$.¹ Now here is where things get real tricky. Take the function g that does much the same as f, only it takes one (which you will note is a finite number) infinite sequence of reals. For example, $g([1, \frac{1}{2}, \frac{1}{4}, \cdots]) = 2$, where the input of g is the sequence $[1, \frac{1}{2}, \frac{1}{4}, \cdots]$ rather than each individual element.

This seems super cursed; f and g are fundamentally the same and yet they are classified entirely differently for a theorem who we will see only cares about this one classification. The point is about finiteness of the number of inputs though, we don't need the members of the input to be finite or countable or whatever.

(Inductive Structures) An inductive structure is an ordered pair (X, F) where X is a set and F is a set of finitary functions.

This definition is very general. There is no restriction besides that every function is F is finitary. F itself may be infinite, uncountable, whatever. So may X be. In fact, not every function in F needs to even admit any member of X or tuples of members of X as inputs.

A standard inductive structure is (ℝ, {+, ×}). Note + and × are functions that take two inputs in ℝ and return an output in ℝ. For instance, +(2, 2) = 4. Yes, this is cursed notation, but you can think of the binary operator + as a function with two inputs. And in fact, you have to when strictly considering inductive structures. Also, this inductive structure feels closed. This is a word we will define later, but + takes two members of ℝ to a member of ℝ. So does ×. You should intuitively feel that this inductive structure feels closed-ish.

Here is a more cursed inductive structure. Let X be the set of odd integers and Y be the set of even integers. Then let f be some function from Y to Y. Then (X, {f}) is an inductive structure, despite the fact that f's domain does not include any member of X. Cursed, isn't it? But it still falls under the definition, even if this inductive structure does not actually do anything interesting.

Now for a bit of notation that will make it easier to refer to f's arguments for arbitrary f ∈ F. The motivation for this notation is vectors, because in some sense vectors are just lists.

We denote the list a₀, a₁, …, a_n − 1 as a⃗. We will refer to an arbitrary element in the list a⃗ as a_i.

Now for our final definition:

(Closure) A closure under F is a set X such that for every function f ∈ F and every a⃗ in the domain of f such that a_i ∈ X, f(a⃗) ∈ X.

Say X is a closure under F. Then you can naturally construct a closed inductive structure (X, F). We won't worry about that in this post.

The argument

Now for the theorem.

For any set X and any set of finitary functions F, there exists a set Y such that X ⊂ Y and Y is a closure under F. Furthermore, we can find a least upper bound Y, meaning that there exists some Y such that for all Z satisfying the same property, Y ⊂ Z.

The main difficulty is showing such a Y exists. The least upper bound property is pretty easy to see just from how Y is explicitly constructed.

Y exists

We will recursively generate a sequence X_i. Let X₀ = X, and let X_n + 1 = X ∪ {f(a⃗) ∣ f ∈ F, a_i ∈ X}. In other words, we look at what we get from taking f(a⃗) with every input in X, and join it to X to find X_n + 1.

Now let Y = X₀ ∪ X₁ ∪ X₂ ∪ ⋯. We claim Y works, and we show this by proving if each a_i is in some X_i, then f(a⃗) must be too.

Note each a_i by definition is in some X_i if it is in Y. Because X_i ⊂ X_j for i ≤ j, and there is some maximum X_m in the list of X_i that a_i are in, a_i ∈ X_m for all a_i. This is where we use the finitary condition, because otherwise we have no guarantee this maximal X_m exists to begin with. And by definition, f(a⃗) ∈ X_m + 1 ∈ Y.

Y is a least upper bound

This is a standard set theory argument; we just want to show that z ∈ Y ⟹ z ∈ Z for the Y we have just constructed. (Z is an arbitrary closure under F that is a superset of X.)

We claim that if z ∈ X_n for any X_n, we must have z ∈ Z. We prove this via induction on n.

The base case where n = 0 holds true by definition (X ⊂ Z).

If z ∈ X_n + 1, then either z ∈ X_n or there is some a⃗ with all a_i ∈ X_n such that f(a⃗) = z. This follows straight from the definition of X. If z ∈ X_n then we are done. Otherwise we are done by the definition of a closure.

Finitary is a lie

A subset of our discussion with Prof. Cummings after class. This is where I don't really know what I'm saying, which is why it is quite hand-wavy.

The finitary condition should make you feel like something is off; like it isn't really necessary. And in fact, it isn't if you use the right perspective. In the "Y exists" section, we indexed the finite set X_i with a countable set, namely the natural numbers. The reason we cannot let X_i be infinite (more precisely, have the same cardinality as ℕ is because then we cannot find some maximum in ℕ of these elements in ℕ.

But what if we had a number that was guaranteed to be "bigger" than anything in ℕ? Another source of motivation: don't you see how the size of the indexing set ℕ is "just an infinity bigger" than the finite set X_i? If we index with the ordinal number ω₁, then we can find some "maximum" in ω₁ that is bigger than every element in the set of size ω (you can treat this as ℕ here) of our finitary set. And thus, if we commit to using ordinals for indexing, we can have the number of inputs be ordinal too. And thus the finitary condition can be dropped.

Apparently all of this is legal in set theory land. They just tighten their argument up much more than I am right now.

Also, if you are a student reading this and have not received any emails from me containing general advice or an invite to the class Discord, please email me at dchen@mathadvance.org.

• Qualified for AIME
• They (NOT their parents) email me

This is for a couple reasons:

• Only AIME qualifiers have even asked me to join MAST (obviously there might be some selection bias since AIME qualifiers are far more likely to be confident enough to ask). But I don't think MAST will be particularly helpful for non-qualifiers anyway, and we will probably hold summer classes for less experienced contestants anyway.
• Emailing me demonstrates that you show some non-zero initiative of your own. I think communication is a lot easier when you don't have to play telephone through parents (so I typically do not bother).

MAST late apps will be open throughout the entire year! You won't be able to submit Season 4 late applications once Season 5 starts (you can replace 4 and 5 with n and n+1 in general), but otherwise, there are no deadlines.

Obviously this is not a blank check to ignore the regular application period. But if you learn about MAST in May and have already made the AIME, I see no reason to stop you from joining the program.

Also, I really only like shows that are more serious, I do not watch a ton of comedies. But if you liked Squid Game (or at least the first six episodes) this list is probably for you.

• Alice in Borderland (manga, JDrama)

  Note that Season 2 of the JDrama is coming out December 2022, the manga is finished.

• Yugami (manga)

• Liar's Game (manga)

• Psycho-Pass (anime)

• Madoka (anime)

• Kaguya (manga, anime)

  OK, it's not finished and it also doesn't fit with the rest of this list, but come on, I had to include it.

• Anohana (anime)

• Death Note (anime)

• Stein's Gate (anime)

• Tokyo Ghoul (manga)

There is nothing to do but walk forward.

So you walk, feeling the soft sand beneath your feet, smooth from the waves that slowly wore it down. You don't think about it in the moment, but one day you'll realize that the cliff was here centuries ago, and the ground was made of hard, jagged rocks.

Yet you keep walking, and to your surprise, you spot another person in the distance. They are only a couple of inches taller, but it seems as if they tower over you. Once you get a few steps closer, the pleasant smell of the ocean is overpowered by a horrible cacophony of odors. In a few seconds, you realize this putrid scent is coming from them.

They greet you, and you mumble back an awkward hello. You head on your way as quickly as possible, but it seems like they want to tell you something. For a moment you resign yourself, but then you remember that horrible stench. So you run. You run as fast as you can, and they are chasing after you, shouting something you can't make out, and you run even faster, you are running for your life.

By the time you stop running, the skies have turned red and the sun is about to set. You look back and they are nowhere in sight. Now that the danger has passed, you are so, so tired. You think to yourself, no, you can't quite rest, it isn't dark yet, as if you only deserve to rest when it's properly nighttime. But your body, exhausted from all the running, screams at you. So you acquiesce.

You lie down on the sand, and as you are about to fall asleep, you notice your pockets feel a little lighter than before. You think back to what you might have lost but nothing important comes to mind, and you slowly drift back to sleep. ∎

• Trump appointees: Neil Gorsuch, Brett Kavanaugh, Amy Coney Barrett
• Original conservatives (i.e. not Trump appointees): Chief Justice Roberts, Clarence Thomas, Samuel Alito
• Liberals: Elena Kagan, Sonya Sotomayor, Ketanji Brown Jackson

It's also worth knowing that Gorsuch, Kavanaugh, ACB, and Jackson replaced Justices Antonin Scalia, Anthony Kennedy, Ruth Bader Ginsburg, and Steven Breyer, respectively. Jackson was a Biden nominee.

For the February goal, I obviously don't know every common variant yet (I recently learned about non-disjoint, for example). So, I'll just consider the common variants I listed as examples, and any (fun) variants I haven't used in January will count. (Which means that if I set one of the variants I list in January, it doesn't count in February.)

The March goal is actually way too easy. I wrote two easy classic sudoku puzzles already, but they don't use intermediate to advanced techniques (partially because I do not recognize them quickly or consistently enough). So, I want to set a classic with at least an X-Wing.

As for what I plan to do for the remainder of January, I want to explore the following two puzzle types I've invented: the close/distant neighbors variant (more generally, positive global restrictions on the difference of neighbors), and Equations (which I think had too few given digits and too many given equations). Stay tuned for more puzzles!

Let me tell you a bit more about him. When he was in 7th grade he applied to the MAST program, not having qualified for the AIME. But through his solutions to the application problems, I could see that he had a lot of depth and could solve really hard problems, and so I invited him to the program.

I was more right than I ever could have imagined. A few months later, he started writing geometry questions and sending them to me. At first I could handle the number he was sending me --- for about a few weeks or so (if my memory is correct). Then the output ballooned into something I couldn't review alone, so I invited him to the contest-writing branch of Math Advance in the hopes that the rest of the team could help review. And even after the rest of the team was looking at his problems, we still could barely testsolve most of his questions --- and our team has a ton of MOPpers and USAMO qualifiers. That's how hard they were, and that's how many he wrote. He's also helped out with MAST and wrote a couple pieces of advanced number theory content.

Two years later he qualified for the USAJMO. This year he qualified for the USAMO --- as a sophomore. He'd been grinding ISLs and not only did his depth extend, he also became more accurate and more articulate in communicating his mathematical ideas.

All this is to say: this is not particularly surprising news. Given his work ethic and his contributions to our program, it was really just a matter of time. Congratulations again, Aprameya, and thanks for all your contributions to Math Advance. Let's continue making more cool stuff like the geometry contest that we've been planning for literally two years!

1600m

5:44.80, PRed over my Freshman year Dan Gabor time (by one second!)

The 1600m was the first event most of the distance runners were racing. My heat started at 5 in the afternoon, but we came too early --- we arrived at 3, when we only need 45 minutes for warmups. So we lounged around and did nothing until 3:50, when we began warmups.

Warmups also began too early, we had literally 30 minutes between the end of warmups and when our heat got called to race. So that was 30 minutes we had to basically fill up with random strides. But I think I was in pretty good condition before the race.

In the 1600 (and the 800) we started in a double waterfall formation. The inner waterfall can merge immediately but the outer waterfall has to merge after the first bend. I think it's better to be in the inner waterfall so you can see everyone else, but then you might get passed/blocked early on. I was in the outer waterfall.

I started out pretty strong during the first hundred meters, but got flat tired as someone stepped on my shoe AND drove their spike into my heel. The hole in my foot didn't hurt as much during the mile (adrenaline!), but would really suck for cooldowns/warmup between the 1600m and 800m. The flat tire made it a lot harder to kick in the end since I couldn't sprint effectively. Plus, some poor tactical moves in the second half of the second lap and the third lap also contributed to a slowish time.

Goal next time: 5:35.

800m

2:29.38, PR by a lot (did not get a good 800m race before this)

After the race the pain in my foot started to kick in, to the point where I had to hobble towards our team tent. I couldn't cool down properly since I could barely walk, so I decided to sit down and eat some food to replenish my energy.We did nothing for about an hour and in the interim I got a bandaid on my foot.

Afterwards we did some more warmups (not as long as the usual pre-race warmups) and this time we finished right before my heat got called to the bullpen. (So we started a bit late then, especially since we had to cut the warmup run short.) It was cold as hell when I was waiting in the bullpen, and I honestly regret not bringing my jacket there.

Again, I was in the outer waterfall, and I made sure to position myself right at the front of lane 5 this time around. I relied on adrenaline to make my pain "go away". Funnily enough this is the third race I've relied on adrenaline before a race. Some part of me would always hurt enough to make it hard to walk and right before the race I just think "it will be fine the pain will go away".

I started the 800m near the front and was doing quite well for the first 600m. A breakdown thereof: when I closed the first curve I was in 4th and right behind 3rd. The top 3 in each heat get a T-shirt and I was thinking, "T-shirt T-shirt gotta go fast". So I kept running fast without slowing down my pace from the beginning (I always start pretty fast, probably too fast) and passed him at around the 300m mark. Then it was open season for the next 200m, I passed 2nd place pretty quickly too and was pretty far behind 1st but also pretty far ahead of everyone else. Unfortunately between 500m-600m I could feel that I was slowing down and started to go a bit more conservatively. Here 3rd place passed me and I became 3rd place again, and in the final 200m I just ran out of gas. Then I got passed by 2 more people and audibly screamed (not that you can scream very loudly when you're running) "No!" because I knew I wasn't passing them after that/getting the T-shirt.

I don't even think I went out too fast really, just that I didn't have the endurance/mental strength to keep up for the last 200m. So I should start taking tempos & repeats more seriously and pushing myself to the point of exhaustion there (not that I shouldn't have before).

Goal next time: 2:27

I don't know how much this will, say, help with college apps (I predict not much). I don't think much in my life will change, but one important thing has happened: I feel like I've finally won something in life. It's been a while since I really, truly felt like I've managed to best a challenge which really mattered to me, and finally I did it.

By the way, next Friday I also (hopefully) have a track event (representing our school at some fun run). So if that pans out, I am skipping three straight days of school. I am definitely looking forward to that.

Well, it is the year 2025, and Typst is a very ergonomic typesetting language. For almost any reason, it is probably wiser to use Typst to typeset slideshows instead of beamer. I personally am using slydst to typeset the slides for my puzzle course next semester.

That's all from me today. Cheers, and happy new year!

I'm in China and they only do facial anesthesia. This seems objectively better. The rest of the operation was a little sus though. The person doing it stepped out of the room to do god knows what and was panicking about missing something after I got anethesiaed. Also the bottom right tooth seemed like an absolute struggle for them to pull out considering the top was removed in like 30 seconds.

Afterwards since I couldn't talk, I had to pantomime to communicate. It's like I was playing charades but five times worse.

I am typing this with only my left hand on my phone. My right hand is holding an ice bag to my face. Obviously I will be publishing this when I am home, at which point I will probably be fine.

I started writing a SCOTUS post before the op and my language was a lot less wooden then. (You can compare the results when it comes out.) Now it is stilted. I wonder why.

This must be what it feels like to suck at writing. And that, I feel, is far more painful than removing wisdom teeth ever could be.

A caveat: this is based on a sample size of n = 1 (i.e. me), and I was pretty weird as a kid and "pushed" things on people (with varying degrees of success), whereas now I kinda don't. (It's not like I'm exactly the paragon of normality now, but at least now I can pretend to be normal when I have to.)

And I'm noticing this mostly because I'm the one who does this to people more often than I'd like. (In fact, I think being non-committal has been a pretty big issue of mine.) I can think of a couple of times where people asked me to join them in doing something or whatever, and since I don't want to say no, I just say "yes" without following up 90% of the time. So yeah, I want to work on that myself.

As two examples, take Evan Chen's Taiwan TST/IMO/USAMO reports and Drew DeVault's Gemini-exclusive blogposts. (There are a couple on his website that conspiciously jump out at you, by virtue of saying "this is gemini exclusive use gemini to access", but some others are not.) If you know how to find them, it will be very obvious how to get there. But if you're new to a site it won't.

I don't think it is by design that these pages are out-of-the-way. Typically it is the miscellaneous, for-fun, trivia, etc pages that are in this position, and not the "main content". And interestingly enough, because these pages are not as important or mainline, they tend to be the most fun to read.

If you want people to stay on your site, have pages that are non-essential for the user and don't get in their way. Think about all the Wordle clones; the reason they got boring quick was because there is no novelty, due to infinitely many words being generated; it's all the same. You can't churn out unique and organic content quick enough so there's something new every day, but just like a fibre does for carbohydrates, hidden pages can make people digest your site for longer.¹

You don't have to try too hard to do this on purpose. Just by virtue of having enough stuff to say, your website will necessarily have stuff pushed to the side. But if your content is interesting enough, could you use that to get people to make changes for the better?

What Drew DeVault did was putting some exclusive content on his Gemini blog instead of his WWW blog. Another thing that I've been doing is jotting down some notes in a small notepad. But because I haven't been transferring my notes over to a more permanent medium (i.e. a blog), I have subconsciously kept my notepad neat by not writing stuff down when I think of it if it doesn't fit into a certain page's structure. Which sort of defeats the purpose of a notepad where I can write anything I want.

So I can put the two together. I'm going to make a Gemini mirror of (some subset) of my website, and on it I will add exclusive status updates. I'll probably put the first one on my WWW website, just to give people a taste of what's on my Gemlog.

Currently links does not work well with redirects. It's not just my site that faces this issue --- the logout link for SourceHut also does not work. links doesn't handle redirects well, and at some point I just have to shrug and say "not my problem".

Well, that's what creative processes are like. Take puzzle-writing for instance: while much of the process is put a clue down, figure out what it does, and repeat, you do have to use some ingenuity to figure out what clues to place. But perhaps far more importantly, determining what kind of puzzle you even want to write is really not methodical. What ruleset do you want to use? How are you going to make the rules interact with each other? That kind of thing.

For instance, take my puzzle Monster Without A Name. Normally, you'd think that I wrote this puzzle and found a good name for it afterwards. And that is true for most of my puzzles. But for this one, I came up with the title before I had any clue what the puzzle would be about, because I really wanted to force the song lyric.¹

So I first was thinking of just making a really hard puzzle. Problem is, you can't just decide to make a really hard puzzle, and I did make a couple of semi-hard puzzles but they all had obviously better titles, e.g. X-Lines Sudoku. So I thought, "maybe the constraint should actually have Monster in the name, at least".

When I thought of Monsters, I thought of the similar word killers, since you have Killer Cages and Little Killers. And since Killers do stuff with sums, Monsters should do the next simplest thing: products.

Then the question is, how do I do products? Just doing the product of every cell is really boring, since it removes a lot of nuance and degrees of freedom. So somehow, I would want to do region sums.

Now I had to decide how to construct my regions. First I tried using shading pencil puzzles like Yin-Yang or Caves to create a hybrid --- and I still think that approach has a lot of viability --- but after doing a few Chaos Construction puzzles, I realized: the most natural way to do products of region sums, which requires region construction... is to do the variant that uses region construction. The Monster Cages constraint had finally crystallized.

The process I took was definitely not methodical: if the puzzles I did or the thoughts I had changed a little bit along the way, I would've gotten a completely different puzzle or no puzzle altogether. In other words, the system is chaotic: small perturbations to initial conditions create a vastly different result. But far more importantly, the process wasn't even logical. There's absolutely no reason for me to fixate on, "ah yes let's force Monster Without A Name" to work as a title and then go to the lengths I did.

But that's the thing about finding inspiration. You're lost in a sea of possibilities, and having something to anchor you can be really helpful. Even if that thing makes no sense at all. And this, in ~600 words, is how and why I use song lyrics as the titles to my puzzles.

• What is the point of using SSH instead of password-based authentication?
• How does one set up SSH locally?
• How/why you upload a public key to a Git provider (like GitHub)

What's the point?

Let's first discuss why GitHub decided to enforce SSH encryption. The advantage of SSH is that it is more secure. When I send a password, as long as the connection between myself and GitHub is insecure, the password can be intercepted, whether it's on my end of sending or GitHub's end of receiving the password. This means that for common operations (like pushing to a repository), it is best if we avoid sending passwords.

The advantage of SSH (and asymmetric keys in general) is that you do not actually provide anything that can be intercepted; you only prove that you have the private key that you claim you have (i.e. the private key that corresponds to your public key). Also, SSH keys are forced to be randomly generated, meaning that SSH private keys are unguessable, unlike your crappy password of P@ssw0rd.¹

How to set up SSH

I don't use Windows, so I have no idea how the specifics work. What goes will mainly apply for Unix operating systems, like Linux and BSDs. (This will probably work for Mac OS work too.)

Using ssh-keygen, generate a public/private keypair:

ssh-keygen -t ed25519

(The -t ed25519 option specifies the keytype as ed25519, which is faster and more secure than RSA, mostly due to implementation difficulties in RSA.)

Then, just copy the public key's contents (its file extension is .pub) and add it to GitHub (or whatever).

Now, when you clone repositories, make sure to clone using the SSH URL, not the HTTPS URL. So, run

git clone ssh://git@github.com/git/git

instead of

git clone https://github.com/git/git

• #426942: I use this color for my Polybar (X11) and i3blocks (Sway + Wayland, though I hope migrate to swaybar eventually). Out of all the colors listed, this is my favorite.
• #421337: I love this purple.
• #691337: I love this magenta even more.
• #133742: A nice dark turqouise blue. It's not bad, but not my favorite.
• #133769: A decent blue, but not the best color. I personally prefer richer blues, like #003366 (dark midnight blue).
• #694269: Might not look great in isolation, but it's similar to MAT's light purple. Don't hate on it, it could be a good complement for a design!

OK, these were the stupidest color reviews of all time. But these colors are unironically pretty good. I know I'll be trying at least one of these out in my next design.

The mission of Math Advance is to, well, advance math education. (It's in the name.) But it is not one of our values, and the difference is important. Because our mission is something we can compromise or change, something we can scale down on depending on circumstances, etc, while when it comes to values, we refuse to compromise.

If our unrelenting goal was to, say, introduce math contests to more people, we would be writing even easier math contests. But almost as a rule, those "super easy" math contests are almost entirely made of filler. I have the same opinion of those math contests as I do for computer-generated Sudoku puzzles: they can be mass produced, are typically somewhat lacking in quality, and while they can be used for learning, I am not touching it with a 10-foot pole.

That's the same opinion that Math Advance as an organization holds. And our goal is not just to produce good math contests or good teaching materials. It is not even to produce the best of something, because that standard can sometimes be woefully low. Our goal is to produce contests (for now, and maybe something else later on) as good as possible. This is why for the first Summer MAT, we spent almost a year constructing the test despite it only having 12 problems, and why we asked close to a dozen people, completely independent of the test-setting group, to provide feedback. Sometimes we will have to compromise other things to ensure quality: sometimes we have to reduce scope, as we did by putting MAST on hiatus, and sometimes we have to delay the release date of lots of good work (we have so many backlogged problems that we could probably run for 3 or 4 more years if no one proposed another problem from now).

When two values conflict, you are going to have to compromise on one of them. That is why the only value we have is the only value we need: quality. I believe there is a role for people who mass produce math contests and math problems. But if you, just like us, are sick of mass-producing problems for mass-produced contests that become irrelevant after two weeks, we have a place at you for Math Advance.

In summary, the Winter MAT is an easier version of the MAT compared to the Summer MAT. On average you could say it's similar to the Summer MAT shifted one problem down, although the introductory problems don't have that much room to get easier.

The Winter MAT is meant to be an introductory contest. It is intended to fill a similar niche as the Mandelbrot before it was discontinued. As thus, there will be a minimal focus on competitiveness, which means we will not need to focus on breaking ties. Prizes (if any) and recognition will be cutoff-based, not ranking based, though we will obviously use contest statistics to determine said cutoffs. This removes the need for tiebreakers entirely, so in the interest of logistics (shorter contests are easier to run), we'll be dropping the tiebreaker set from the Winter MAT. (It will still stay for the Summer.)

As part of our outreach efforts, some Math Advance members are thinking about holding the Winter MAT at school or other local testing centers. This is because taking contests online is a draining experience, even if it isn't as long as other contests. One of my favorite parts of math contests was discussing problems immediately after tests were collected after each round. COVID has robbed this experience from us for the last few years, and I'd like to try and get it back.

This year we made the Summer MAT significantly easier, particularly for the last problem in each set. While we don't ever intend to return to 2021 levels of difficulty, you can expect a spike next year in the final problems in each set, and a minor increase for 2/5/8 as well. This is for mostly pragmatic reasons: we have a lot of really good hard problems that we've been waiting to use. Part of the reason we're hosting the Winter MAT is to make hosting harder MATs in the summer more feasible: if we already have a dedicated beginner contest, then the Summer MAT does not need to be as accessible anymore and can cater more to stronger contestants.

I'll just show you what an example beam document looks like, since the spec is so simple.

^ \documentclass[12pt]{beamer}
^ \title{beam}
^ \author{Dennis Chen}

\titlepage

# Introduction

~ What is beam?
- beam is a file format to write presentations with
- beam is a program that converts beam files into beamer

~ Why?
beam does for beamer what Markdown does for HTML
- beamer takes far too long to write.
- We only need a subset of beamer's features

# Features

~ Frametitles
Use the ``\textasciitilde'' character to set the frametitle.

~ Bullet points
- You've already seen it in this presentation.
- Bullet points invoke the ``itemize'' environment.

~ Block
> How to make a block
> This symbol makes a block.

~ Exampleblock
< How to make an exampleblock
< This symbol makes an exampleblock.

~ Alertblock
! How to make an alertblock
! This symbol makes an alertblock.

~ Images
Use the ``@'' character to set a background image.

@beach.jpg

# Miscellaneous

~ Stylization
beam should be written in all lowercase, even at the start of a sentence.

Interested in using beam? Read the beam page on my website.

Spoilers ahead for 5, so please try it yourself first! It is GAS level and is probably around 2.5 or 3 stars in difficulty.

This post will be a little different from my last one. Instead of listing the (ultimately useless) trial and error I did, which was not much, I will instead show the order which I added the final clues in. Since the solve process is fairly linear, it was easy to just layer clues on top of each other this time around. (This is in contrast to Alphabet Soup, which was much more intricate... a post on that coming soon!)

You will notice that two sets of clues were added at the same time: 5 and 6. This is because I was near the end and experimented with brute-force to find the most aesthetically pleasing set of constraints to finish the puzzle off.

So a couple of days ago, I decided I wanted to try writing a German Whispers puzzle and playing around with the number 5 (German Whispers also cannot have a 5 on them). I wanted to pay homage to this number 5, so I just drew a fat 5. Originally it was not as tall (1 row less on each side), but on paper it didn't look as good so I stretched it. I also knew it would be more constraining and make writing the puzzle easier.

Conveniently, XV pairs also cannot have a 5! I was going to use V's at first, but if I needed an X, I knew I could use it later as well. So I'm like, "Hmm let's force my first 5" and placed clue 2. Now 5 was fairly constrained: it must appear in row 4 for region 5 and so must appear in row 5 for region 6. Then I pushed the puzzle to its logical limits and got the 6 at the bottom, and some other good stuff

There are some very constraining things to note: though 8 and 9 cannot resolve at this stage (there is no differentiator between the two at the moment), the 6's and 7's on the German whisper only have one degree of freedom. Also, a V in regions 1 and 6 would nicely differentiate where 5 was, and the V in region 1 would also take care of the 4 in region 4.

So hopefully it is quite clear why I added clues 3 and 4. I had to do something to differentiate columns 1/2 and columns 8/9, because at the moment those pairs are symmetric. Determining some 5's and the 4 seemed fairly natural. (As to why I chose to place them on the outside, well, I just thought it looked aesthetically better. I could've just as well placed the V in columns 2 and 8 instead of 1 and 9.)¹

After adding each of clues 3 and 4, I logicked out the puzzle to take stock of where I was. After clue 4 I figured I was close enough to the endgame. I decided to place clue 5 to forcefully differentiate the 14 pair in column 3 and the numbers in the V of clue 3. Then I realized I really was close, started playing around with online sudoku solvers (essentially it was just a faster way to fill in candidates for each cell), and after trying a couple of potential XVs I settled on clue 6.

Then I decided to check that clue 6 actually made things satisfying to solve and didn't cause any unnecessary cheese. It did: it gave solvers a early 5 disambiguation in region 5 (the X clue), made the middle and end of the puzzle less convoluted (the V clue), and the combination of the two also somewhat quickly determines the location of 4 in region 5. (I mean, technically it determines everything, but you know what I mean.)

Summary

So yeah. This puzzle was almost entirely constructed on paper and took about 3 hours. This was a much easier puzzle to construct than Alphabet Soup. (Alphabet Soup took my soul out of me. I consider it a miracle that I managed to finish it within a week, and that's a long time for one puzzle, I think.)

My general thought process for constructing a puzzle, and what happened for this one in particular, was

• I thought of a general theme (the 5, the German whisper, etc). From my experience writing math contest problems you don't just decide "OK I will write a problem", you have to stumble your way into the core idea. (At least, I can't just decide to write a problem --- I'm sure there are people who can.)
• I put down a couple of probing clues to get started.
• In the middle, I aggressively constrain things that are almost constrained.
• At the end, I use some computer assistance to determine which set of final constraints is most viable.

In particular, there was a lot less floundering than when I was writing X Marks the Spot, and since the puzzle was not as intricately connected as Alphabet Soup, it was not too hard to finish.

Hopefully this made my thought process behind the puzzle clear, and helps you in writing your own puzzles! If you do write a puzzle, whether it is because of my blog or for any other reason, please send them over --- I'd love to solve some nice puzzles :)

By the way, if anyone knows how to make German Whispers and XV sums in Penpa, please let me know :)

The reason this happened was because I didn't have auto-renew on for God knows what reason, and I just let it get to today without paying any attention. Maybe I should have Namecheap send emails to the addresses I actually check (i.e. dennisc.net and mathadvance.org). Well, hopefully that's never happening again.

My impression is that these puzzles are in some ways easier to set than sudokus of similar difficulty. When setting a puzzle, you have two problems you need to simultaenously solve: you must make sure there aren't no solutions and there aren't multiple solutions. With a sudoku, you are not really guaranteed that solution at the start. But with a suguru, you start with the solution: a blank board. You just need to make it unique.

Also, since not every square has to be filled out, Suguru is a little more forgiving, since while the geometric interactions are plentiful, actual numbers do not interact that much unless you specifically design them to.

At the same time, I wouldn't recommend a first-time setter to try a Suguru in particular. It is much bigger, and there is less solver support, meaning that verifying a puzzle must be done entirely by hand. Also there's a lot of scanning when you check, and Sugurus feel easier to break.

There are a couple of quirks of this genre too, especially when solving. Since the solution is unique, say you have a cell completely enclosed off, so it can only be a 1. By uniqueness you know it can't be a degree of freedom so somewhere in that row there must be a 1. More generally, you can be sure that only squares that interact in some way with clues --- maybe not directly --- will have a digit. While using uniqueness as a logical basis for solving is cringe, sometimes it is useful to know what must be true and find the logic to confirm it. I got lucky in that uniqueness didn't actually help with my puzzle, and it doesn't really play a big role in any potential solve routes, but it is a minor pain point that exists.

Anyway, the Gmail web client is a disaster, so I wanted to get my CMU email on aerc. And since CMU uses Google Suites, the problem essentially reduced to getting Gmail to work on aerc.

Tutorial

TL;DR follow the commit message for the aerc patch that introduced aerc support. However, note that the token endpoint --- when URL-escaped--- is now https%3A%2F%2Foauth2.googleapis.com%2Ftoken instead.

Step-by-step:

• Go to console.cloud.google.com and create a project.
• Save the client id and client secret somewhere. (You can just copy it into a temporary text file or something.)
• Go to developers.google.com/oauthplayground.
• Press the settings button and check "Use your own OAuth Credentials". Then, copy the client id and client secret that you saved.
• Authorize the mail.google.com API under "Gmail API v1".
• Exchange your authorization code for tokens and save your refresh token somewhere.

Now that you have all the information you need, go to aerc's accounts.conf and create a new entry with the following source:

source = imaps+oauthbearer://{username}:{refresh_token}@imap.gmail.com:993? \
client_id=XX&\
client_secret=XX&\
token_endpoint=https%3A%2F%2Foauth2.googleapis.com%2Ftoken

The outgoing entry is identical: just replace imaps with smtps and imap.gmail.com:993 with smtp.gmail.com:465 and you're good to go. Full example:

outgoing = smtps+oauthbearer://{username}:{refresh_token}@smtp.gmail.com:465? \
client_id=XX&\
client_secret=XX&\
token_endpoint=https%3A%2F%2Foauth2.googleapis.com%2Ftoken

Blog posts are timely --- they largely rely on context from events around that time. As of now, some of my posts reference my anime taste in 2021, the 2021 MAT, an English essay I was really upset about having to write, a track meet, USAMO 2022, and MAST's indefinite hiatus.¹ This doesn't necessarily apply for really stupid stuff, but hey, a man's got to shitpost.

Essays are more timeless and usually don't need the context of a specific event to be understood. That's why To parents: stop emailing me ultimately became an essay even though it doesn't feel like one. In retrospect, maybe these posts about the distinction between my blog and essays should be in the essays section too.²

I don't make this distinction for tech posts because all tech posts become obsolete over time, though the rate of obsolescence differs (for instance, I along with the rest of the world stopped caring about Gemini, but the morally correct way to make a static website will likely remain the same for the next decade at least).

We’ll unfortunately still be using Javascript, because I am not rewriting the frontend.

which implies that the marketing website and the contest portal would still be intermingled. Now, I plan to separate the MAT website into the marketing website and the contest portal.

I intend to use Rust to server-side generate the pages (just like with Glee) and only use the bare minimum of Javascript required (timer and autosubmit when time runs out). Then, I'll rewrite the marketing website using just HTML and CSS, because we really don't need Javascript for that. (Maybe I'll pawn the marketing rewriting off on someone else.)

In semi-related news, I am developing multiple choice support for mapm, and I will be integrating mapm with the contest portal (i.e. you can upload mapm contests). So the portal will be able to hold mock AMCs, and I'm considering opening up a community portal for anyone who wants to host their mocks in addition to the official Math Advance portal. But this is all pretty far off into the future.

These variants are worth further exploring, but at the same time, it'd be silly to have a puzzle on this website titled "Close Neighbors 10" (whereas it might make more sense in a book). So I tentatively plan to write a small book of puzzles with these custom variants.

I will try to make it suitable for beginner/intermediate solvers while still providing fun for more experienced solvers. Some of the things I'm planning to add are:

• brief explanations of the implications of the rules at the beginning of each section (the reader can choose whether to read it and "spoil themselves" or not)
• possibly puzzle hints for harder puzzles
• two challenge times (one for beginners, one for experienced solvers) for anyone who wants to time their solves
• answers to each puzzle, and a worked solution for each section.

Anyway, here are the puzzles I'm planning to include:

• Classic sudoku, depending on how approachable I want to make this book for absolute newbies (7)
• Consecutive Neighbors, which is what I'm calling the "Close Neighbors" genre (7)
• Dutch Neighbors, which is what I'm calling the "Distant Neighbors" genre (7)
• Averaging circles, which is what I'm calling the "That's What I Mean" genre (7)
• Digit Neighbor Sums, which is what I'm calling 7-Eleven (and any puzzles with a similar constraint, just with the digit and sum changed) (7)
• Thermo Sudoku, as an introduction for thermo placement puzzles (5)
• Thermo placement puzzles, like Alphabet Soup (2)

This adds up to 28 or 35 puzzles, depending on whether I include classic sudoku. These numbers are not strict, and is just a guideline for about how many of each type I will include. I might write more of a type depending on how I feel, and I will probably post final specifications once the book is ready for purchase. I will probably include the publicly available puzzles, because that way I can reasonably generate a preview of the book without revealing any more book-exclusive information.

In the off chance someone would like to submit puzzles for this book, email me and we can discuss details.

Can't set up Arch Linux? Maybe that's just because you're a dumb-dumb. Maybe you just aren't smart enough, and deserve to use Noobuntu for the rest of your life. You'll never get to use the AUR, muwahahaha...

Until you learn how to use Bedrock Linux, that is...¹

My brl setup

A long, long time ago, I installed Arch manually on my laptop, so it was the distro that I hijacked and it is the distro I boot into. For my desktop, Ubuntu is the distro I boot into, because I accidentally nuked my system by doing a sudo rm -rf in the wrong place at the wrong time.

On my desktop, I use Ubuntu to work with my hardware (including, amazingly enough, an XP-Pen Pro). I'm sure I could set it up with Arch as well, but I do not know nor care how. On my laptop, which is a Lenovo IdeaPad, the hardware has been very cooperative with me and easy to set up, so after adding a couple of scripts to my i3 config it was up and running.

For most of my software, I use the Arch package manager. It's incredibly fast and between the official repos and the AUR, pretty much everything is there. If I'm really particular about a particular program (say, one I use often for work like Vim), then I will compile it myself with Gentoo.²

If I'm feeling especially braindead one day and want to copy some Ubuntu tutorial on the internet, or get something that's really annoying in Arch (looking at you, printer drivers!) I have an Ubuntu stratum on both my desktop and laptop.

Installation

Because I liked the idea of Bedrock almost immediately, have some of the biggest cajones on planet Earth, and have literally no important files on my computer (my dotfiles and code are in Git repos AND synced with Dropbox³), I just went ahead and installed it on bare metal. So I don't get flamed, I'm going to pretend that I totally have a backup of my old system somewhere, even though I literally have never used it in forever.

Maybe you don't want to commit to installing Bedrock just yet and want to use it in a VM or something. You can do that and see how it goes, but from my experience if you're using Ubuntu/Arch it really makes no difference anyways, nothing breaks. Running recovery scripts, though, does get a lot more annoying since you can't just chroot in /, you gotta chroot into /bedrock/strata/<some_strata>.

If you have an already existing machine, installing brl is so simple that the official documentation is actually all you need for once. Since you'd already have a desktop environment (and I'm counting the likes of twms like dwm and i3) setup, there's not much to say on that front. I think what's most interesting is how to set up a new computer.

What I do with new machines (and indeed, how I setup my desktop after bricking my old partition and giving up) is install an Ubuntu variant called Regolith. It's like Ubuntu but with i3 on top of GNOME. It's a bit annoying if you already have your own i3 dotfiles and want to use them instead, and you will have to Google where Regolith puts its own i3 config files so you can symlink yours there. Then I just clone my dotfiles, run dotbot's install command, fetch Arch and Gentoo, and just setup whatever apps I need.

Fortunately all my hardware cooperates with Linux more or less (though pulseaudio volume scaling is annoying to deal with and is the only thing that doesn't work out of the box), your experiences may differ.

I want to make my website accessible so I make it really easy for anyone to go to the footer, find the mailto link, and send me an email. That's why I haven't employed tricks such as "dchen at dennisc dot net". Unfortunately, accessible to anyone includes spammers. Of the spam I've received, the most interesting is stuff from web/app developers, probably because a lot of keywords in my site suggest I'd be interested in that.

Let's take a look at this gem from Priyankshu, sent in 2022-05-05:

Hi there,

My name is Priyanshu, and we are a team of Professional IOS & Android Apps Developer with 7+ years' experience based in India. Do you need a great App adjusted or built from scratch?

Reply me if you need any kinds of help in mobile Apps.

Let me know what you think.

Kind regards,

Priyanshu

The very next day (2022-05-06), Ameli the Wordpress expert (ugh) jumps in:

Hi,

I'm a web developer who builds amazing looking websites for small businesses. I'd like to have the chance to show you how I can improve for your business.

I'm an expert in the Wordpress website platform and many other platforms and shopping carts.

I can upgrade your existing website or build you a new one completely that has all of the modern features and functionality. Would you be available in the next few days for a quick consultation?

I'd like to share with you my ideas and some expert advice, share me your Requirements and any direct contact number.

Thank You.

Ameli

I'm sure these people are copy-pasting emails because there's no way they have the technical skill to run a script to automatically spam random people with emails. I'm pretty sure these emails aren't just spam but also (attempted) scams: it is likely you'll never see the app or website these guys promise you. (I mean, what kind of developer cold-emails people without a portfolio?)

If I'm bored one day, I might scambait one of these poor chaps (or figure out if they are, contrary to all appearances, legitimate). I'd set up a burner email like notspam@dennisc.net to do it and then delete it immediately after (unless I found more people to scambait).

Oh, and a ton of people are trying to tell me that my databases for dennisc.net have been hacked. Yeah, sure...

For anyone who doesn't know yet, "5" was featured on Cracking the Cryptic:

I left a comment on it, but because I linked back to my website for my construction notes, it probably got auto-deleted by YouTube's anti-spam bot. Sad.

Anyway, a couple of puzzle-constructy things & my overall goals for the next 4 months (let's be honest, I am going to give up on "New Years resolutions" way before that, so let's not pretend for the next 8 months).

I really enjoy setting puzzles with my own homemade variants (PARTICULARLY 7-Eleven and Close Neighbors), especially variants with simple rules. But I also think there is some value in just exploring more conventional variants, like thermos, anti-knight, etc. Also I really want to set a renban and a between line someday.

January goal: volume

Release 8 puzzles, i.e. two a week (biweekly).

Nothing too much to be said about that, I think it's doable, but also I will have to balance puzzle writing with USACO and updating my blog and doing taxes for Math Advance and aaaagh

You will probably see a lot of ideas I've done before since I have experience with them. For instance, rulesets like that of "Close Neighbors" will probably appear a few times. ("Distant Neighbors", anyone?)

February goal: coverage

Use at least three of the following five common variants I haven't used before:

• Renban lines
• Between lines
• Anti-knight/anti-king (for the purposes of this challenge I will consider them the same)
• Quads
• Palindrome lines

I can invent my own rules if I want or not. Either way is fine.

March goal: classic

Write a GOOD classic sudoku. I dunno, I feel like this is hard, but I also should just do it. (For the purposes of this challenge, I will count an anti-diagonal sudoku as pretty much meeting half of my goal.) Also, I will better know wtf I am doing, since my classic sudoku knowledge is... not exactly the greatest? I just have "intuition."

I should probably watch some CTC classic sudoku videos to get a feel for them. Although I probably will not be using any difficult techniques in mine.

April goal: Non-sudoku pen puzzle

Probably a logic puzzle but crosswords are fair game too. (Although I probably won't be doing a crossword quite yet? I dunno, four months is a long time.)

As a matter of analogy let's say you have an array of true/falses and want to determine whether every value is true. What I did was essentially (and this example code is written in C, though it is so basic I would be shocked if it didn't work in C++ and maybe even Rust) the following:

void allTrue(bool bools[], int size) {
  for (int i = 0; i < size; i++) {
    return bool[i];
  }
}

and obviously this isn't what you want to do, you want

void allTrue(bool bools[], int size) {
  for (int i = 0; i < size; i++) {
    if (!bool[i]) {
      return false;
    }
  }
  return true;
}

It's so obvious an error that there's no way this should've slipped past me, but it did. And this is the price I pay for getting lax on doctests. (So reminder to future self tomorrow, write the goddamn test for this function. Ya?)

I've finished the January goal --- in fact, I've pretty much blown it out of the water. With 8 puzzles posted to my website and 6 more kept private for various reasons, I've been able to write at a pace of around one puzzle every two days.

Puzzle-book

For the puzzle-book I discussed earlier this month, it might behoove me to reconsider which variants I include more of. Equations is particularly easy to set, while Consecutive/Dutch Neighbors is incredibly hard to set. Also, if I want to make this book entirely approachable (rather than targeting both beginners and experts), Clue Placement (i.e. Alphabet Soup + maybe other puzzles I'll set later) might not be appropriate. Every other puzzle (so far) seems like it'll be 2-3 stars of difficulty, so it'd be weird to force in a 5 star puzzle.

Also, I don't know how milkable the 7-Eleven concept is. I tried setting a puzzle entitled "24-7" where the orthogonal neighbors of each 7 summed up to 11, but no such grid exists. (Exercise for the reader: why?) Instead, to preserve the theme and many of its ideas, I forced the 7 to have orthogonal neighbors of 2 and 4.¹ The Averaging Circles variant seems much more tractable, since we don't have to deal with global constraints, so global digit neighbor constraints will likely cover a broader range of puzzles in the book.

In general, I'm thinking that as long as my puzzles have unique, semi-related constraints, it will be fine. It is probably going to be difficult (and somewhat boring!) to literally write 7 puzzles with the exact same constraint, so the sections of puzzles in the book will likely be roughly grouped by theme, rather than strictly organized into constraints.

New Year Goals

As for the rest of my goals, I think for February I should probably up it to four new variants. I originally was thinking five, but between writing this post in my notebook and typing it up I managed to use Renban lines, X-Sum, and Little Killers (speaking of the latter two, I really ought to complete my sudoku rules intro). So the goal will probably be a bit harder than initially anticipated, particularly since some of the constraints I have left are annoying global constraints like Anti-knight.² But there are a ton of other variants I didn't even mention, like Kropi, Sandwich, Odd/Even, Diagonal, Anti-Diagonal, and Indexing.³

On the other and, my goals for March and April were grossly easy. (Which is a good ting, because if I set low standards I am virtually guaranteed to meet them and not get discouraged, etc.) As I've already discussed, I've managed to set two Classic Sudokus, and a few days ago I also set a Star Battle. So maybe I should up the ante a little bit and change March/April goals to themes. Try to set 4-ish classic sudokus in March and 4-ish pencil puzzles in April, and increase the focus in those genres for those months.

May Goal

I've only set one 5-star puzzle so far (Alphabet Soup), and I think it's much easier for me to set 2-3 star puzzles than 4-5 star puzzles. (1-star puzzles I can do, but I don't do it as often because the space to explore is smaller with easier puzzles.) So a new goal for May: publish puzzles on Logic Masters Germany such that I have at least one of each rating (1-5 stars). I say publish because I'm a pretty bad judge of difficulty (I originally thought Bird was a 2-star puzzle at most). So depth/range is one of my goals now.

(Of note: I've never written a 4-star puzzle.)

Cracking the Cryptic

I know it's been a while, but Close Neighbors has been featured on Cracking the Cryptic too. I think the debut feature was a much bigger deal for me (since, y'know, first time), and probably I will get less surprised with each new puzzle featured.

Of course, it's always going to be a nice thing to see my puzzles make it, but I think it does not make sense to make a post announcing each new feature. So I will probably mention features when they happen if I'm already writing a puzzle blog post, and otherwise, I'll probably just silently update my puzzle page.

Taking a break?

Maybe I should take a break from setting or reduce the pace to get my life kind of together. (Right now I'm barely even a functional human being since I spend so much time setting.) I say this, but I know with pretty high probability I won't intentionally do this. A couple of tings that will naturally make me slow down, though:

• Track and Field is about to officially start
• In February I'll be spending 10 days grinding out my hardware project for Hack Club
• In March I will be attending WARP and milking as much as I can out of that experience (which means I cannot spend hours setting --- in fact, I expect to set 0 puzzles in that timeframe)

There are other things I want to do, like finally finishing my Git server⁴, pivoting back towards teaching for Math Advance, etc. So yeah, a steady pace of about 3-5 puzzles a month (and not 14) is probably what I'll converge towards.

Thanks for reading, and thanks for trying my puzzles.

What I want on top of Pandoc is

• no complex theming system (Hakyll CSS is good, my static/ directory is good, Jekyll is bad)

• generate RSS and HTML themes (something like a Pandoc template is probably best for the latter, the former could be made much better than Pandoc + Bash)

• Have LaTeX and Asymptote just work. Pandoc can do the former (well, kind of, it can't statically compile KaTeX into HTML which would be ideal) and cannot do the latter.

  Note that under the hood, you'd want to replace an asy environment with the generated SVG which can trivially be generated with asy -f svg file.asy.

Maybe this will be a future project sometime later.

I don't really care if commits are feats or breaking (and in early development, many commits are breaking), so I omit these tags from my commit messages. Later on, I do care if a line of changes is breaking. I totally see the value in saying "this is a breaking change" in the commit description, but when I'm scrolling through all I need are the versions tagged. In practice breaking releases will typically not include changes that are really compatible with the previous major version, because otherwise why wouldn't you release those changes separately first? Also, most commits are features, so I'm saving the extra room in my commit header and not writing feat.

Instead, I use the absence of a tag to indicate "this is a normal commit that adds some functionality". When I want to fix a logical error up, fixup. For formatting errors (which shouldn't ever slip by since I have a pre-commit hook that runs cargo fmt --check among other things), I use the fmt tag. Tags should be used sparingly and only for commits that are "weird" in some way, i.e. commits you ideally never would have needed to make to begin with.

Of course with Rust, you have to make commits where you just bump up the version number. Here, a release tag would be appropriate.

Tags are things that should be used sparingly for out of the ordinary commits. The fact of the matter is that most commits add functionality of some sort to your program, and indicating that with feat dilutes the value of your actual tags. Conventional commits are a good set of guidelines to consider for your repository's Git history, but for me they hold little value as specifications: they are too intrusive on the commit history. Especially since I haven't ever used any conventional commit tooling and honestly don't see the point: I just search for fixup: or whatever in my Git shortlog.

Overall I think my approach is working out for me, and I find my history a little more readable than a history where everything is tagged. However, there are cases where tagging can present a more concise way to describe the scope of your changes, which is one thing conventional commits do well as is.

In doing so, they reveal that they have little interest in an independent judiciary. It's just a policy tool, one that happens to be Theirs and not Ours at the moment. Of course, conservatives played politics with the judiciary too: they blocked the appointment of Merrick Garland and still reel over David Souter not ruling exactly as they expected him to. But currently the judiciary's opinions please the conservatives, so they do not cry foul.

An independent judiciary is vital. Who else is supposed to interpret the law? The legislature, whose response to any Constitutional challenge would be "trust us"? The executive, who would then shape the law according to his whims? If we let those who make or execute the law interpret its Constitutionality, then the Constitution merely becomes a series of suggestions.

Living in a constitutional democracy has costs. It means a church of homophobes can picket the funeral of a gay Marine (Snyder v. Phelps). It means cakeshops and website creators can refuse to express support of a gay wedding for religious reasons. (Masterpiece Cakeshop, 303 Creative LLC). These results are detestable. Even most homophobes would side with Snyder.

But the costs of not having a Constitution are far greater. If we do not apply the Constitution impartially and abrogate the rights of those we dislike, the minorities we seek to protect today could very well become the targets of tomorrow. The Constitution admits no exceptions¹, and for good reason. It is rights for all or rights for none.

To be fair, the Supreme Court has not always held true to this principle. Famously bad examples of judicial overreach are Dred Scott, Plessy, and Korematsu. More recently, Bakke/Grutter and Roe² are examples too. (Given this, one can hardly fault the current Supreme Court for overturning these last two decisions.) These examples of judicial overreach do not show that the SCOTUS is "broken". Rather, they show the dire consequences of the judiciary being used in ways it was never meant to: as a partisan tool to buttress policy outcomes.

Given this, what are we to make of those who claim the justices' impartiality is a sham, and that the judiciary is just a political tool? There are those who want the judiciary to become a political tool --- just one that works for Us, not Them. They do not believe in an impartial judiciary whatsoever and do not understand its importance. These people lack an understanding of the law, and frankly, common sense.

How about the rest, who wish the judiciary was "impartial" but think it is not? Their concerns are entirely based on policy: they do not like the outcome of SCOTUS cases, so they cry foul. Never mind that they only pay attention to the "big cases" --- the ones they are personally invested in --- and ignore the cases that get less press coverage, where many "conservative" justices vote just as they would.

There are legitimate arguments that some of the judiciary's rulings are incorrect. I certainly do not agree with every SCOTUS outcome. However, any such argument would have be to based in constitutional law, rather than naked policy preferences. And that is how conversations about the Supreme Court should be. If you have a problem with a Supreme Court ruling, find where your interpretation of constitutional law disagrees with theirs. "But muh outcome" is no reason to disparage the legitimacy of one of the most important institutions in America. It is when they apply the Constitution inconsistently, rather, that the alarms really ought to be sounded.

• Typst is trivial to set up on your machine. LaTeX is a pain in the ass. Sure, you can install some partial distribution of LaTeX on your machine, but you are in for a world of hurt the second you want to install or upgrade a package. You can also install LaTeX as a user with the install script, which can be quite daunting for a complete computer newbie.
• Typst compiles much faster.
• And you don't have to use a single brain cell to get the compilation pipeline working. This is in stark contrast with LaTeX, which requires installing and configuring the external latexmk script just to get things to work right.
• No \documentclass or \begin{document} needed. Just type some text and equations and it will compile.
• Typst is less of a pain in the ass to write. Sure, I have to write $x y$ instead of $xy$ because otherwise Typst will think I am referring to a variable xy, but the commands make a lot more sense (e.g. union over \cup), I don't ever have to type a backslash, and writing a fraction does not make me sad because I no longer have to use \frac.
• It is much easier to design a simple document. My LaTeX document designs would take two days; my Typst design took me two hours.

Now, life isn't all sunshine and rainbows with Typst. I admit there are some pain points. #show rules behave somewhat strangely, especially if you are used to LaTeX. I also do not think it was trivial to do some more complex things, such as setting up footers to reflect the current chapter/section and then not having my footnote counter resets screw up the references. (By the way, the footer counter doesn't reset at all, which is frankly ridiculous.)

But here's the thing. You can install Typst right now, import my package¹, and compile a good-looking document in under ten minutes. Good luck doing that with LaTeX.

Cleanliness

This will be much angrier than my usual post --- even my usual angry post. If you don't want to deal with that, you've been warned.

Jesus. Fucking. Christ.

How hard is it to just clean up after yourself?

Yeah I get it, you're too rational to clean this shit up. You choose to defect rather than cooperate and let some other poor chap clean up in the mess, and in the same day, you manage to mess the entire commons up again. I'm honestly impressed.

If you're reading this, and you know that you're one of the people who regularly contributed to this, then know that I think you're generally a very cool person --- everyone at WARP is --- but please, make it a habit to clean up after yourself. Please make "leaving the place better than I found it" an actual value of yours, even if it's 2AM or whatever.¹ I know no one is doing this out of malice, but if you "just forgot" to clean up, it's because you just don't care enough. I know, for a fact, that you learned a ton of skills to train habits at this camp. Put the "Applied" in "Applied Rationality": please make this a habit you decide to train because it is really important.

Because how the fuck are you going to solve hard problems like AI alignment if you can't even solve the mess in the commons?

In any case, it was mostly for my own learning experience. And learn I did:

• File i/o
• Used malloc() and free(). Also learned about realloc(), though I intentionally did not fix segfaulting for slides/presentations that were too long (since long presentations are bad)- Learned about structs/enums/unions in C (and how to make a tagged union), though I never ended up using them
• The equivalent of str.split() in pretty much any other language
• Wrote a Makefile for the first time, learning about .PHONY and the all convention, including files like config.mk, and the ${DESTDIR} variable.

And this is only a subset of what I applied in the code. Since it's my first program in C, the code probably sucks ass. If you want to make sure something works I'd use the Rust implementation because I don't suck as badly at Rust, this was just an educational exercise for me.

Well, there is. And it's git add -p.

Commit Discipline

The way you are supposed to write a commit is by staging what you want to commit with git add, and then actually writing the commit with git commit. In some sense, the fact that git commit has any more arguments (like filenames, but especially -a and -m) is really dangerous. But dangerous in the sense that an idiot or a lazy person will use them improperly --- they have reasons for existing.¹

Sometimes you know that all you changed was refactoring some module by renaming it, in which case git commit -a is justified. Or you just changed one file, initialized your project, etc. But in these cases you knew from the start exactly what you were going to change.

More often, when you're writing the most important or complicated features, you'll have a bunch of semi-related changes uncommitted. The wrong thing to do is git commit -a. The right thing to do is ask, "What's the smallest set of related changes I can commit?" And then commit those.

So my workflow looks like

• Run the appropriate git add and git commit operations to get, in some order, the changes I want to push to upstream. Typically, I do the smallest/simplest commits first.
• git rebase -i and swap the order of commits if necessary.

How does git add -p help?

What the workflow I described above requires is being able to separate changes in one file into separate commits conveniently. Running git add -p FILENAME lets you pick which edits you want to stage. It gives you an interactive prompt, and you can choose options like y,n,q,e,s (these tend to be the only ones that matter). They stand for yes, no, quit (aka answering no for every remaining prompt), edit (more on that in a second), and split, which splits a hunk into multiple sub-hunks. Split only works if the current hunk is not all in one chunk of consecutive lines.

Here's an example of what pops up when running git add -p. Notice there is no s because you can't split up these changes, since they're on consecutive lines.

diff --git a/api/src/accounts/verify_email.rs b/api/src/accounts/verify_email.rs
index abf1a7d..d870c98 100644
--- a/api/src/accounts/verify_email.rs
+++ b/api/src/accounts/verify_email.rs
@@ -2,7 +2,7 @@

 use lib::{
     models::{
-        table::Table,
+        sql::Table,
         token::{RedisPool, RedisToken},
         user::User,
     },
(1/2) Stage this hunk [y,n,q,a,d,j,J,g,/,e,?]?

Now let's talk about e, which is edit. Here you can directly choose which lines in a hunk you want to stage and which you want to ignore. When you open up the edit menu, you'll be met with all the changes in the hunk, and what you want to do is remove the changes you do not want to keep.

• In order to remove a deletion, simply remove the - character. (You will have to insert a space in the line to compensate.) Do not remove the rest of the line.
• To remove an addition, remove the entire line.

That's pretty much for git add -p. It's a pretty simple tool, but it is very powerful. It makes it that much easier to keep your commit history clean, and that is why you should use it more often :)

Now, this isn't quite the same when it comes to academic endeavors.¹ Take programming, for instance: except for some throwaway scripts --- which don't make you significantly better at programming anyway --- there is no "practice code." All of your code is supposed to be neat, organized, and checked into Git. And you're not supposed to write bad on purpose.

But training isn't supposed to be bad. You're still trying your best, whether it's with a piece of writing or a workout. Training is just different --- different in the way it's performed and different in its goals.

The reason I don't do daily writing prompts (and I always despised them in English class) is because almost all of the prompts are stuff I don't care about. Writing is actually quite similar to track and field: on the surface the events all look the same. But dig a little deeper and you'll see the intricacies behind them. Why a 100m runner cannot run the mile. Why an essayist like me can't describe his first pet.²

So why separate my writings in the first place, instead of just throwing them all in the same place? To answer this, it's worth identifying the consequences of neglecting to do this separation.

Around the spring of 2020 to the fall of 2021, the COVID pandemic was in full force for us, and we had not reopened or returned to in person school. Naturally, this meant that every athletic competition was cancelled. This totally eroded any separation between training and competing. Maybe my willpower isn't strong enough, but I totally stopped caring about a month in.³

Another reason is because it encourages me to follow through with the writings that are less likely to pan out.⁴ There's a certain standard when it comes to my essays: there has to be a point.

But even if there isn't a point, it can still be entertaining. I still read other people's updates on the last however many days of their lives. And hell, separating my blog from my essays gives me room to mess around from time to time. Even though this isn't my serious writing, I like to think it makes my serious writing better.

Was there a point in writing this post? Not really. It doesn't make any salient points about the world, or even writing. But I wanted to write this anyway, and that, to me, is enough to justify it. That's why I have a blog.

Back in the start of 2022, I said in Separation:

The reason I don’t do daily writing prompts (and I always despised them in English class) is because almost all of the prompts are stuff I don’t care about.

And typically I don't really care about my daily life a lot, but as college applications are coming up, I've begun to realize that maybe I should. I've written about a lot on this website in the past two years: about my teaching philosophy, the way I approach programming and technology, how I like to write, and more. But I've never really written about myself, sans qualifying for USAMO 2022 and a track meet. Looking at my writing in the last two months, almost all of it has been focused on or related to the work I've been doing: it's either about technology or Math Advance. Hell, even in the last four, the only exceptions have been shitposts and a post complaining about an English essay I really didn't want to do.

Before journaling I couldn't really write about myself. You know, it's a little embarrassing: writing about yourself means to be presumptuous enough to think anyone else cares about who you are. It also means you think that to some extent the way you think is right. I think we all secretly feel this way though, and it's just that some of us are confident enough to admit it outright.

But the reason I journal is usually not that deep. Part of it is for the generic reasons, like processing the day, decompressing, or whatever. But I really do it to have a log of my thoughts (or at least a subset of them) and what I did. This is part of why I wrote even before journaling: sometimes I read my past writing and think "wow this guy was onto something". But as cliche as it sounds, I think sometimes I can find value in the everyday things I do years in the future. Knowing what I did and how I thought will make it much easier to be introspective in the future, which is a skill that would come in real handy right now.

Talking about yourself is a rite of passage that everyone goes through, and it's my turn now. So I'll lay all the cards down on the table.

I'm an obstinate and stubborn person who really cares about doing things the way I like them. I don't really thrive in situations where I don't have a lot of agency, but give me the reins and you might get something amazing. I have a kind of foul temper and can get irritated easily, but it's rare that I actually get angry at someone, and when I do I don't stay mad for long.

I like to be sarcastic, but it's probably because I feel kind of awkward when I'm trying to be nice.That's something I want to work on: I want to be a person that makes people feel good and I wish it came more naturally to me.

I'm really picky about the kinds of jokes I tell but I'll laugh at pretty much anything. Speaking of my laugh, I don't actually laugh most of the time. Instead, I wheeze like a dying banshee. Multiple people have expressed concern for me because of this, stop asking questions and just enjoy the show.

I'm also pretty clumsy, but it's usually because my attention is always focused on something else when I walk. I've nearly walked into poles while reading a book, noticing at the last second that something is appearing right in front of me and rapidly expanding in my peripheral vision, and I recently did run into a trash can because I was admiring a tower on the other side of the street. For me it makes life more fun and I wouldn't have it any other way.

The things I like? Hmm, what most people know about me is that I'm good at STEM, where STEM is this nebulous thing that they "don't get". I'm actually not that good at STEM as a whole, believe it or not: I'm really just good at math and programming. (Not CS, at least not yet.) But these are some boring things to like for 99% of people (although if you're on this site, you're probably interested in math, eh?) and I've already talked enough about math and programming.

So let's talk about architecture! I don't really know that much about it: Italian architecture goes Romanesque, Gothic, Renaissance, Baroque. Romanesque is characterized by semicircular arches and Gothic architecture has really ornate windows/decorations and pointed arches, not smooth ones. The Santa Maria in Florence is really impressive; droves of people flocked over there to get in line, and I was lucky that I found a chance to get in when the line didn't wrap around an entire side of the church (and that church was big). Nobody was in the St. Lorenzo though (probably because the ticket in costs nine euros), and that was just as impressive. Speaking of churches, my hobby in Europe was going inside churches and admiring the interior and the paintings. I think I went in 20 churches on my trip.

Architecture was a new obsession I picked up recently (although the roots of it can probably be traced to the Stanford campus --- I mean, have you seen the arches and the marble floor, man?), but something I'd liked for a long time was running. I'm no good at it and it really frustrates me: my 5k is barely under 20 minutes and my mile time is around 5:40. I want to get serious about it since it's my last shot --- I don't see myself competing in college for a myriad of reasons. Cross country is uniquely difficult for me because I can't run up a hill for long. I mean, no one can, but for me it's worse. So I've got to work on that if I want to see any substantial improvement. I love the feeling of stopping at the top of the mountain and admiring the view though. A picture of a lake really can't compare to seeing it in person, both from the summit and right by the shoreline.

I really like writing too --- you can probably tell --- but I feel like what I write is not really diversifying. I write a lot about technology because pretty much nobody knows anything about it and it's really easy for me to write about: just document my workflow. But I want to write more teaching materials, rather than just occasionally writing about teaching. I want to write fiction, too --- I've got a couple of ideas throughout the years and if I come up with something good I'll consider working on it. But the last several times my excuses have been "well I have to worry about grades since it's junior year", "I'll do it after I'm done with all the programming projects I care about" (which seems like it'll never happen at this rate), and now "I have to worry about college applications". All of which are true, but reasons that I'm "too busy for writing" will never go away. I want to confront the fear I have about writing, because from past experience you never really get rid of it, you just learn to get over it when you're writing. (Like now!)

What are some random fun facts about me that I can share? Remembering them all is a lot harder than I expected it to be, which is why I wish I kept better records of myself --- in other words, journaling. Either way, let's give it a go: I like watching anime and reading manga (2022 recs coming soon!). Minecraft is a fun game, but for the last year I've been more hooked on Spider Solitaire than I care to admit. I'm actually getting bored of my mobile games so gaming is kind of taking a hit in the list of things I like. I read random tech blogs in my free time on my phone, which might not be too surprising. Everyone thinks I'm an extrovert but I'm actually really happy when I'm alone, as long as it's not all the time. I think they have a word for that now, "ambivert". I like to sing a lot but I'm at best average.I prefer paper books over ebooks because of the feeling of the pages on my fingers, and if I get stuck on a page for a while that page will tend to crease a bit. My favorite fruit right now might be the honeydew, but apples are really good too because they're so convenient.

Everything I listed before was stuff I remembered about myself, but I'm going to cheat now and look at my journal because I know I wrote more. I want to learn more about military tactics, history, and technology. (I'd watch an anime on the Three Kingdoms if there was a historically accurate one, rather than stuff based on the Romance of the Three Kingdoms.) I wish I knew more about natural history so I could recognize plants and animals when I encountered them in the wild. Speaking of which, mushrooms are actually really interesting, and I discovered this because of a mushroom museum. (Did you know Greece has 9 mushroom societies?)

Was this oversharing? Probably, but I think I might as well get used to it for the next few months. The thing about writing about yourself and sharing it to other people is that all the doubts in the back of your head are right. Most people aren't particularly special and lead fairly ordinary lives, and I'm part of "most people". It takes a special sense of self-confidence to write about yourself, reveal the unflattering parts of yourself, and still do it in an interest way. Let's just say autobiographies impress me a lot more now.

Future

I don't think I'm shameless enough to write down "I did XXX with YYY and feel ZZZ around them", even in a private diary. My journaling has looked like my previous two public posts, although surprisingly it has been more polished, not less. I think I'm going to post some of it to Gemini (finally I've found a proper use for my capsule). Yeah, I know, Gemini is inaccessible, but that's kind of the point.

Let's go through it, day by day.

Disclaimer: I know basically nothing about economics or economic history.

Friday (Sep 8)

Technically you are supposed to register for the conference, and the fee is nothing to sneeze at: $75. We just snuck in, and no one really seemed to care, despite my babyface and lack of a name tag giving away that I was definitely not a grad student.

We were busy taking our fake (21-341 Linear Algebra), so we missed the first half of the presentations on Friday. But our other fake (21-300 Basic Logic) got cancelled, so we could make the second half of the presentations. We also met one of cursed schedule guy's PhD Twitter mutuals (that's how he found out about this conference).

We went to Session 5: American Political Economy.

• The first paper was about the New Deal Era and how the legislative branch increasingly delegated its authority to the executive. As the New Deal is the part of US History I've studied most extensively, I found that I could mostly keep up... until the discussion started involving statistics. The methodology is not something that I could understand in a span of 30 seconds, especially since I wasn't used to the way economics papers presented their statistical results.
• The second paper was about fencing laws. For a bit of background, there is a principle called Coase's Theorem that stipulates private individuals can negotiate an optimal solution, given preconditions that never happen in real life, such as negligible transaction costs, etc. Fencing laws concern ranchers and farmers: should ranchers be responsible for fencing animals in, or farmers responsible for keeping them out, or some combination thereof? The paper analyzed the effect of these fence laws and their changes on farmers and ranchers.
• The third paper concerned the spread of Confederate Culture to the North and West. After the South lost the civil war, many former slaveholders migrated to the North/West for various reasons, and they brought their culture with them. The paper examines its presence (via a number of metrics, e.g. Confederate statues) and its effects. One of the important facts the paper established was that former slaveholders disproportionately found themselves in positions of power, which is how their culture had such an outsized negative influence.

Then we had a plenary session (plenary means "everyone attends") about how housewives provide much value to the economy even though they do unpaid work, and how measures to the productivity of the national economy (e.g. GDP) can be adjusted to better account for this sector of the economy.

After a short coffee break that we spent reading papers and talking to professors (note: I did not do much of the talking), we went to a different hotel for dinner and probably collectively bummed off at least $80 worth of food. (It was really high class.) Then we almost tagged along for the grad students dinner, but then the PhD Twitter mutual said "free-rider problem (also the grad students are paying for the food)", so we were like "I get it" and left. I was perfectly fine with it because I was absolutely exhausted.

Saturday (Sep 9)

The plan was to attend the Historian's Breakfast at 6:30--8:00 AM, because history is something I know a lot more about (broadly) than economics. But due to a series of bad decisions, I ended up sleeping at around midnight last night, and decided that my health was worth more than attending the conference. So I barely made it to the second session of the day.

We went to Session 10: Innovation, Finance, and Urban, but because the bus was running late I could only make it to the second half.

• The first paper analyzed the effects of the loss of men in Japan after World War II. The methodology was very interesting: they used the change of the gender ratio and the death of male soldiers to find how many people died in the war. Also, because the US employed the strategy of island hopping, and regiments were made of people from the same prefectures, some prefectures saw heavy losses in men while others saw lighter losses in men. This lent itself easily to a comparative analysis by prefecture.
• The second paper concerned the effect of trade on US ports. As most history books will tell you, an increase in trade during the late 1800s-early 1900s coincided with industrialization and a decrease in agriculture.
• The third paper was a case study of Red Vienna, a large-scale public housing complex with cheap rent.

I was then going to attend the dissertation session as well, but then I got a splitting headache and realized I was just a couple of minutes away from collapsing. So I went back to my dorm and collapsed on the bed instead.

Sunday (Sep 10)

I deliberately woke up late again to get breakfast. The bus was late again, and this time it didn't even let me off on the correct spot despite stopping there and my pressing the "Stop Requested" button. So I only made it for two papers. I attended Session 16:

• The second paper was about female hypergamy (a fancy word that means "marrying for higher social status"). The conclusion drawn was that, at least in England, hypergamy did not happen. The methodology was twofold: first, marriage registrations were used, and then for those with rare surnames not in the records, the average level of wealth for a person of that surname was used instead. The problem with this is that averaging data out tends to screw up correlation factors, but I'm not actually sure how impactful that is here.
• The third paper was about the link between the spread of electricity and birth rates. There were two opposing factors: electricity made it easier for children to be raised (since many manual tasks were replaced with machines, e.g. dishwashers), but electricity also created new skilled jobs, which increased the opportunity cost of having children as women had a comparative advantage in skilled jobs.

And that was the entire conference! It was a lot of fun, and I learned many things through osmosis. I probably need to study up on statistics, but economic history papers do not seem to have that high of a barrier to entry when it comes to reading them. Any layman could probably understand a good bit of the presentations and conclusions, even if the methodology is mathematically challenging.

You can just run curl URL -O to save it as a file, and then open that file in Firefox. This works, but it requires some manual intervention. Unfortunately piping it into Firefox also does not work (because Firefox is a browser, not an actually good application that respects pipes --- ugh!)

But we can tidy it into a script:

FILENAME=/tmp/$RANDOM.html
curl '$1' -o $FILENAME
firefox $FILENAME

Now, the Firefox add-on API does not just let you run a shell script. In theory, you could run a script (using JS --- fucking browsers) to

• intercept certain URLs (you'd need to create such a list, and unfortunately it doesn't seem there's a Googleable list in plaintext)
• run the shell script shown above.

I don't know if it's worth writing a Firefox add-on when the API changes so fast. Maybe when I have more time --- right now I shouldn't be on these news sites anyway.

Well, that sentence told you exactly nothing. So instead of asking what latexmk is, why not ask why we use it?

Cross Refs

Read A High-Level Overview of TeX, particularly the "Build Process" section.

So LaTeX has to run multiple times in order to get crossrefs to work. But how many times?

With smaller files you can just keep spamming pdflatex until you've gotten all of the references. But

• with bigger files each compilation is fairly expensive,
• and it's nice to just run one command and know it will run exactly as long as it needs to.

Latexmk runs exactly as many times as you need it to. And it doesn't get itself stuck in an infinite loop if a reference is just missing, it stops after seeing that it is missing twice.

Why twice? Well, the first time the reference is supposed to be missing, as it might get written to the aux file after it is called. The second time, it should already be in the aux file, so if it is missing that means it was never written.

Asymptote

If you've ever used Windows + TeXWorks the official asymptote "solution" is

pdflatex FILE
asy -f pdf *.asy
pdflatex FILE

where the second line is replaced by some script in the TeXWorks compilation options.

This is stupid, we should have one command that does the entire process, and optimally, if and only if there are any asymptote files generated. That's part of what latexmk does (at least, if you configure .latexmkrc properly).

Run

# apk add rclone

to install rclone.

Then run

$ rclone config

and you will be greeted with a series of interactive prompts.

First you start making a new remote.

n) New remote
d) Delete remote
q) Quit config
e/n/d/q> n

Then you need to give this new remote a name; it hardly matters what it is, but for this guide we'll just call it "dropbox".

name> dropbox

Then you will be asked what kind of storage you want to use. Look for the option that says "Dropbox" (as of the time of writing, it is 12), and input that.

Option Storage.
Type of storage to configure.
Choose a number from below, or type in your own value.
...
Storage> 12

Then you will be asked about "OAuth Client Id" and "OAuth Client Secret". Just leave those blank by pressing Enter as rclone instructs. You'll be asked if you wanted to edit the advanced config and use auto config too. Just press Enter to use the defaults, unless you have a reason not to.

Finally, you will be prompted to allow rclone OAuth access to Dropbox. Do this. If all went well, you should get a success message afterwards.

To sync, run

$ rclone sync dropbox: ~/dropbox

As long as this command is running, your local and remote Dropbox directories will be synced. If you want, you may set up a daemon to keep this persistently running.

More detailed usage instructions are also available at rclone's Dropbox page, and of course, the man page for rclone.

It will be group lessons, once a week for about an hour. It will be held online on some platform that is NOT ZOOM. There will be 10 official lessons a semester, all taught by yours truly. The difficulty will roughly be mid and late AMC/early and mid AIME. Tuition is $100 a semester (10 lessons)¹, slides/handouts will definitely be provided, and recordings maybe (if I remember to record). The timing will probably be 3 PM Sunday EST (hard start) to 4 PM (soft stop).

The material will be old MAST contents for everything except geometry. For geometry, I will be piloting the draft of a textbook called "A Brief Foray Into Euclidean Geometry". It is easier compared to the content a prospective student might be used to working with for the other subjects, but I think people really suck at geometry. (And it is hard to expect otherwise, given the lack of any good intro level geometry text.) Old MAST materials may have to supplement the text, depending on how advanced the group I get is and how far I get into the draft.

I will not be the sole teacher; however, the amount of instruction other Math Advance staff will do is also to be determined. The only thing for certain so far is I am teaching intro geometry (i.e. the lecture associated with Foray).

Something a bit experimental which might not pan out: I want to try having guest lecturers. This is where other Math Advance staff might be able to help out, or any interested other people. If it happens, the content will be whatever they want, maybe keeping in mind your level of experience. No promises this will happen or to what extent though.

Where I would want staff help is the following:

• Answering questions
• Hosting guest lectures

If you are interested in helping as staff, email me at dchen@mathadvance.org. Otherwise, if you want to apply as a student, hang tight (and spread the word!) Details will come soon in another post, probably within a week. (Said details include finalized timing, the registration process, and a tentative syllabus.)

Details

These details will vary as my and the other instructors' lives shift. (For instance, we don't know when we have midterms.) But here is what we know.

• Read the application document and follow its instructions to apply by September 3rd.
• Classes begin September 9th.
• After 4 or 5 classes we take a week-long break for midterms. I will try to time the bulkiest lesson, the intro geo lesson, to be right before that.

Syllabus

Tentative. No promises what order content is covered in, but the subjects will be interwoven and lectures will be in roughly increasing order of difficulty.

Algebra

• Basic algebra, including symmetric manipulations and logs
• Polynomials (division, remainder theorem, etc)
• Summations (and janky miscellany, e.g. AIME I 2023/10)

Combinatorics

• Introduction to Counting
• Perspectives

Potentially you will see a guest lecture on efficient casework.

Geometry

The primary text is Foray. If I magically finish Ch. 4 by the time I teach Triangle Centers, it will use Foray too. A copy will be provided, free of charge, for students.

• Angle Chasing (Ch. 1, 2, 5 of Foray)
• Triangles (Intended to be Ch. 4, 6 of Foray, but no way I finish that fast)
• How to do trigonometry (for real). sin (a + b) and cos (a + b), complex numbers. Sort of a hybrid of algebra and geometry.

Number Theory

• Mods (definitions, FLT, and Chinese Remainder Theorem)
• Prime factorization (incl. divisor count, sum of divisors, and ν_p)

A Disclaimer

We take our work seriously. This class is meant to be well-designed and teach you things. However, "perfect is the enemy of good", and as much as I don't like this philosophy in general, it does apply to this class. If I were to wait for our team to develop perfect content for an AIME class before we taught, we'd be completely out of touch 30-something year olds trying to teach competition math. (Foray, which does not abide to this philosophy, has taken a breathtakingly long time to write: the first two chapters took two years.)

Nor is this an incredibly battle-tested curriculum. Yes, we have used various versions of the materials with an AMC class, an AIME bootcamp, and MAST. But this is not a class that has been taught a hundred times; it is the first time we are using this curriculum. We have made changes after every iteration, and we can only hope they have been for the better.

By all means, I think you will learn a lot from this class if you elect to take it. However, it is not my intention to mislead you as to what this class is. Our team consists of USAMO and MOP participants. Our students have gone on to participate in these programs too. Yet no class, least of all this one, will be the magic bullet that gets you there. It is not some gospel. It is just meant to shift your thinking a little bit and hopefully give you the boost you need to improve.

There are two things about English essays that really bother me: the lack of paragraph breaks and the boring titles. These two things should be alarming to any English teacher, particularly because even the best writers/papers suffer from this.¹

This suggests a problem endemic to English classes. You can't get rid of the disease just by treating the symptoms, but on the other hand you need the symptoms to figure out what the problem is.

Press Enter

A lot of people have been given totally wrong ideas about a paragraph since elementary school. And this is because teachers tell you, "write a paragraph about your thoughts", and students always ask, "how long does a paragraph have to be?" So then the teachers, thinking they are doing their students a favor by not letting them get away with the minimum possible work,say four to five sentences.

Later a "paragraph" will mean 300 words, or however much space you need to fit in your mandatory 3 quotes per main idea. And even though no one would say the definition of a paragraph is "whatever the teacher says", the way people are writing sure sends a different message.

In a 4 or 5 paragraph paper you have paragraphs sprawling across the entire page, obscuring the development of separate ideas and jumbling it all into one giant clusterfuck. If you go on arxiv and look at a random paper, chances are that each page has 3 or 4 paragraphs, even when some of the space is taken up by diagrams. So even math research is easier to read than a high school English essay.

Chances are that everyone knows when to start a new paragraph: at the place that most optimizes readability. Length and relatedness as two of the largest factors, but there are plenty more.

If everyone knows when to start a new paragraph, why aren't people doing it in English class? The answer is pretty boring: because the teacher told them not to and they don't care enough to.

Title

I suspect the only reason English papers even have titles is because the MLA format requires them. And the titles are usually pretty boring: "An Analysis of Luce's Rhetoric" or what have you.

Typically in English class you struggle to pick a title because you can't think of any. When I struggle in my own essays it's for the opposite reason: there are so many good choices I can make.²

A title is just a really short version of a hook. And a hook explains why you wrote your essay. For instance "Force Multipliers" is about how tools can have multiplicative rather than additive value. "Hook" is how you get someone to read what you wrote. And so on.

If I were really honest in my titles for English class, it would be "Please give me an A I want to go to MIT." That won't fly, so "Into the Unknown" it is.

1. every submodule of a free module over a PID is free, and the proof involves the well-ordering principle,
2. given N ≤ M where M has finite rank, there exists a basis {m₁, …, m_k} of M and ring elements r₁ ∣ … ∣ r_j such that {r₁m₁, …, r_jm_j} is a basis of N. (Of course N and M are over a PID.) The standard proof is over the Smith Normal Form and requires us to rely on the fact that the required row/column operations we do to perform a change of basis are finite.

Relationship to the Axiom of Choice

Relevant StackExchange

What is the relationship of the first fact to the Axiom of Choice? For context we know "every vector space has a basis" is equivalent to Choice. So it does not seem implausible that the first fact is equivalent to Axiom of Choice too.

Nice bases of a submodule of a free module of infinite rank

Relevant StackExchange

Is the second fact true if M has infinite rank? If not, which M disprove it? Alternatively what extra conditions do we need to put on the underlying ring of M? (For instance we know making it a field is sufficient.) So on and so forth.

This is a medium-length (113 pages) introduction to abstract algebra. Intuition is emphasized, fully formalizing arguments takes a back seat, the theory is extremely streamlined (so many important and interesting ideas are omitted).

There is not much to say here that I have not already said in the primer. This covers much more than a standard one-semester undergraduate course and is probably closer to the content of a graduate course, but a lot of details and exercises are not included since brevity is one of the main goals. I imagine the level of detail is somewhere between Evan Chen's Napkin and Dummit/Foote.

Errata

If the date displayed on your PDF is later than the date listed, the error has already been fixed.¹

Only errors are listed here. Other changes, such as additional exercises, streamlined notation, etc. are not included. To see all changes, you may peruse the Git log.

Aug 05, 2025

• Homset Adjunctions: The direction of arrows (namely, f versus f^op) were incorrect/inconsistent. This has been fixed (in Figures 10 and 11).
• Homset Adjunctions: The roles of f and h were inconsistent in Figure 10. This has also been fixed.

Jun 27, 2025

• Coproducts: The inclusion arrows i_c should be of the form i_c : c− > ⨁, rather than i_c : ⨁− > c.

Apr 12, 2025

• Algebraic Closures: To show that a maximal element of ℱ is an algebraic closure, we show that any non-trivial algebraic extension has an isomorphism to some field in ℱ which fixes $\overline{F}$, not F.

Feb 13, 2025

• Natural Transformations: Fix some inconsistent notation. In particular, any time "a₁" or "a₂" was mentioned, I really meant to write c₁ and c₂.
• Concluding Remarks: No group of order 56 is simple. Previously the text said "finite" instead, which is clearly not true...

Nov 25, 2024

• Products/Coproducts: All mentions of a function f were supposed to be ϕ.

Oct 31, 2024

• Exercise 2.11. Significant typo; meant to write g₁⁽ − 1)g₂inH rather than g₁g₂⁽ − 1)inH.
• Exercise 2.29. Significant typo; meant to write N ∩ H ⊲ H rather than N ∩ H ⊲ G.
• Proof of Theorem 2.31 (Second Isomorphism). Minor typo; meant to write N ∩ H instead of n ∩ H.

Collaboration

mapm is meant to be used with other people, but it is not opinionated on how you do this. There are two main ways: Dropbox³ and Git. I would recommend the former.

The problem directory for a profile foo is ~/.config/mapm/problems/foo. So you can symlink whatever directory contains your problems to the aforementioned path. For example, Math Advance uses ~/Dropbox/MAC/problems as our problems directory, so we would run

ln -s ~/Dropbox/MAC/problems ~/.config/mapm/problems/mathadvance

as our profile is called mathadvance.

You can do something similar with Git; just don't forget to push and pull when appropriate.

Multiple contest files for the same contest

mapm was designed for Math Advance contests, first and foremost --- this means it works best with MAT as well as "smaller" contests like the AMCs and AIME. Something expansive like HMMT would not fit well in a single contest file. Fortunately, there is a sane way to compile contests in mapm.

Let's use HMMT as an example. The Combinatorics and Geometry rounds have nothing to do with each other --- how hard problem X is on combo has no effect on how hard problem Y is on geo. Therefore, the two rounds can be considered independent, and thus should have separate contest files. It may also be worth noting that they do not need a separate template file; a subject template variable would suffice.

Here is a heuristic for this independence: do these rounds exist in separate PDFs? For ARML and MAT, where the subject and difficulty distribution of rounds should be taken as a composite, the entire individual test is in one PDF. Thus they should have one contest file.

No solutions

Sometimes you might be lazy and not want to put solutions in a problem. If you're using a release of mapm-cli before v2.1.0, you can pass solutions: [] into your problem file. If you're using a release after v2.1.0, the field is optional and the library will interpret no solutions field as an empty array.

cli tricks

Change default editor

When running mapm edit, the EDITOR environment variable is used. It is strongly recommended that you set it to a terminal editor, like nano or vim, so as to minimize context switching.

List problem names

There is no built-in feature to list problems in the command line interface, but mapm find show should suffice; it finds every problem, and shows no tags. You may also pass filters into find for a filtered list.

You probably learned that matrices were arrays of numbers in high school pre-calculus. Arrays that you can add and do this magical "multiplication". And then later down the road, you learn that matrices are actually linear maps.

That, too, is a lie. A white lie, given that matrices and linear maps are deeply related. (In fact, there is an isomorphism between matrices and linear maps.) But they are not quite the same.

Let me use "change in basis" as an example. You have this "array of numbers" T, and you can "magically" "transform" it into something "similar" called P⁻¹TP by "changing the basis". Note all the air quotations; you're not exactly changing the basis. You're just doing something analogous.

When you're considering a linear map T: V → V, you analyze what happens to basis vectors for different choices of bases. So in a "change in basis" you don't actually change the linear map T whatsoever. You are just considering the behavior of a different set of vectors.

Meanwhile, when you have a similar matrix P⁻¹TP, it is actually different. Yes, it's different the same way $\frac{4}{6}$ is different from $\frac{2}{3}$, but it is different. These are different representations; the array of numbers is literally not the same. Sure, these similar matrices belong in the same equivalence class (where two matrices are equivalent if they are similar), but two similar matrices are not actually identical. A "change of basis", from the perspective of a matrix, is actually a change.

These distinctions are subtle. Obviously there is still a very close connection between matrices and linear maps. But --- especially as you go further --- the distinctions between a matrix and a linear map can be important. Rather than blindly knowing how to do matrix calculations or turning a blind eye to concepts that more naturally originate from matrices (yet still have solid grounding in the theoretical, "only consider linear transformations" side of things), it is most fruitful to understand how both perspectives work in tandem.

...but that won't change the fact that, the first time you learn linear algebra, you should try to avoid matrices as much as possible. Learning the matrix perspective after you learn the theory perspective is not so hard. The other way around is much worse.

Polynomials vs. Polynomial Functions

For more details see StackExchange.

You are likely familiar with these things that look like a₀ + ⋯ + a_nxⁿ, where each of the a_i are an element of some field F. But what is x? It turns out x is just some meaningless symbol.

Then you have these things called polynomial functions that look like f(x) = a₀ + ⋯ + a_nxⁿ. And initially you might think, "Okay, this is just a distinction without a difference; surely a polynomial like x² corresponds to the polynomial function f(x) = x²." And yes, this intuition usually holds; in any infinite field, there is a bijection between polynomials and polynomial functions. But this is not the case in a finite field. Consider the polynomials 0 and x² + x in 𝔽₂. Clearly these are different polynomials, yet the polynomial functions f(x) = 0 and f(x) = x² + x are identical; they exhibit identical behavior.¹

So our notions of equality among polynomials and polynomial functions are not always identical. (Although if you see the StackExchange link above, modifications can be made so that the two notions do once again coincide.)

Prerequisites: have some knowledge of at least one programming language. Not mandatory but helpful, know what a struct is. Also know that && and || are used for the AND and OR operators.

Data Types

Each data type in a struct is logically joined together by the AND operator. For instance, let us say we have items in a store represented by the Item type. Each item has a name and a price. If you have a struct (code not written in any specific language)

struct Item {
    int price;
    string name;
}

then every Item must have a price AND name. Whereas if you're trying to identify an item in a store, you can either use the name or the barcode:

union Identifier {
    int code;
    string name;
}

Unlike a struct like Item, with both a price and name, a union like Identifier must contain only one of an int code OR a string name.

So we've established that there are two data types: an AND data type and an OR data type. In functional programming, the AND data type is called a "product type" and the OR data type is called a "sum type". Why?

To rigorously and mathematically define our product and sum types, where each variable of a type could have infinite possibilities, we'd need to turn to set theory. Since we're trying not to do that, we'll instead build intuition by only considering the case where each variable has finite possibilities.

As an example, take our Item struct. Say that each item in the store is sold for one of 10 possible prices and takes one of 10 possible names. Then we have 10 × 10 = 100 possible Items with distinct data. This is why the AND data type is the product type.

For the OR data type, take our Identifier union. Say that each item in the store has one of 10 possible barcodes and takes one of 10 possible names. Then, in the case we describe an item by its barcode we have 10 possibilities, and in the case we describe an item by its name we have 10 more possibilities, giving us a total of 10 + 10 = 20. This is why the OR data type is the sum type.

Aside: why are unions called unions?

I promised that I wouldn't talk about set theory, and I lied. (If you don't understand set theory feel free to skip this part!)

In set theory, if we have a set A of prices and a set B of names, then the set of items described by a price and a name would be the Cartesian product C = A × B. Basically, each element of C would be an ordered pair of a price and a name.

Now, if we want a set C of all the elements that are either in set A or set B... well, that would be a union. If A is the set of barcodes and B is the set of names, then C = A ∪ B is the set of possible identifiers for an item.

Aside 2: Unions are actually called tagged unions

Actually, in some lower level languages like C and C++ (and for some reason, in some higher level languages like Java too), there is no native support for checking that exactly one field of a union is filled out. Possibly multiple fields could have been assigned values, and there's no way to check, especially since when initializing the union, all fields could be null. In this case, the programmer must indicate somehow which data type the union actually is. This is done through a tag, and this is why sum types are strictly called "tagged unions".

Using our Identifier example, to actually create a tagged union, we'd need to write a struct like this:

enum Tag {
    CODE, NAME
};

union Identifier {
    int code;
    string name;
};

struct TaggedIdentifier {
    enum Tag tag;
    union Identifier identifier;  
};

(This code is written in C. An enum in C is a datatype that must take the value of one of several constants. These tags are said constants in the case of a tagged union.)

Booleans

Consider the boolean function x && y. For x && y to be true, we need both x and y to be true. Similarly, if we want x && y && z to be true, then we need x, y, and z to all be true.

This motivates the following observation: if we do the hack where we assign true as 1 and false as 0, then x && y = x * y, and x && y && z = x * y * z, etc. So the AND operator is very clearly equivalent to multiplication.

The OR operator is not as direct an analogue to addition in this case (the better analogue is exclusive or, aka XOR). In this case, x || y = x + y > 0.

Asymptote is a language totally separate from TeX. You should treat it as a stroke of fate that it happens to support some aspects of TeX like denoting labels in math mode. Asymptote is a standalone language and it can compile without the help of a LaTeX document or compiler, thank you very much.

Typically I see people who use Overleaf leveraging TeXeR, because Overleaf is buggy and can't compile Asymptote properly. If/when they move to a local editor, bad habits carry over for Asymptote, because Overleaf has obscured stuff so much that no one even knows how Asymptote works.

Here is a brief summary of how the LaTeX asymptote package, which should not be confused for Asymptote the language itself, works:

• When you run pdflatex on your .tex file, the asymptote package extracts the code inside every asy environment and outputs it to a set of .asy files.
• You run asymptote (and this time, it is asymptote proper) on each .asy file.
• When you run pdflatex again, the asymptote package detects the .pdf files that you got by running asy on the .asy files, and inserts them into the proper location in the PDF.

Actually, this is one of the reasons you need latexmk: to automate this asymptote process whenever necessary. This is because --- and I cannot stress this enough --- Asymptote is not TeX. LaTeX cannot build Asymptote for you because LaTeX doesn't implement the compiler, Asymptote does. This is in contrast to TikZ, which does not need a whole separate build process because TikZ is part of TeX.

This means that there is a separate compiler for Asymptote. And it is fast. If you want to test whether an asymptote diagram compiles correctly, and if so what it looks like, here is what you do:

• Copy the asymptote code.
• Open up a terminal.
• vim test.asy
• Paste the asymptote code inside test.asy.
• :wq
• asy test.asy
• zathura test.eps&

Bonus points if you make an asymptote-specific keybinding for the compile+preview process.¹

If you use Vim you have no excuse for using TeXeR. Doubly so if you are also using a tiling window manager. Why take the time to open a browser, wait for TeXeR to load, manually add the BBCode nonsense that AoPS expects from you, and have TeXeR fail on you for no good reason with about a 20% chance?

So what are the reasons you would use TeXeR? There are two I can think of.

• You can't run LaTeX on your machine for some reason. Perhaps your only computer (maybe it's a phone) is super weak, and you need to leverage the browser to perform expensive calculations. Or perhaps you're stuck at the public library and forgot to bring your laptop.
• You don't know how to use your own computer.

So, instead of throwing out the entire laptop, the rest of which worked fine, I did the sensible thing and just replaced the SSD. We were (as of the time of writing, "are") leaving for Chicago and so I wanted my SSD soon. Thus we went to Best Buy, even though I could've gotten a fully functional SSD for around $20 less if I bought it online. Since I was using a 256GB NVMe SSD, I figured I might as well get another NVMe SSD. The cheapest option was a 512GB SSD, so I ended up getting that.

Replacing the SSD was pretty straightforward: unscrew the cover, unscrew the SSD, replace the SSD, and put all the screws back in place.

If I wanted to, I could've cloned the old SSD onto the new one. However, I decided against it for several reasons:

• The old SSD is failing, so I had no idea how or even if the cloning would succeed.
• I'd have to put both SSDs inside the same machine (my desktop), and I didn't want to go to the trouble of disassembling that too.
• My old Bedrock system was pretty complicated, and I didn't feel like I gained anything from this extra complexity.
• All my important files are in Git repositories somewhere off my laptop.

So I decided to install Alpine Linux on my new SSD. In terms of how easy to setup Alpine is, it's a mixed bag in a couple of interesting ways. It's by no means "beginner-friendly", but for an experienced user the setup is so much easier than Arch Linux.

• Installation is straightforward: you just run setup-alpine. A little bit of Googling might be involved if you don't know what some of the options mean.
• Actually getting a graphical desktop environment and setting up networking is a bit hard, especially if your only experience is systemd/your distro took care of it for you before. I didn't try installing X11, but it seems fairly involved to get i3 working; getting Sway and Wayland to work, however, was much easier.

I like Alpine's apk a lot more than pacman. pacman has arcane letters where apk's operations are very simple to understand. I also like that apk also doesn't ask you to confirm where every other package manager does. I believe apk is also a bit faster, but in terms of how fast package managers are, apk and pacman are approximately as fast with everything else being a lot worse.

Alpine also takes a lot of cues from the BSDs. They use doas instead of sudo, and OpenRC instead of systemd. I like doas, and don't really care about OpenRC vs. systemd.

I've had pleasant experiences with Alpine, sans OpenSSL not working (I've replaced it with rustls in my projects). Someday I might migrate the Math Advance server to Alpine, in which case my dev and server environments will be identical.¹

So it's no surprise that a lot of programs break on musl libc. calc, for instance, does not work on musl libc. (A good alternative is libqalculate, for everyone else on musl libc.) Even my programs had issues; bootstrap for my dotfiles and the MAT website¹ used OpenSSL, which does not work well on Rust + Alpine. Thus I switched to rustls, which is better than OpenSSL anyways and works on all the hardware I need it to.

There are plenty of merits to using musl libc which you can easily find online. The wisdom of using musl libc on a desktop environment may be debatable, since some software (mostly propreitary software that's been built against glibc, like Dropbox or Steam) won't run. This means you'll have to get glibc, one way or another. Thus Alpine, through no fault of its own, may not be suitable for casual desktop use if, for instance, your goal is to play games with minimal hassle: Ubuntu is a better choice for that.

In any case, if your programs do not support musl libc, it should not be a decision made out of ignorance. It's usually less of a big deal if you don't write a lot of C, but a lot of things in other languages can refuse to work for subtle reasons involving glibc/musl. And if you're, say, running Rust servers on Alpine, it's also important to understand that the reason your program is segfaulting is because Rust + OpenSSL does not work well on musl. If you want your programs to be reusable, part of that is making it portable, and if you want them to be portable, don't have them rely on glibc extensions.

Sidenote

This post was almost called "Stop relying on glibc", but besides musl I know of no other libc's that are part of a major libre operating system.² There are things that work in glibc and musl but not, say, dlls, but I could frankly care less about them.

For those of you who don't have experience with musl libc distributions or proprietary software, this is a big deal because most proprietary software is compiled with GNU libc (glibc for short). To vastly oversimplify, these programs will be expecting functionality provided by glibc and glibc only.

I don't love the official Discord client. It takes up way too much space and I like working in the terminal regardless (which is why I'm often on 6cord). But each of these unofficial clients have their own problems. 6cord takes a while to switch between channels and the voice chat doesn't work.¹ gtkcord4 is doing some weird stuff with keyrings (i.e. not remembering my login) and despite many attempts, it refuses to respect my theming preferences (maybe this is just a GTK problem). It's technically against TOS too (not like I really care though). Still, I'm looking for any excuse to ditch Discord, or at least its official client. Hopefully one day this will happen.

Anyway, here's how I did it:

• install gcompat and patchelf through doas apk add gcompat patchelf
• install Discord as a tarball through wget -O discord.tar.gz https://discord.com/api/download?platform=linux&format=tar.gz (or click on the link)
• cd to whatever directory contains discord-0.0.x.tar.gz (x is the version number). This file will be called discord.tar.gz if you fetched it using wget, in which case replace discord-0.0.x.tar.gz with discord.tar.gz.
• tar -xvzf discord-0.0.x.tar.gz
• cd Discord/
• patchelf --set-interpreter /lib/ld-musl-x86_64.so.1 Discord
• patchelf --add-needed libgcompat.so.0 Discord

You may need to run ldconfig and/or reboot for this to work.

To run Discord, do ./Discord --no-sandbox. I'm not sure if the --no-sandbox flag is necessary on Xorg-based systems, but it probably is required on Wayland (definitely is for Sway). Blame Discord for shipping an out-of-date version of Electron /shrug

I use a tool called sshuttle to tunnel all my traffic through a server which I have SSH access to. (Installation for sshuttle is straightforward and can be found on the project's README.) This set of conditions is not very common since "people who need to bypass school wifi and have SSH access to a server" basically just describes high school sysadmins.

But if this describes you, here's what you do:

sshuttle -vvvv -r user@0.0.0.0 0/0

where you substitute user@0.0.0.0 with your server and the user you want to SSH into (the user doesn't actually matter).

Background

The MAT's tiebreaker format is as following: there is one tiebreaker set, and ties are broken first on points then on time. Another way to think about each tiebreaker problem is worth $\frac{1}{3}$ of a point.

To understand the problems with tiebreaker formats, I have to first present the Fundamental Theorem of Tiebreakers:

If two people have identical performances, then there is no way to rank them differently.

It is not necessarily true that it is always possible for two people to have identical performances; it depends how you measure it. For an in person contest, you might use "submission order" as a tiebreaking method. (Indeed, we use "time" as a proxy for this.) However, for our purposes,

It is always possible for two people to have an identical performance.

In fact, for MAT it is quite feasible. For two people to score the same on the main sets and the tiebreakers, and for them both to forget to submit until they've used all 30 minutes, is not impossible. Though we avoided it the last two years, this year the jig is up: we had two ties (3rd and 7th).

"Submit earlier, stupid"

This was our initial reaction. If only someone decided "hey I'm not going to get the problem in the last 10 seconds, let's submit at 29:50". But unfortunately that is not how real life works; people are not always optimal strategists. Even more so in a contest like MAT.

The MAT was designed not to require much strategy. That is why we have split the contest into several sets. You do not need to wonder, "hey should I check questions 1-5 or take a stab at question 9 which I left for later or reach for questions 11-15"? All you can do is solve what is in front of you.

Contests like HMMT Guts require a lot of strategy, and that can be fun. But focusing on solving problems is much better when it comes to developing logical skills in the long run, and personally I think it is more enjoyable. We're not the only ones who think this way: top contestants have directly contacted us to express the same sentiment.

So tiebreakers requiring a strategic decision (time or problems?) is already a jarring break from our philosophy. Realize it or not, but the first three sets lower peoples' guard. It gets them thinking, "thank god for a contest I can just relax in". Which is a great mentality to have, but the downside is it increases the risk of ties after tiebreakers.

However, I think a reminder on the contest page during the tiebreaker set ("Remember that ties are broken on time if scores are identical.") might help with that, since people have told me they forgot tiebreakers was even time based.

Estimathon

When originally designing the MAT, I considered using an estimathon (or even just one estimation question) as a tiebreaking method. In theory, if you make the upper bound for the answer around 10⁵ ("how many primes are less than 10000?"), then it will be very unlikely people guess the same thing. Add multiple such problems and add contrived formulas for the estimation score and bam. No more ties.

There's a couple of problems with that. One: the more complicated your scoring formula, the more complicated the optimal strategy. Which means that contestants will either have to perform suboptimally, or shift their focus away from solving problems. Two: people gravitate towards certain numbers. Whether they are seen as funny, or look nice (multiples of 5, multiples of large powers of 10, etc). So someone is more likely to guess 2500 than 2583. Although with multiple problems, the second point is more moot.

Furthermore, estimation problems tend to have less clean solutions, if any at all. The skills we are trying to reinforce do not include "throw out wild guesses at random questions and guess less badly than everyone else". It would be such a stark departure from our contest format too, and it would drastically reduce the problem quality (even if it is quarantined in tiebreakers). An estimathon would be jarring and feel out of place. So we bite the bullet and accept that our tiebreaker format might lead to ties.

Breaking less ties

I floated an idea around recently, though it is one we are unlikely to adopt: who cares about ties? After biting the bullet once, it seems much more acceptable to think that way now. Instead of making a compromise on our philosophy with the tiebreaks format --- ties being broken by time --- why not just ignore time?

Because it could lead to massive ties, too large to be acceptable. Last year (not this year), we had a four-way tie for 1st if one only considers main set scores and tiebreaker score (i.e. if one ignores time). And looking back, it was a disaster narowly averted. If people decided to submit a couple minutes later, we may have had ties with a tiebreak time of 30:00.

But think of it another way: four people all score 8 points on the main sets and ace the tiebreakers. The only difference is, their submission times are 25, 26, 27, and 28 minutes. Are their performances really qualitatively different? No. We only acknowledge this difference because they all came in with the shared expectation, "time matters". Yet that mostly awards strategic skills, not mathematical ones --- and strategic skills based on extrospection, rather than introspection. The role of a math contest is to develop problem-solving introspection. So a contest format with an optimal strategy based on "what would other people likely do here", even if very tangentially, is not desirable.

Yet it too is a bullet we must bite for prize reasons. Many of our prizes are discrete items that cannot just be "split" between massive ties. And on some level, it is good to have a winner --- the adrenaline of winning a math contest is part of the fun. So we will likely continue using time as a tiebreaker method. It is imperfect. Indeed, it is the worst tiebreaker method, besides all the others that have been tried.

Data that a user manually edits makes the most sense in config. If you look at .local/share on your machine, chances are that it's filled with incomprehensible machine-readable but not human-readable data. (And that it's all from GUI prorams.) And even though problems are not config files in the traditional sense, you are writing key: val pairs anyway, so in spirit it really is a config file.

Also, it makes things ten times simpler if all mapm data is stored inside the same directory. This is why profile, which arguably does belong in $XDG_DATA_DIR, stays in .config/mapm just like the rest of mapm-cli's files.

As for mapm-gui, comments etc will not be directly edited by users, but will be interacted through the mapm gui interface. So I am thinking that mapm gui information will go in $XDG_DATA_DIR. That makes the setup process a bit more complicated for people working together, but whoever's in charge of tech in an organization using mapm should be competent enough to write an install script that abstracts it all away. (That "someone" in Math Advance is me right now.)

A little off-topic, but thoughts on mapm-gui: since LinkedHashMap doesn't implement Serde in Rust (or is it the other way around?) there are no guarantees for the structural integrity of problem files, particularly with respect to comments, if you edit fields through an interface. So I am thinking that similar to a forum post, you see the "rich text render" when you are not editing, and when you are editing the problem yml itself pops up in a pane somewhere. Trying to work out manual text replacement/editing seems too buggy to be worth it, especially since YAML is pretty simple/readable anyway.

Due to some poor initial technical decisions by our tech lead (read: me), Math Advance has been straddled with a bit of tech debt. To summarize, we have stuff written in crappy not-Rust languages (typically Node JS) and want it written in Rust. It will be largely from scratch, though we will keep some legacy components of the MAT website (due to me not wanting to reimplement it all), so the tech debt is not something any new developers will be straddled with. There will be no crappy codebases to struggle through.

A list of Math Advance related projects, from my most to least recommended:¹

• Math Advance logo (not actually software): We have one for MAST and MAT but not one for MA as a whole. Currently I'm working through some prototypes but if you have any ideas yourself let me know!

• mapm: mapm is the software we use to compile our contests. This is my top software recommendation because it's the one that sees the most real-world use today (read: I'm not the only one that uses it) and has the most flexibility in contributing! Some ways of contributing do not even require any Rust experience.

  The mapm ecosystem is currently in an alpha-state, where it is usable for our purposes and few others. Naturally, I would like that to change! Some areas of research include a GUI (which is helped by the fact I planned ahead and separated the core mapm library into another crate), a mapm-specific package manager for templates, more templates², and manpages/shell-completion for the CLI.

• mathadvance.org: When you go on our current landing page you learn absolutely nothing about Math Advance. Sexy, but impractical (and also JS! Blegh!) It's just going to be a Makefile and some custom config since a) it's gonna be a static site and b) I hate every static site generator out there. It's very simple, but design help is appreciated!
• mast.mathadvance.org: Currently only works with JS. I intend to rewrite this website so that everything is server-side generated, i.e. no JS. It will be one monolithic codebase written in Rust, much like what Glee is intended to be. No design skills necessary, we'll stick with the same design; it just needs to be re-implemented in not Tailwind.
• sysadmin: I've got a couple sysadmin scripts for MAST I should write/migrate. Improving the mailserver and setting up SourceHut mailing lists are on my to-do list as well; I'm currently using Mailcow and I don't like that it uses Docker, that kind of blackboxes things for me. Nothing super permanent can happen with regards to sysadmin though, because so much of what I need to do is based on GitLab which we'll hopefully replace with Glee soon anyways.
• glee: See my initial post and followup. I did mention Svelte in the followup but my position has become more hardline: no JS shenanigans, which means no SvelteKit. We're going to use Rust (probably axum as web framework), plain old HTML, and CSS, tied together with a healthy dose of NGINX. The way I'm going to approach this project is very opinionated, so unless you are experienced, know my opinions, and share them, I recommend you look at something else. There's a lot in the Math Advance world worth looking at, eh?
• mat.mathadvance.org: It's currently a living nightmare; I don't recommend you work on this.³ We'll unfortunately still be using Javascript, because I am not rewriting the frontend.

Of course, the nature of open source is that it is distributed. You do not have to join the Math Advance team itself to send patches! Total commitment is not necessary, especially for mapm --- which is why, if you want to help but don't know what to pick, I seriously suggest you pick that.

If any of this interests you, shoot me an email! Alternate forms of contact OK too, if you have them.

add/commit/push

Commits are the smallest unit of reasonable change. In between two commits should lie the smallest possible meaningful change that doesn't break anything. Notably, it should not be reasonably possible to break a commit into two smaller commits.

("Commit" as a noun means "a specific version of the repository history", while "commit" as a verb means "create a new version with the smallest reasonable difference from the current version.")

git add is the "scratchwork" as you are making the edits you need for your next commit. git commit, well, actually creates the version. git push is irrevocable¹ and updates the rest of the world about your changes.

Merge conflicts

Merging/rebasing is a way to reconcile the history of two divergent branches.

There are two types of divergent branches:

• branches whose changes commute,
• and branches whose changes do not.

As a concrete example, let us say I create file A and you create file B. Then, no matter what order we do that in, the end result is the same: files A and B have been created.

Git uses diffs (the difference between two branches) to calculate the changes made. So, for instance, if we start with a file that says

b

and I add a character a above the b, it will look like

+a
b

while if you add a character c above the b, it will look like

b
+c

and Git is smart enough to realize that the change isn't, "Dennis overwrote the file b to ab, while you overwrote the file b to bc" --- rather, Git recognizes that I added a before b and you added c after b. Therefore, even if we edit the same file, there is a possibility that git merge spits out

+a
b
+c

because under a diff based system, the order in which we apply our changes does not matter. You can write c below b before I write a above b and we will still end up with the same result. (By the way, diffs are performed on a line-by-line basis: it will not work if I change b to ab while you change b to bc.)

However, if I do

+a
b

but this time, you do

+c
b

the order which we apply the changes matters. If we apply my change first, we get

+a
+c
b

while if we apply your changes first, we get

+c
+a
b

Git can automatically merge for you if the order does not matter; any reasonable interpretation of the changes (my changes first or your changes first) gives the same result. However, when the order of the changes matters (i.e. the order of lines is not necessarily fixed), that is when you get a merge conflict and must manually resolve it.

A more abstract example with mathematical functions

To make an analogy, consider the functions f(x) = x + 1 and g(x) = x + 2. Then f(g(x)) = g(f(x)) = x + 3. In other words, it doesn't matter what order you apply the changes (i.e. f and g) in: the end result is still the same.

However, if f(x) = x + 1 and g(x) = 2x, then f(g(x)) = 2x + 1 while g(f(x)) = 2x + 2. Here, the order in which you apply the changes matters.

Here's a trick to remember that j is down (and thus by process of elimination, k is up). j looks like a fishing hook, so you can visualize a fishing hook going down.

With this, the number of times each file's metadata (i.e. the title and date fields) is parsed has increased to three: one for the All Writing page, once for whichever one of the Essays/Blog/Tech pages it falls into, and one for the RSS feed. It would likely be possible to fold it all into one Rust script, which reads only the metadata (i.e. doesn't try to parse LaTeX as Pandoc does, which would already make it a lot faster), and only reads it once. This will probably decrease the build times of my website, at the cost of some additional complexity in the form of another language. (On the bright side, the code will look nicer because I have fmt strings.) But my build times right now are longer than 10 seconds, which is unacceptably slow for a static site.

Puzzle fixes

I fixed a couple of small things in my older puzzles recently too. In Starry Sums, there was no mention of the positive diagonal restriction in the rules, and I just now added the solution to the example puzzle in Alphabet Soup. (By the way, I really recommend you give Alphabet Soup a try.)

Roughly speaking, there are two things you need to know:

• The server has no "memory" of you, it cannot recognize if two requests come from the same person/machine/etc. (They can technically use your IP but that's not reliable for authentication --- consider a public library!) Thus the client must authenticate for each request.
• Anyone can be a client and send an arbitrary request for your server, so you must verify all requests.

Either way it boils down to the same thing: verify all your requests and make sure you make no assumptions about them when handling them server-side. In particular, this means do not rely on any client-side form controls: Inspect Element or just curl can trivially disable them. Client-side checks exist for convenience, not security.

Authenticate for each request

The client will make separate requests for each distinct action it wants to take with the server. For instance, if I wanted to write a new topic in a forum that would be a POST request (POST requests are generally used for write operations to the server's database). If I wanted to then reply to that same topic, it would be a separate POST request. And if I wanted to read said topic and reply later, that would be a GET request (generally used for read-only operations). Each action the client takes is one request. (This applies for loading a page too: when you go to dennisc.net, you're making a GET request to the route /.)

To protect against malicious requests the client has to prove within each request, via means of a cookie or a token, that they are who they say they are. This is authentication.

There are two common ways to authenticate on the web that we will cover: username/password and session tokens.. To simplify, a website usually works as thus:

• use your username/password combination to generate a session token and set it as a cookie (this is what happens when you "log in")
• for any other GET or POST requests, the client should send the session token to verify its identity.

The browser automatically sends the session token to verify its identity by storing it in a cookie, which it then automatically sends on every request. This is what cookies are: just information the browser sends on your behalf for every request to a website.

Anyone can make an arbitrary request

Whenever people first try to write a full-stack web "app"¹, they inevitably fall into the trap of thinking that only officially approved clients (i.e. only the frontend of your website, and not any external actors malicious or otherwise) will send requests to your server. This leads to embarrassing things like "authenticating" requests based on username rather than something verifiable like a session ID.²

Personally, downloading Postman has been really informative. Not only is it possible, it is also easy for any user to send whatever they want to your endpoints. cURL should be similarly enlightening.

In any case, you can either only allow your website to act as a client (this is done for forms using a CSRF token, but you should not solely rely on this) or accept that clients can send whatever they want. This also means that any clientside checks in formdata must also be done serverside.

Summary

The server has no memory of clients, so they must send identifying information on each request. Anyone can act as a client and they can send anything, and you must verify the validity of each request server-side.

Basically, assume every request is made with malicious intent until they prove otherwise.³

When you were a little kid your teachers and parents probably emphasized the value of hard work. So much so that the act of putting in effort itself was seen as virtuous, rather than working as a means to an end. Certainly motivation is a problem for most people, but chances are if you're reading the blog the opposite is true. I would be incredibly shocekd if someone unmotivated decided to spend their time reading essays instead of, I don't know, playing BTD Battles 2 instead. Which would be fair enough, since I would do the same.

A lot of people do a lot of their work on a computer. If you're anything like me, you probably get distracted easily when you're working on a computer and encounter resistance. After all, the friction of checking your phone only needs to be less than the friction of the problem you're working on for you to check it. Checking your phone is easy and worthwhile work is hard, so it's really easy to waste a lot of time.

Quotas

We think of work quotas as being "spend at least X hours on Y task." This turns out to be a very wasteful approach. It might be more helpful instead of set a maximum time limit instead; say, only spending up to two hours of free time a day on your work. Instead of thinking "I want to do vague X, how much time do I need?" think "How much can I squeeze out of T time?"

There's a reason the people best at what they do are minimalists. Novice writers think "How many pages can I stretch this out to?" The best think "How little space do I need to fit everything in?" Computer noobs think sexy is "Wow, look at how pretty my Windows screensaver is." Power users think "Wow, look at how much I can do with so few resources and such little clutter."

Experiment

I'm going to go full circle and try something: a screentime limit. Here are the details:

• At home I can only use the computer two hours a day. Homework counts (incentive for me to do it faster).
• Using my computer at school doesn't count, since my butt is already in a chair, it would be too hard to count, and I actually spend my time at school programming instead of playing games (mostly).
• I'm recording time on little slips of paper. I think it'd be ironic if I wrote scripts on the computer to prevent myself from using it. Also if something critical happens (read: the English essay I don't give a shit about is due when I get to class and it's 5 AM in the morning), I want to be able to use the computer. (Or if I'm just transcribing an essay from paper to computer, I really don't care if I go over for that.)

In principle I think you could omit the general ban altogether and just limit your work. After all, the point of this restriction is to make work more efficient, not because the computer is a boogeyman that will kill you if you stare into it for too long. That being said, I still think a general computer limitation is better:

• It's harder to get distracted. Especially for power users like me, I can just press $Mod+d to open dmenu, then run steam, and enter into some game from there. It's not as easy to put down my pencil, walk over to my computer, deal with the guilt of committing not to work midway through my work, and also commit to wasting my two hours of computer time. It's also hard to switch back to working, so it's less likely I fool myself into thinking "I'll just finish it later, how hard could it be to switch to Workspace 1 where my Vim terminal is open?"
• Health. This really requires no explanation, it's just common sense.

Note this only works for academics --- limiting your exercise time is not likely to work the same way, even though in principle some people do need stop exercising as much (injured people. I learned that the hard way.) This is because when you're seriously studying there's only one pace --- a full-on sprint. Whereas when you're exercising there are long days, hard days, recovery days, etc. all with different paces.

Sucks

There are some things that really suck about limitations like this though, at least for me. They are somewhat specific so someone else trying this probably won't face these.

• Writing. There are some times when I can think much faster than my hand to move. This problem is mostly solved when I'm typing because I can type really fast and also am an absolute Vim[^vim] lord. This is why I always do revisions by typing, because I can basically think on the fly and translate it to words. (Though forcing me to write on paper has gotten me to write more, even if it takes longer.) [^vim]: I actually kind of suck, I still don't use hjkl in Vim (though I do know how to use it in programs with vim-like keybindings, thank you very much).
• EdPuzzles. Imagine if your teacher forced you to watch an unskippable ad, except instead of being 30 seconds long it's 30 minutes long. And instead of being about Google's new phone it's about the Civil War. I guess I can use it to get some gaming time in?
• Music. Although I already know how I'm going to handle this: I'll just use the TV to play whatever song on YouTube while I'm working out. (Obviously it won't count as "computer time.")

These really won't be issues, I'm already thinking of them as features and not as bugs.

I would say on the whole that this experiment is probably a good idea. Part of me is slightly embarrassed that I'm publicly admitting I use the computer so much that I need to give myself a screentime restriction, grade-school style, but in the end I think it would be more embarrassing not to acknowledge this problem.

Part of why people compartmentalize the web is because its standards and features are so expansive. With almost a hundred status codes (and 418 I'm A Teapot isn't exactly helping), media queries for printers², and a million different Javascript-based frameworks, it's no wonder that the frontend devs don't want to deal with the backend on top of all this. And for the same reason, backend developers understandably stay the hell away from this abomination.

But rather than learning everything webdev related, you should try to do as much as you can using as little as you can. You don't need Tailwind when CSS variables exist, you shouldn't POST to ajax.php when you can just set up an endpoint for the same page you sent the GET request to for a form (come on, forms aren't proper APIs anyway), and you shouldn't use cloud-based databases like Firebase because that surrenders control and incurs extra complexity. Web development can be a simple process --- you're just requesting and sending data according to established protocols --- you just have to commit to making it so.

I've been saying "frontend" and "backend" developers this whole time. However, a more accurate way to characterize them is "people who don't fully understand the web". There's no shame in admitting it! The internet is filled with awful advice, bloated and unnecessary Javascript frameworks are presented as the norm, and servers are blackboxed instead of being treated as just another Linux machine, just with certain ufw rules set. But if you commit to using a small subset of the web's features, avoid Javascript unless absolutely necessary, and understand the principles of client-server authentication, creating full-fledged websites on your own will not be too far off.

For example, taking 3 = {0, 1, 2}, we may take ℱ = {{0, 1}, {1, 2}, {2, 0}}.

I claim the maximum is $\binom{n}{\lfloor \frac{n}{2} \rfloor}$. Equality is obvious, just put every subset of size $\lfloor \frac{n}{2} \rfloor$ into ℱ.

To show the upper bound, we will prove that

$$\sum\limits_{A \in \mathcal F} \frac{1}{\binom{n}{|A|}} \leq 1.$$

This suffices because

$$\sum\limits_{A \in \mathcal F} \frac{1}{\binom{n}{|A|}} \geq \sum\limits_{A \in \mathcal F} \frac{1}{\binom{n}{\lfloor \frac{n}{2} \rfloor}} = \frac{|\mathcal F|}{\binom{n}{\lfloor \frac{n}{2} \rfloor}}$$

and multiplying by |ℱ| on both sides would yield $|\mathcal F| \leq \binom{n}{\lfloor \frac{n}{2} \rfloor}$. Now we show this bound indeed holds.

Take a random maximal chain 𝒞 ∈ 𝒫(n). For example, taking 3, we might get 𝒞 = {∅, {0}, {0, 1}, {0, 1, 2}}. And obviously |ℱ ∩ 𝒞| ≤ 1: remember that no members of ℱ are subsets of each other. Thus E[|ℱ ∩ 𝒞|] ≤ 1.

But by Linearity of Expectation,

$$E[|\mathcal F \cap \mathcal C|] = \sum\limits_{A \in \mathcal F} P[A \in \mathcal C] = \sum\limits_{A \in \mathcal F} \frac{1}{\binom{n}{|A|}}.$$

To elaborate, $P[A \in \mathcal C] = \frac{1}{\binom{n}{|A|}}$ by symmetry: 𝒞 contains exactly one member of cardinality |A|, and there are $\binom{n}{|A|}$ such sets in ℱ to choose from.

And that's it!

Currently the easiest way to do this would be to share my password with him. Well, drats. How do I share my password with him? I'd have to send it to him (bad idea), he'd have to copy it from his texts (mildly annoying), then he'd have to switch to the Google Fi app (unnecessary context switch), and finally he would be able to paste it.

Technically in some cases it is possible to exercise proper access control, like with Google Brands (for YouTube and presumably Google Plus before it died), but Google sucks so much at documenting their services that even many veteran Google users (read: pretty much the entire world) don't have any idea something this essential exists. We need a culture shift where it's expected that different people can do something on the same account without sharing passwords.¹

Maybe we shouldn't be sharing passwords. It's a terrible idea for security and it's inconvenient to boot. The same way that multiple people can SSH into the same user of a server, maybe we should allow multiple people (using separate identities) to perform actions with the same account (same permissions). This is what Glee does: there are multiple users (identities) which are assigned a role: admin or user, depending on the permissions I want them to have.

Access control is a hard problem. GitHub and GitLab manage to mess it up royally and make it a major pain to use, which is why you have people spending hours and even creating entire teams just to manage access control. It's also why I'm building my Git service with access control in the center of my mind. It might not be easy to design a solution that works elegantly, but can we please, please not have the default solution to "two people want to do things on one account" be "your use-case is unsupported (so you have to share passwords, deal with it)"?

Additionally I posted a tutorial called From Zero to mapm, to bootstrap anyone who is generally competent with computers (you should know the basic shell commands and Git commands) with the tools needed to install mapm. The only runtime dependency is LaTeX and the only buildtime dependency is Rust; the tutorial also instructs the user to install Git because templates should be hosted with Git.

Currently the tutorial only has Mac and Linux, but I plan to write one for Windows soon! I'm just quite rusty with Windows, what with it breaking every convention known to man.

I hope this is as exciting for you as it is for me. Even though we do not have a GUI yet (optimistic estimate, mid-summer), the command-line interface is expressive though a bit awkward at times², and the documentations, tutorials, examples, etc are quite rich for such a simple program. mapm has been battle-tested against some members of Math Advance, who've gone through the grueling work of setting it up manually, writing documentation themselves (thanks especially to Prajith for the Mac OS documentation, which I used as a reference for "From Zero to mapm"), and it works without major hiccups on all major operating systems.

If you are writing a math contest and really want it to be good I suggest you try mapm, if you've got a tolerance for dealing with any issues that might arise. The ability to recompile drafts in the matter of seconds is super helpful, especially since we can be confident that all problems are coming from the same source, the mapm problems directory. (I'll probably shill harder in the future once we've got a GUI: for now we are sort of in a beta stage. If that doesn't sound like your cup of tea yet, please come back to mapm in a few months!)

P.S. mapm is always stylized with all lowercase letters, even when it starts a sentence. I sort of implicitly set that convention for myself and now explicitly follow it after I realized. You can feel free to respect or break this in your personal conversations, but any official docs should abide by this. Let me know if they don't.

College decisions

• REA Princeton => Defer
• EA UT Austin + Turing => Accept!
• RD MIT/CalTech => Reject

I think --- or perhaps, thought --- I have the kind of personality that aligns itself with MIT/CalTech best. Now I am rethinking this, and this is only partially me malding and seething over the rejections, because the reasoning against MIT/CalTech applies to some other schools (most notably CMU).

For people who don't know, UT Austin is a Texas state school that is really good at computer science for undergrad (top 10 on US News, or whatever other metric you want to use). Turing is an honors program that selects less than 100 of the incoming freshman CS class, and this is very biased, but I think it's really good.¹ There's a tight-knit community already forming within (some subset of) the admitted class of 2027, and apparently Dr. Lin (the director of the Turing program) has dropped bombshells like "The difference between an Ivy and Turing is a Lambo, pick the Lambo". I recommend that the class of '24 applies to this program!

Anyway, the point of this post isn't about college decisions, it's just about a fun visit.

Visit

We left WARP for the airport at about 12 PM. Looking back, the cohort of us who wanted to travel should have requested an earlier group, but until then I wasn't even aware there were people with ridiculously late flights (and were interested in coming to Austin). But anyway, I vaguely remembered (and fortunately, my memory was correct) that we had to catch the bus on route 20, and as luck would have it, we caught a bus right as it was about to leave.

I didn't have enough coins to pay for all 5 of us, so the bus driver just said "forget it" and let us on the bus after I inserted some number of dollar bills (probably because we were holding up the line). The bus took much longer to get to Deen-Keaton than I would have hoped (45 minutes, as opposed to ~25 on Uber), but since we didn't have to wait for an Uber at the airport it was probably okay.

After scrutinizing Google Maps for the entire bus ride to know when we were supposed to get off, we found ourselves at the perimeter of UT Austin, which means that it was the intersection of UT and downtown. None of us had lunch, so the first thing we did was get In-N-Out. Three of us were European and never had In-N-Out before, and suffice it to say, their entire worldview got shattered.

Then we walked around campus with the limited time we had left and took a couple of pictures near weird structures. We went past Moody, the biotech(?) and chemistry sections, and saw the sign for the Cockrell School of Engineering. Funnily enough, I was not able to find the Computer Science building, which is ostensibly what I visited for in the first place! (If I had more time, I definitely would have gone looking for it.)

Impressions of the campus:

• Seems lively! Integrated with downtown, lots of cool stores and stuff to do in Austin. Also there is an In-N-Out right outside of campus, which is very important and awesome.
• The architecture is on point. Unlike Northwestern, the campus actually looks consistent; every building has the same vibe, color palette, and materials. And unlike Harvard, it's actually really pretty.
• I'm concerned about the lack of trails that I saw (which maybe is a problem with how little we explored). Google/Reddit says that there's stuff nearby, but you actually have to run pretty far to get there or take the bus, neither of which is a great choice.

Everyone else had a flight at ~6 PM while my flight was at ~8 PM (or it was, until it got delayed a stupid amount), so we headed back pretty early and had to get to the airport fast. The bus was too slow, so I decided to Uber instead, but Uber was being stupid and kept reserving taxis at 17:45 when I clearly inputted 15:45, twice. When I realized, I went "fuckfuckfuck" and just reserved an Uber to come pick us up, NOW. Our driver was initially going pretty slowly but once we texted him that we had flights to catch, he put in the full stops to make sure we got to the airport as fast as possible.

Then when I passed through TSA Pre, I got mistaken for a SXSW attendee, then correctly identified as class of '23. Right as I was about to leave I said "Longhorns" and the agent was like "that's what we're talking about". Everyone in Austin (and even Woodcreek, where WARP was held --- which is quite far from Austin proper!) seems to be very proud of their flagship university, which is a far cry from what happens in California.

Airport

FUCK SOUTHWEST.

So "because of weather" (which is somewhat reasonable, fine, I guess), the first leg of my flight got delayed by an hour. Then Southwest put my layover in Las Vegas instead of Dallas, which seemed reasonable enough, and initially it would've taken off at 8, as originally planned. Then it too got delayed until 9. Trouble is, the second leg of my flight decided to come on time, so now I had to catch a flight that left at 5:25 AM instead. Well, that's great!

It would've been OK if Southwest actually, y'know, provided accommodations given that I paid for a continuous-ish flight from AUS to SFO, but OK! Apparently they do not provide accommodations "because the delays were solely due to weather", which accounts for... pretty much every delay? Never mind what the customer is supposed to do about this situation --- either sleep on the floor of the airport or pull an all-nighter and die of sleep deprivation because Southwest ALSO decided to change my boarding group from A to C, which means that basically the only seat left was a middle seat next to a guy who... did not leave a lot of room left for me, let's just say. (My other neighbor decided to try and steal my armrest --- the only armrest I had because my other neighbor couldn't fit without putting the armrest separating us up --- until I just laid down the law and put my arm there.)

So now I am stuck in Las Vegas, after pulling two consecutive all-nighters, with no food in my stomach and shitty water fountain water. I'm not happy about Southwest's response to this situation, but as for the situation itself, it's pretty acceptable. This is the risk I took when I decided to book a late flight in order to visit UT Austin, and I think it was a risk worth taking --- the flight part just panned out really poorly for me this time. The only lesson for me to learn is to book flights a little earlier because they will probably get delayed and certainly will not leave early.

Overall it was a fun trip worth taking, and a nice way for me to close out WARP by hanging out with the boys even after we left camp premises. Scout mindset.

But people forget this frighteningly often specifically when dealing with middleware in web servers. (An example of applying middleware is altering every response to return the appropriate CORS headers, because it would be a massive pain in the ass to manually do that for every response.)

Let me give you two examples. First I was trying to update Glee's dependencies. You may as well just read the commit message:

Update dependencies

Important changes:
- fallback (i.e. 404) is no longer into_service()
- axum::Server is gone, need to use tokio::net::TcpListener as in the
  axum example

      https://docs.rs/axum/latest/axum/index.html#example
- as Form is an Extractor that might consume the request body, we must
  put the Form extractor *last*.

  https://docs.rs/axum/latest/axum/struct.Form.html
  https://docs.rs/axum/latest/axum/extract/index.html#the-order-of-extractors

  Important to remember from second link: *the order of extractors
  matters*. They are applied left-to-right and then the request is
  fed, and *generally extractors do not commute*.

You don't say, right? But I managed to chase this error for two hours before realizing that my middleware was non-commutative.

Today I was working on a web project for K-Scale Labs. We had been facing perpetual, mysterious CORS errors. I spent several hours wondering if, despite the configuration for CORS being piss-easy in FastAPI, I had somehow managed to mess it up. It turns out that my API calls were incorrectly formatted. Fixing them magically made my CORS errors go away. What on earth was going on?

My leading hypothesis as of half an hour ago was "some inexplicable bullshit is going on that I will never understand". Only then did I actually decide to look at how the CORS middleware was set up in the context of the rest of the app, not just the one or two lines that set up the CORS middleware itself.

Oh. The project was passing every request to the appropriate handler BEFORE applying the CORS middleware, and those requests would return appropriate error responses by RAISING AN EXCEPTION. My CORS configuration was perfectly fine. Instead, every actual error was being swallowed by a CORS error because the webserver raising an exception stopped execution before the CORS middleware could be applied.

And I overlooked all of this because this code and the structure were not set up by me.¹ I was just taking it all for granted. There is a lesson here: confusing things are very often the cause of non-commutative middleware, especially when you don't understand the structure of the overall web application.

In some ways this is an obvious principle. If I asked you explicitly, "do your middleware commute", well of course they don't. But you don't have a fairy godmother hovering over your shoulder asking "do your middleware commute" at the exact right times. You are bound to forget once or twice, which is precisely what makes this class of problems so irritating.

But you can do the next best thing: anytime anything is going wrong and you can't figure out why, you should always ask yourself if the cause is non-commutativity. Be your own fairy godmother. Non-commutativity in general is the most rich source of bugs and unpredictable behavior. Even more so when it comes to web projects, which tend to have a lot of complexity.

• you have a branch with a bunch of garbage WIP commits;
• you can edit the default commit message and write something non-terrible.

But let's be real. We already had a tool for squashing garbage commits: it's called git rebase -i. Just fix up the garbage commits, because surely if you have made a change non-trivial enough to require WIP commits, it can't be trivial enough to fit in a single commit, now can it?

But git merge --squash is not the target of my ire today. No. It is GitHub, and it is the people who enforce only using the squash and merge option on pull requests.

Listen. The only reason I have ever heard for enforcing squash and merge is "it makes for a cleaner commit history". Cleaner my foot. The only way to have a good commit history is to commit with intention and rebasing to clarify and clean up. Squashing will not get you there. Sure, your two-thousand line diff is now compressed in one commit with a terrible message like refactor backend, rather than ten commits with even worse messages like wip and done and oops. But that is still a terrible commit history, only now the terribleness is compressed in less commits.

Worse, squashing will actually delete any commits made with intention. And unless you really expect every commit to be a PR, which is patently ridiculous, you are actually removing good history and actively making it worse.

Furthermore, say you want to squash and merge and actually have a good commit message. Suppose you rewrite the entire commit message or whatever. A good commit message has appropriate line wrapping, something my editor (vim) is perfectly capable of handling properly. GitHub does not even give a whit of an indication when a line is too long. So you can't even write a proper commit message. Ridiculous.

So what should you do? Just merge. Merge like a normal person.

git merge --no-ff

This makes a merge commit even when you could have fast-forwarded, because it explains why you are merging a certain feature.¹

While I'm at it, GitHub pull requests are a disaster. As usual, the Linux Kernel is the gold standard.

When you want to merge feature into master you should run git merge feature while in the master branch. And obviously you take care of any merge conflicts in this merge. In particular, you do not "back merge" master into feature without good reason because

Merge branch 'feature'

Merge branch 'master'

is a really stupid commit history! But GitHub, especially with a lot of its automated CI/CD stuff encourages maintainers to put the responsibility of fixing conflicts on the people writing the PRs. So now the only thing they can do is rebase on top of master, and then re-test every commit because otherwise they'd be lying.

• not self-hosting a Git server,
• not having the necessary build tools on your server (and there might actually be good reason for this: do you really want to install TeX Live on your server?)

automatically building stuff on your website becomes a major PITA.

So let me suggest another approach: instead of automating "server receives stuff => server builds", why not automate "client pushes => client builds => client sends build output to server"? Particularly for smaller sysadmins like myself, the server is not actually more powerful than the client: I'm running a low-resource Vultr cloud instance, but other people might be self-hosting with their own hardware like a Raspberry Pi. For us, building client-side may actually be the better solution.

Here's how it works. I write a pre-push hook (I would've preferred post-push, but it doesn't really matter and post-push hooks don't exist) that looks like this:

make
tar -czhf dennisc.tar.gz build/*
scp dennisc.tar.gz HOST:dennisc.tar.gz
ssh HOST "tar -xvf dennisc.tar.gz && rm -rf dennisc.net && mv build dennisc.net"

where HOST is the IP of my server. In order to automatically authenticate into SSH, you need to set up an SSH key and configure HOST to use said SSH key in ~/.ssh/config. What my SSH config looks like is

Host mathadvance
    Hostname 45.32.70.22
    IdentityFile ~/.ssh/mathadvance
    User mast

(I'm revealing the IP of our server since it's public information anyway: ping dennisc.net will get you it.)

Anyway, what this hook does is it builds my website using the Makefile, creates a zipped tarball from the build output, copies the tarball to my server, and extracts the tarball/moves it to the proper location. It's nothing complicated, though it is mildly annoying that I have to set this up on all my devices.

dennisc.net

glee
dennisc.net
dennisc-gemini
dotfiles
    sway
    i3blocks
    ... (nvim, etc.)
    bin
        musiclist
        latex-lint
        ... (dlatexmk, etc.)
math
    mvc-am-gm
    linalg-notes
    ...
misc
    musiclist
    rust-beam
    beam (planned)
    min-tex-install
tex
    bounce
    nature
password-store (private)

mathadvance.org

mapm
    cli
    lib
    gui (planned)
mast
    units
        APV-Log
        ...
    unit-drafts
        ...
    diagnostics
        season1
        ...
    lectures
        ...
    mast.cls
    mast.mathadvance.org
contests
    mat.mathadvance.org
    archive
    summer-mat.cls
    winter-mat.cls
books (private)
    mabook.cls
    foray

If this course were to happen, here would be the details (many of them are still quite flexible). The course would probably last 6-8 weeks, meeting twice a week 75 minutes each time, with a focus on groups and fields/vector spaces (we will do some linear algebra). Rings will likely be mentioned too though. Maybe some categorical theoretic ideas. It will be an introductory course, which means we will spend a good amount of time on first principles. However it will be designed to be as hard as possible while still keeping students interested in the material and without being overwhelming. Most importantly, the course will either be free of charge or low cost (think on the order of 100 dollars with financial aid).

The intended audience is largely high-school students (rising college freshmen included). No math contest experience is expected, required, encouraged, etc (everyone with sufficient mathematical background and curiosity to learn is welcome). Here's a quick pitch for the course that aims to answer the following questions:

• What is abstract algebra about?
• Why should I take it?
• Why should I take it with you?

Structure begets structure. This is what abstract algebra is about. Doubtless you are familiar with the integers modulo p. Consider the set of all the non-zero residues G = {1, 2, …, p − 1}. Clearly they are closed under multiplication modulo p and we also know each element has an inverse. Given a fixed g ∈ G, the map h ↦ gh (recall both g and h are non-zero residues modulo p) is a permutation (i.e. a bijection from a set to itself). Furthermore we know g^p − 1 ≡ 1 (mod  p) for all residues g, and in fact, if k is the smallest positive integer such that g^k ≡ 1 (mod  p), we must have k ∣ p − 1.

It turns out that neither of these facts are all that unique to multiplication modulo p. For instance, take addition modulo p: with G = {0, …, p − 1}, h ↦ g + h is a permutation too and the smallest integer k satisfying gk ≡ 0 (mod  p) also satisfies k ∣ p. Or consider the permutations of {1, 2, 3, 4}, of which there are 24. If we repeatedly apply the same permutation k times and get back to where we started --- {1, 2, 3, 4} --- and k is minimal, then k ∣ 24.

What on earth do integers modulo p and permutations have in common? How can we abstractly capture these commonalities and use them to derive maximally general results? And there are differences too between the integers modulo p and permutations. For one, multiplication is commutative. Composing permutations is not. When do these differences matter? And in what ways?

Abstract algebra also connects well with combinatorics. Take Burnside's Lemma for instance, a good way to take care of overcounts caused by symmetry in counting problems. It is precisely because of facts in elementary group theory (buzzwords: "orbit-stabilizer theorem", "class equation"). There are many examples of mathematics you know that will provide clarity to abstract algebra, and many parts of abstract algebra that serve to clarify deeper properties of things you know.

Hopefully you are convinced abstract algebra is an interesting subject. But you have plenty of interesting things to do, and you can take abstract algebra at any time. Why now? The obvious reason is that it gives you a leg up. If you've gotten exposure to algebra and have a decent understanding of the motivations behind the ideas (even if you don't quite remember the ideas themselves), with a bit of self-study you can set yourself up to take a graduate-level algebra class your first semester of college.² But there are subtler reasons. Abstract algebra as a first course leads well into a lot of other subjects: examples are category theory and topology. You could technically study those without algebra, but it is so much better when algebra gives you (part of) the reason to care.

And in my personal opinion, algebra is the hardest subject to do wrong. You could study QR/LU/SVD decompositions for months in a standard linear algebra class, to say nothing of row reductions. (Also I think the study of matrices, which is a common interpretation of linear algebra, is not particularly interesting.) And most the proofs in analysis are long and painful, and all it takes is a teacher without good judgment as to which proofs to skip to make it mindless computation. Algebra, meanwhile, pretty much only has cool stuff. It has combinatorial arguments, number theoretic arguments, and direct ties to both subjects too. It develops the widest repertoire of skills while being the least painful.

My teaching philosophy, which I will expound on in a later post, is simply to give people a reason to care. Instead of saying "Here is Theorem 238, with a random setup and random conclusion," my aim is to say, "here is a concrete example, generalize it and you will get Theorem 238." The words concrete example may instill dread in you, perhaps because your previous teachers used it to mean "hyperspecific nonsense no one cares about." Instead I mean a natural example we already care about. For example, take the integers ℤ, something we all have a good feel for. One of the reasons we care about integers is because they uniquely factorize into primes (e.g. 120 = 2³ ⋅ 3 ⋅ 5), and this allows us to do things like define the notion of a GCD. A natural question would be: which definitions/properties/procedures work because we're working with integers in particular, and which ones hold just because we have some notion of unique prime factorization?

These questions are quite abstract and nebulously defined, and that's the point. Ring theory and the notion of Unique Factorization Domains came about because we wanted more precise answers to questions like these. And when the student is aware of this, it is much easier to notice the nuances of the general theory at hand, because now we have some idea what we should care about! Much better than saying "a unique factorization domain is a ring with properties X, Y, and Z" right?

And finally there's the mandatory spiel about how I've been teaching for however many years: this is far from my first rodeo. I've been teaching every year for the last 7 years, I think my students enjoy my lectures and get a lot out of them, and empirically they do quite well for themselves afterwards. (Not that I think the causal link is super strong though.) So if circumstances permit, I do think this class would be worth your while.

Are you interested? In the Pittsburgh area this summer? Email me at dchen@mathadvance.org to discuss details! This class will only run if enough interest is expressed (something like at least 5 people).

f(x) = x + 1;

f(1 div 0)

SML will say, "OK let us look at 1 div 0 first". And then it will raise a Div exception because hey, you can't do that!

A language like Haskell, in contrast, will use lazy evaluation. What Haskell will do is first expand f with no regards to its input to get something like

f(1 div 0) => (1 div 0) + 1 => Div error

(OK, Haskell actually gives you Infinity. But pretend that we're dealing with SML except it does lazy evaluation instead.)

Why does this matter?

Often we will want to prove one piece of code is equivalent to another, via structural induction or whatever. If we have a function like

f(a, b) = a + b

then we should be able to say that

f(g(a), g(b)) = g(a) + g(b)

But no! We cannot! Because the way equality works is (roughly) that two pieces of code have to evaluate to the same thing. So you actually have to prove that g(a) and g(b) even evaluate to something, as opposed to not raising an exception.

It turns out that in this case it is equal, regardless of whatever you set g, a, and b to, because + is a simple enough function. But it is annoying to prove, which is exactly my point, and if we set f to something more complicated we would not be able to prove this.

A case where (with eager evaluation) mathematical equality breaks is with

f(x) = raise Fail ""

(A quick definition: raise Fail "" just raises the exception Fail.)

You'd want to be able to say that f(x) evaluates to Fail --- in other words, f(x) = Fail (in the mathematical sense) regardless of what x is, right? It'd only make sense for f(x) = raise Fail in code to mean the same thing as f(x) = Fail.

But no! Plug in 1 div 0 and all the sudden f(1 div 0) = Div. Arghhh!!! This function had one job, to just raise the exception Fail! Why can't it do something that simple?

This makes it annoying to mathematically reason about functions. Now you have to handle edge cases (i.e. where errors pop up), or just work in the world where everything behaves as you want and never handle said edge cases. The former is unappealing, and the latter misses the point. For a paradigm of programming all about formal verification and relying on mathematics, not being able to rely on the very definition of a function isn't very mathematical.

Eager evaluation has its advantages...

but mathematical verification is not one of them. I understand that when writing real-world code, which is chock-full of side effects, lazy evaluation might make things a bit harder to work about. But also running real-world code should not require an eternity and then some, so I'm not sure "practicality in the real world" is exactly the main concern of functional languages :)

Appendix: run it yourself!

If you have SML New Jersey installed on your machine, run

fun f (x : int) : int = raise Fail "";
f(1);
f(1 div 0);

in the SML REPL. You should get Fail and Div errors, respectively.

Git, GitHub, and GitLab

For a well-informed discussion about these two tools, it's necessary to get a clear understanding of what they are first. (If you don't already know the difference, this is just a brief introduction --- I recommend you look this up in greater detail on your own time.)

Git is a version control software. If you're using the command line to interact with Git, then all the commands you run are built on Git. They have nothing to do with GitHub or GitLab.

Git hosting services, like GitHub and GitLab, are just places where you can host Git repositories so as to centralize them, to some extent. This way people actually have a codebase they can consistently reference, which is especially helpful for beginners.

The important concept to grasp is the direction of dependency. GitHub/Lab rely on Git, not the other way around.

GitLab

The term "GitLab" is even trickier to define because it can mean two sort-of different things. Because GitLab is open source (hooray!) you can host your own instance of GitLab. So when I opened with "I moved all my serious projects to GitLab," it was actually ambiguous: did I mean the website gitlab.com, or my own private GitLab instance?

It turns out that for people like me (developers with their own server), it doesn't really matter. I can migrate my GitLab projects from the public GitLab website onto my own private Git server, and vice versa. The reason I'm hosting on the GitLab website is because of accessibility. This way, all my personal code is publicly accessible, and whatever non-sensitive Math Advance stuff is also publicly accessible. It hasn't been long enough after the switch for Google to start caring about my GitLab projects, but I'm sure that over time it'll eventually start displaying something vaguely related.

The Good

Every comparison of GitHub and GitLab I see lists features of each tool and does a superficial side-by-side of them. The conclusion is always the same: there are reasons to choose one or the other, it depends on your circumstances. This is fair and all, but not very helpful if you don't actually elaborate on which tool better fits which circumstances. In my eyes, GitLab is vastly superior when it comes to organizations that are creating a large number of discrete things.

Keep in mind that I didn't say a large organization, just a large number of discrete things. For instance, MAST has a lot of units (around 30 at least at the moment), even though the team actively working on them is quite small. In contrast, if your organization is really only working on one project or website or whatever, GitHub will be fine, though I don't think GitLab is inferior for these purposes either.¹

The real nice thing about GitLab is groups/subgroups. Groups function similarly to organizations in GitHub. But you can also think of a group as a root directory, and subgroups as nested directories. Learning this is what finally got me to make the switch over.

Back when MAST was still using GitHub, we had one repository with all the units inside of it, along with some extra garbage. Why in tarnation would we ever do that, when the LaTeX for two separate units has absolutely no relation to each other? There was absolutely no reason for it --- not even when using GitHub --- but I believe that one reason was because there were no nested directories in GitHub. If I made a separate repository for each MAST unit, I'd end up flooding the organization with mathadvance/CQV-Perspectives around thirty times over. Oh, and I'd also have to grant access to the MAST team for every single one of these repositories.² Maybe keeping everything in the same repository wasn't a mistake, and that is horrifying.

In GitLab we have mathadvance/MAST/units as a subgroup, and unit repositories in the subgroup. Much simpler, and navigating through the group actually makes sense now without external READMEs or whatever, which helps with DRY --- no need to note in separate places how your projects are organized if it's self evident.

Differences

These are not issues with GitLab. Rather, they are challenges that I faced because I was used to GitHub, or just things that are plain different.

First off, GitLab doesn't have the equivalent of GitHub for students, or whatever it's called, that gives students free GitHub PRO. As far as I'm concerned there is no significant difference --- if your organization or codebase is big enough that the limits start posing an issue, you can afford to pay for whatever you're using. (Or self-host on a server.)

Something I really miss is a personal README (a la chennisden/chennisden on GitHub). This is totally a vanity feature, and I don't need it whatsoever because I have a website to show off my tech stuff. But it'd still be nice to have it on my GitLab profile too.

I think GitLab by design encourages a lot more caution, which probably makes it more suited for organizations whereas GitHub might be better for individuals who are just starting out.³ Repositories (or projects, as GitLab calls them) are private by default. And by default you have to wait seven days for a project to soft-delete in a group. Then there's how GitLab interacts with Git: by default, you can't delete protected branches (or any branches, depending on who you are), you can't force push, etc. It's nice to have this here, actually, because I only really need the ability to force push in certain repositories anyway.

The one real struggle I had with GitLab was the lack of a central CLI tool. There are plenty: there's one in Go, several in Ruby, one in NPM as well --- and depending on what you search for, your first result will be totally different. And some of these CLIs have incredibly abtruse documentation; I couldn't figure out how to specify a (sub)group to make a project in for one of them, for crying out loud. In contrast, GitHub's official CLI is very self-explanatory, and you can use it without having to search for documentation.

I will be explicit and say that I think GLab is the best of these by far. In fact, it's might be better than GitHub's official CLI tool is for GitHub. And, at least on Arch, it's just as easy to install --- one yay -S gitlab-glab-bin and you are done.

Concluding Thoughts

Contrary to what it might seem like, this isn't me trying to get everyone reading to start using GitLab instead of GitHub --- just some people. Specifically, if you're doing organization-style stuff you might want to check out GitLab. But if you're just a beginner with two wholly unrelated repositories, the granular level of control GitLab gives you is probably not necessary yet.

Something I never mentioned yet is that GitLab makes it very easy to import your GitHub repositories. This made it a lot easier for me to migrate over, but if you have more projects you may want to consider scripting with GLab instead and doing everything on the terminal.

Addendum

Update from Dec 5 2021: Apparently you cannot create subgroups under a user. This kinda sucks, and apparently I'm not the only one who thinks this. The workaround I'm using is just making a group called 'chennisden' but that's kinda nasty to be honest.

Spoilers ahead for X Marks the Spot, so please try it yourself first! It is 4 or 4.5 stars in difficulty and is a thermo/arrows sudoku puzzle with no numbers given. This post won't make much sense until you've finished it anyway.

Many thanks to all my testsolvers! Without you, I would've posted a broken puzzle --- thanks for taking the time to do the puzzle even after the first version was broken.

Recently I've been doing a lot of sudoku puzzles on sites like gm-puzzles, and I noticed that long thermos/long arrows are particularly constraining. (I mean, duh.) So I thought, "huh, what if I slammed down a long thermo and arrow, and had them intersect?"

This has the added bonus of creating the theme, "X Marks the Spot", and cool stuff probably happens because of it, so I'll build the rest of the puzzle around this. (This cool stuff ends up becoming the logic in the cross in the final puzzle.)

(You might see that the thermo doesn't match what's on the final draft. I only realize it doesn't need to be this long later.)

Now I have some ideas of bounding the center, determining a large portion of the thermo. I know this isn't enough though, so I just throw some random crap at the wall.

You will notice that the two rightmost arrows seem very constraining --- and that is exactly so! Even though I randomly added them, I had the instinct that they would be the most helpful. So I do a bit of logicking and discover the following:

This is based on the idea that the two thermos have to add up to at least 16. There are 5 numbers on in arrow in the rightmost column, and the five smallest numbers add up to 15. Add the other number on the arrow which is at least 1, and the sum of the circles is at least 16.

At this point, everything besides the thermos and the rightmost arrows is irrelevant. Note that A and B represent 1 and 2 in some order. The important problem is this: in the rightmost row, the smallest numbers on the bottom arrow are B, 3, and 4, meaning that the sum in the circle is 7+B. But also note that the sum in the center arrow is B+A+4+B=7+B, contradiction. Crap.

Well, what if we make the thermo a little shorter?

(I removed the arrows that weren't doing anything at this point, because now I'd honed in on some actual ideas I could work around.)

Okay, now we have some numbers determined, but this isn't nearly enough. Time to constrain things a little more. This time, instead of throwing crap at the wall, I deliberately force 1 and 2 to not be in the center row.

Now I come up with the idea to restrain the locations of 1 and 2 in the center row. I notice the 1 and 2 in the third column and immediately realize the most natural way to do it is with a thermo.

After some more logic, however, I find that it doesn't work.

There's nowhere the 5 can go in the center row (check the thermo). The solution I eventually come up with is flipping the thermo a la the final puzzle, but first I shorten the thermo, forget that I can have 2 in the row now that the thermo is shortened, and send this ill-fated first draft to my testsolvers.

However, this failed draft did give me something --- the ideas with the arrow on the bottom and the diamond arrow on top (since 7, 9 were restricted on the second row, 1, 2, on the third row, and 1, 2, 3, on the third column) which I thought were cool and still tried to preserve in the final draft.

Eventually I settle on doing the right thing --- flipping the thermo.

At this point, I bring back the arrows, because I still think it's underconstrained anyway (it is) and want to add them back and adapt to the new circumstances. I quickly realize the bottom arrow is not constrained enough.

So I bend the bottom arrow to constrain the first number to be 1 or 2 again.

Now some logic:

There are a lot of steps, many of which align with the final puzzle: the fact that 3 is in the bottom of the bottom arrow and so on. I won't lie, a lot of these deductions went over my head when I was constructing the puzzle at the end until I used a computer to find some common similarities between all the possible solutions.

After some more computer analysis, I concluded that adding a thermo would finish the puzzle.

The end.

Future

I'd always intended this to be a one-off thing, at least until I finished. I wanted to make the most out of the ideas I had... and I would've followed my plan, had I not thought of an even more crazy puzzle idea. This one will take a while longer, as the logic is so intricate that even I'm currently stumped by it. Look forward to at least one more release!

If an honest writer drops the modifiers in their writing, just like their teachers tell them to, they typically also drop its nuances. Any essay consisting only of absolute, unqualified truths is not worth reading. It's not even worth writing. That's why every high school English paper looks like it was written by an elementary schooler. It's not because the author is a dunce. It's because they don't allow themselves to say anything a dunce doesn't already know.

Besides the obvious function of modifying a statement, modifiers have another use: pacing. When you say "I think" or "maybe" it weakens the statement. When you omit modifiers and pointedly deliver a line, you emphasize it. But why would you want to express weakness in your argument? Because it helps you communicate clearly. What do you think, and what do you know? What's concrete advice, and what's just a potential option?

Any advice telling you to discard a tool that creates variety is almost always wrong. You might not think removing modifiers would change the substance of your writing all that much. But if you want to write better, you have to change the way you write. (Or in this case, you have to stop letting your English teacher change the way you write.) Sometimes a modifier might not seem to convey any additional information. But modifiers aren't just a matter of practicality, they're a matter of style.

And I think it makes for damn good style.

TL;DR use reychen@dennisc.net from now on

Well! After some failed experiments with self-hosting on bare metal (which I may try again at some point in my life, once I actually have stable housing), I realized I haven't been able to send email, only receive. I had no idea what was happening and didn't really dig into it because I was a bit busy dying at college. But now I know what's happened, and for now I'm using Zoho to have a functional email address. The self hosting experiment may continue at some point in the future, probably after I graduate.

To anyone whose emails I haven't responded to: my apologies! Particularly if it concerns teaching. I have been really busy lately, but now that my email actually works, please feel free to send your email again. (Any emails sent in the past year or so have forever been lost to time, so please paste the full body of any relevant emails.)

so what happened?

It's a pretty boring story. I could receive emails and not send them, and that was because Vultr silently blocked port 587.¹ I eventually realize this is the case when I try actually reading the messages and pinging port 25/587, and getting no response. A quick internet search confirms that my suspicions were right. A month or so later, I decide to actually get down to fixing it.

However, Zoho is... more than a little buggy. For some unknown reason, it believes that dchen@dennisc.net is a used alias when it is not. And I really cannot be bothered to figure out what is actually going on, so here we are.

Today a student in my Fall 2023 AIME course submitted some solution writeups which contained the following line:¹

We can simplify x² + xy into x(x + y).

And the gist of my feedback was,

Instead of saying simplify, it's better to say factor.

Imagine you are teaching a class to students. And you say "do a thing to this thing and get a thing" instead of saying "apply the function f to the integer x and receive a string". Very unspecific, right?

In my eyes, "we can simplify" is like that. It could mean a bunch of things: you could be expanding a factored polynomial, factoring an expanded polynomial, cancelling terms, etc. It's basically the same as "we do a thing to the equation". What thing?

When the word "factor" is used, it is much clearer.

To be clear, the proof was written very well (which is why I bothered to be nitpicky like this). But this is a good segue into some thoughts I have about proofs and communication, so I'll use it as an excuse to talk about them.

Being clear in a proof is the most important thing. Sure, it should be reasonably correct, in that it shouldn't claim anything that's untrue and should be reasonably² complete. But the primary responsibility of a proof-writer is to make it clear, "Hey this is what I am doing right here and now," at every step of the proof. Yes, compromises need to be made between being explanatory and being concise. But using the most specific word possible is being more explanatory without being any less concise. In other words, it is an improvement you can get for free, so take it.

Pedantic? Yes. But when you're proving something, your job is not to say "and the things I said are right". It is for the reader to say, "I see what you did, I am convinced it is correct, and I get what you're doing." So even if being more specific doesn't make your proof more correct, it is still just as valuable.

Last time we discussed the limitations of mathematical logic and how it leads to many worlds of mathematics, regardless of whatever system of rules we choose. This time, we will discuss a theorem known as compactness which allows us to do some powerful things in mathematics, as well as demonstrate some more limitations of first order logic.

To motivate this, let's start with a theorem:

If every finite subgraph of an infinite graph is k-colorable, then the entire graph is k-colorable.

At first glance this is not obvious. But with compactness, we will be able to show this very easily.

Terminology

Before we dive into compactness, let me first informally define some terminology.

1. A theory is a set of rules. It is a purely syntactic object.
2. A model of a theory is a concrete mathematical object that satisfies the theory's rules.

For example, the vector space axioms form the theory of vector spaces, and ℝ³ is a model of this theory.

Compactness

Suppose we have an inconsistent theory. That means we may derive a contradiction. The important thing about a contradiction is that it is finite, as it is a proof of a false statement, and every proof is a finite sequence of steps. This means that the contradiction must reference a finite number of rules.

Therefore, if a theory is inconsistent, it has a finite inconsistent subtheory. Taking the converse of this statement, we get the following:

A theory is consistent if every finite subtheory is consistent.

How does this prove our theorem? Here is how we do it: given a graph G, we construct a theory that describes k-colorings of G. If it is consistent, then Godel's Completeness says G is k-colorable.

Let me be a little more descriptive: our theory describes functions from V (the set of vertices of G) to the set 1, …, k. For every edge (u, v) ∈ E (the set of edges of G), we have the following rule: the color of u must not equal the color of v (more formally, f(u) ≠ f(v)).

To show the consistency of this theory, we just need to show the consistency of every finite subtheory. Each finite subtheory only references a finite number of vertices, say S. And by assumption, the finite subgraph S is k-colorable, so the finite theory has a model. Therefore, it must be consistent.

And we have just proved the theorem!

Upwards Lowenheim-Skolem

Aka: Blowing up models of a theory.

There is a theorem that roughly says the following:

If I have a theory T and an infinite model M of T, then I can construct a model M^′ of T that is as big as I wish.

Here is how you prove it: consider the theory T^′ that consists of the rules in T, as well as some rules expressing "I have λ many elements, and all of them are distinct." (Here λ is a cardinality.) The sentences we use to specify they are distinct is "element c_i and element c_j are not equal" for every pair of elements c_i, c_j whose existences we mandate.¹

If we have a model of T^′, then it is also a model of T with λ many elements. By Godel's Completeness, we just need to show T^′ is consistent. And by compactness, we just need to appeal to finite subtheories.

Note that any finite subtheory references a finite number of elements c_i. Therefore, all we need to do is have a finite number of distinct elements in our model of this finite subtheory. But since M is infinite, it already does the trick.

Limitations of first-order logic

You may know about the Archimedean property of an ordered field, which states

A field F is Archimedean if for every x ∈ F, there is some n ∈ ℕ such that n > x.

In English, it states that for every element of the field, there is a larger natural number. Most ordered fields you know are Archimedean; think the rationals and reals.

In propositional logic, we can write things such as "p ⟹ q". First-order logic allows us to access the entire universe of our model: for instance, if we are working in the theory of ordered fields, then we can write statements such as "for all elements x in our field F, if x is not 0, then x has an inverse; furthermore, that inverse has the same sign as x."

The Archimedean property is what we call a "second-order property". This is because we have two universes we are quantifying over: we are writing ∀x ∈ F, which is quantifying over F (first-order), but then we have ∃n ∈ ℕ, and quantifying over ℕ is not something you can do in first order logic.

You may try to express the Archimedean property in first order logic. You may even try to establish a theory with an infinite number of sentences that is equivalent to being Archimedean (i.e. if it is satisfied, then the ordered field is Archimedean, and if it is not satisfied, then the ordered field is not Archimedean). Both of these will fail. Is there a way to prove this?

It turns out Upwards Lowenheim Skolem gives us a very easy way to prove this. There is one important thing you need to know: every Archimedean field is isomorphic to a subfield of ℝ. In particular, this means that no Archimedean field has a bigger cardinality. Just take this fact for granted.

Great. Now assume for the sake of contradiction that there is some theory of Archimedean ordered fields. We have an infinite Archimedean ordered field: ℝ. Now blow it up to a model with cardinality larger than |ℝ| that satisfies this theory, contradiction.

There are other properties that cannot be expressed in the language of first order logic. Two of them are the phrase "this field is algebraic" and "this ordering is a well-ordering". And the way we prove neither of these are expressible is quite similar to the proof for the Archimedean property.

EF --- Linux Made Simple

List

• Partition your disk
• Boot into ISO (can be swapped with partitioning disk, not recommended for beginners)- Mount the appropriate filesystems
• Pacstrap into the mounted filesystem
• Chroot into the mounted filesystem and perform setup operations (timezone, keyboard, locale, etc)

Why?

Partitioning

You are installing Arch Linux to somewhere, and it has to be somewhere on your hard drive. (Or SSD if you use that.) When you are partitioning your disk you are telling your system "an OS is installed here". So you basically create a place for the OS to be installed, before you actually install it.

Boot into ISO

The ISO itself is an operating system, though incredibly minimalist.¹ What's useful is the scripts inside it. The scripts don't even have to be contained in a separate OS --- you could just package a set of scripts that installs the OS to the appropriate partition. (In fact, if you are already using Linux, you can do this!) The reason that installing through an ISO is popular is because of accessibility. Most people, especially those installing Linux for the first time, are using a shitty OS like Windows or Mac (which tends to be why they switch). Said operating systems make it harder (particularly Windows) to install the necessary tools than it needs to be, and packaging everything in an ISO is just better for uniformity.

Mounting

This one is the hardest to explain.

To access a filesystem in Linux you have to mount it. Your partition that you want to install Linux is a filesystem. By mounting your filesystem, you expose it to the current directory tree.

Pacstrap

By running pacstrap you are installing some base packages into your mounted filesystem, i.e. where your new Arch operating system will reside. pacstrap is one of those utilities provided to you by the ISO, so it should not be surprising that it doesn't exist anymore when you chroot (since you no longer need it).

Chroot + Setup

Chrooting is largely for convenience, theoretically you could probably install Linux without chrooting.² (It would be really annoying to do so.) Basically, by chrooting, you are entering the filesystem of your new OS, exposing its commands to you. For instance, the essential commands provided to you by pacstrap.

You should know that when you chroot, you are essentially running inside the OS that you installed on your desktop. Beforehand, you are running inside the live boot environment, which is an OS that you installed on your USB stick in order to install Arch Linux to your desktop. So you will not be able to get access to the live-environment tools once you arch-chroot (which is typically why you do that last; you are supposed to arch-chroot once you no longer need the live environment).

This is also why you can't run iwctl after chrooting or when booting into your Arch Linux install without the USB stick, because that tool wasn't installed into your OS; it only exists inside the live install environment.³

Some of the setup (locale, timezone, keyboard) is non-essential, especially if you use a QWERTY US keyboard by default. You are just setting some system preferences, it does not really matter if you do it now or later. (Best to do it now though, so you can take care of everything at once, but if you somehow find yourself screwing it up that's ok.)

Other parts of the setup are essential. For instance, the bootloader is what helps you boot Linux properly. And the hostname file is also important since it helps you connect to the internet.

After Install

Don't pay too much attention to this section until you have finished your Arch Linux install.

After the installation process you have a very minimal installation of Arch Linux. All it does it boot, you literally do not even have Vim. At best you set up wifi before you rebooted your system.

At the very least you want a graphical interface which allows you to display windows of programs. To do this, you need a window system at the bare minimum, like X11 or Wayland. You probably want X11, though Wayland is the future.

If you want a traditional desktop environment (Windows and Mac both have traditional desktop environments), you can install Gnome, KDE, XFCE, etc. I recommend you pick something lightweight or else boot times are going to start to resemble Windows machines. If you do decide to do this, you don't need to worry about X11/Wayland.

I strongly recommend you get a tiling window manager (TWM) instead though. For X11, you have i3, Xmonad, dwm, awesomewm, etc. Wayland has some window managers too, like Sway (based on i3), dwl (based on dwm), etc, but the selection size is smaller. It's hard to explain why a TWM is so good, and I certainly won't try to, google it yourself if you're interested.

You need a terminal emulator too. If you are using a desktop environment, one is typically provided for you so you can ignore this. But if you are using a tiling WM, some popular choices are Alacritty, kitty, urxvt, tmux, etc. I personally use Alacritty because it has sane defaults + I don't need tiling in Alacritty because I use a TWM.

You are also going to want to install some programs, like Vim (to edit text), Firefox/Surf/Links2 (for browsing), TeX Live/Tectonic (typesetting TeX files), etc. But this is increasingly swaying further away from the install process, so I will end this post here.

Puzzle Goals

It's midway through February and I think this challenge is going to be pretty hard because of what I've enjoyed setting lately (Killer Cages). As a refresher, the original goal was to use 4 new variants, whether classic or my own invention. I'm weakly considering changing it to 4 common variants that I haven't used before, like Quads or Sandwich. But either way you count it, I'm behind.

On January 31st I sniped myself by setting Genome: I used Renban Lines and Kropki, two easy constraints. (Hey, it's good to make the setting challenge a bit hard for myself!) And right now, I've mostly been setting Killer Cage puzzles along with a few Star Battles, so I'm not exactly treading a lot of new ground variant-wise here. But honestly, I'm not too concerned about failing, because I can just arbitrarily extend the challenge or move it to a different month or something. (Not like I've figured out a challenge for every month anyway --- not even close...)

Under the most generous interpretation, I've set puzzles with two new variants: Diagonals (classic) and X-Lines. Right now I'm trying to set a Thermo Quad¹, and Fortress, Extra Regions, and Sandwich also seem promising.

Oh, and check out Class of '23.

Puzzle Book

For the puzzle book, I will probably include some classic sudoku and killer cage sudoku even though the focus is on "homemade variants". Two reasons: one, a lot of my puzzles use cages or at least arithmetic (and obviously all sudoku puzzles use sudoku, so classic is always good), and two, it helps pad the length of the book :)

Also, I'd like to think that I'd include Highway, Stairs, Class of '23, etc, because otherwise that means I've made no progress this entire month. (Well, that wouldn't be true, I invented X-Lines two days ago so now I have more flexibility.)

Math things

Objectively speaking, I don't know much math. Most of my friends that do math have finished at least abstract algebra and real anaysis, and I have not really done either. So I plan to read a book on one of these --- probably linear algebra --- and type up a few notes. On the bright side, I think linear algebra has finally clicked for me. Turns out, that all I had to do was keep repeating to myself that matrices are linear functions from ℝ^m → ℝⁿ, that's all.

Math Advance has been in a lull lately, especially since our leadership is concentrated in the class of '23. I also need to collate Winter MAT results, hopefully that will happen soon. The Summer MAT is taking longer to start writing than usual (mid-February and still only initial drafting stages, as opposed to sending out serious drafts in December the last two years), so we need to get that going fairly quickly.

Code stuff

mapm

Yesterday I adjusted mapm to support multiple choice tests and today I released mapm 6.1.0, which has MC support. I also fixed the bug where mapm preview and mapm preview-all don't care if the problem doesn't satisfy the preview template's variable requirements.

As for how many people (or rather how few) people are using mapm, I think most people are uninterested in setting up mapm on their own computers, especially the people who are not good with computers (and math contest kids, by and large, are bad with computers). Even asking people to install a GUI would be a pretty tough ask, so I think I'm going to write a web host to import/export problems from instead.

The benefits of writing a web host besides increased adoption is that LaTeX in non-HTML GUIs is hell, and I don't really want to use Tauri for a GUI because that would be pretty cursed. I'll do the research (and maybe some of the work for) LaTeX in GUI but this will probably be after Glee is stable and the mapm web client is done.

Glee

(This is mostly hardware stuff.)

I'm participating in Hack Club's Winter Wonderland, a hardware event that lasts for 10 days. Right now I need to get my Orange Pi set up in some trivial way, so that I can move on to getting the temperature/humidity sensor, establishing a static IP address/routing through a static address that hopefully our ISP is providing us (more research needed here), and migrating some less mission-critical projects, like mathadvance.org, dennisc.net --- mostly static websites --- to the physical server.

Glee itself will obviously take more than 10 days, but I might try to get something rudimentary set up during this 10 day span. Of course, I can and do intend to continue working on it even after the event.

Crosswords

Just something miscellaneous that I wanted to add: crossword construction seems pretty interesting. Unlike Sudoku, whereI think I have a handle on what makes a good puzzle, I don't have the slightest clue about crosswords.²

For comparison, I started solving Sudokus pretty intensely for six weeks or so before I tried setting my first puzzle, and even then I had some experience prior to those six weeks.³ Whereas my crossword experience mostly consists of failing to solve them on airplanes and trying a couple these past few weeks.

Well, I think that's all for now. See you next time!

Yesterday, my last college decisions came out.

Yesterday, my school's spring break started.

It seems fair to describe this month as an interregnum, before high school final exams and AP exams begin ramping up for the last time. Before I have to make the one decision that really matters off of very little information.

A long time ago, I had the idea to share my college applications and reflect on them in their entirety, hopefully providing useful advice to future classes applying for college. But now, I don't have the energy to do this, and nor do I think my college results are good enough to justify giving advice. In fact, given my stats, I did really badly. So I will merely describe rather than prescribe.

Acceptances:

• UCLA math
• UCSD undeclared (rejected from CS)
• UCI CS with honors
• UCSB CS
• UIUC math (rejected from CS)
• UT Austin CS, Turing Scholars¹
• UWash "pre-major" with $5,400/yr scholarship (rejected from CS --- so I'm good enough to get a scholarship but not good enough to study CS...)
• CMU CS

Rejections:

• UCB (applied for math)
• Cambridge math
• All Ivies except Dartmouth and Penn
• MIT/CalTech
• Stanford
• JHU
• Harvey Mudd
• GaTech (applied CS)
• Duke

So yeah, I had about a 33% acceptance rate, and taking out safeties/CS rejections (because I'm not going to UWash to study not-CS) plummets it to 10-15%. Interestingly, I expected to get screwed for CS and decently for HYPSM, and the exact opposite happened. Surprising, given my total lack of CS achievements. (No research/significant projects --- where "significant" means sufficient reach/fame --- and only USACO Silver, before difficulty inflated.)

In some sense I've won. If I knew in August that I'd get into CMU CS, I would've been really relieved --- relieved enough not to care about the absolute slaughter I went through. (Also I'd be really confused why future me is telling me about this Turing Scholars thing so enthusiastically.)

In another sense, I've lost. I expected to get in some of HYPSM, and there are definitely people who have a lot more choice than I do when deciding where to go.

But the thing about college apps is it always feels like you're losing. As long as I don't have to wonder, what would've happened if I got into a top college? everything will be fine.

CMU is apparently really strong at AI. Unfortunately I don't have much interest in it (mostly because I despise applied math).

Maybe I'll change my mind in 4 years. Hopefully I do.

For the last 4 years, I've been convinced that I was going to study math and major in math and become this professor guy in math that does Important Research.

But why was I so convinced? Probably because I figured out I was good at math a long time ago, decided "OK and now I will like math", and then went with it. So what if I got really good at CS in the next few months and the next few years, and just decided to myself, "now I will like CS"?

I find the theoretical stuff really interesting at least, even if I don't know a lot about it. I hear it's basically just mathematicians pretending to do Not Math. At the minimum, I should enjoy that...

I'm probably going to CMU. But whichever of CMU and Austin I went to, I'll feel like I'm missing out on something. CMU's core curriculum aligns really well with my views on CS education, and it's ranked #1 for CS --- that's gotta count for something. But Austin is a tech hub and sends a ton of people to quant and will be a boatload of fun that I will, with all probability, not experience.

If only I had two lives. If only I could experience both of these universities.

If only I could study math at UCLA too. Once upon a time I was really convinced I wanted to do math at a UC. Wouldn't it still be worth doing? What fundamentally changed in the last 4 months?

I am very excited about going to CMU. But some part of me knows that I'd also be really excited to go to Austin. What if I end up sitting alone in my room for god knows how many days and asking to myself, "what if I went to Austin instead, what if I didn't have 50 hours of work every week? What if I valued my mental state more than my superficial ego?"

The scariest thing is, I won't ever find out if the grass was greener on the other side. And there's a 50% chance that it is.

I don't know a lot of CS. Scratch that, I don't know a lot of anything, including math.

I want to study ahead over the summer.

I want to work on the things that interest me. Finish up some projects before school starts, because god knows I won't have the time to do that in school.

I want to get the ball rolling before it's too late.

I'm about to graduate. Some part of me is having trouble really believing this.

I have two to three more months to make some final memories with people in high school. Aside from a couple of outliers, I probably won't ever talk to them ever again.

Near the end of WARP, I was intellectually aware that I'd never see most these people again. But emotionally it never registered.

And I really mean never, not even now. I just said, "huh OK WARP is forever over", and went back to my normal life. For a week, normal meant sitting in the Lodge saying stupid shit with my friends. Climbing rocks. Swimming in the pool. But now it's just a distant memory, something that I know happened but don't really feel anything about.

I feel the same way about high school. Next year, I'll be looking back at all four years of high school, and hell, maybe even middle school, and just think "huh I guess these things happened. And I knew these people." I don't think I'll really miss high school, even though in the moment it was really fun. Junior year already feels like an eternity away. Hell, even first semester senior year. I feel like it was just yesterday that I was in middle school geometry, showing off like the immature brat I was. Intellectually I know that all of high school happened and I should probably treasure these memories or whatever. But on an emotional level, I'm finding it hard to feel much of anything. I mean, I'm excited for college and excited for more than 50% of my classes not to be a steaming pile of fucking nonsense. But I don't feel sad at all that I'm going to be leaving high school, even though high school has been nothing short of great.

It's weird that I'll cry over a fucking imaginary pink elephant dying in a kid's movie, but not over literally leaving everyone I have ever known in my entire life.

Y'know what? I think it's probably OK.

But if you're from the area and I know you, and you're reading this, let's hang out or something.

Before it's too late.

Before it's too late. Before it's too late. Before it's too late.

Why is it, when I've had for years to do these things, I decide to wait until it's almost too late?

I should study math and CS.

I should finish my geometry book and Git hosting software.

I should say goodbye to some people.

But thinking and doing stuff is hard, and this Whispers Skyscrapers puzzle seems a lot more appealing.

For beginners: git add and commit

Git's history is built on commits. The way you tell Git what you want to commit next is through the staging area, and to add something to the staging area you just run git add. Sometimes when a file isn't being tracked by Git, git add will also make Git track the file. This is why people run git add --a and then git commit -a -m "msg" whenever they want to add a new file. (Don't do this, by the way, it's terrible practice.)

You can do git commit FILE as a shortcut instead of running git add FILE; git commit. But a lot of beginners do not get the difference between git add and git commit. If that describes you, try this for a day: never add any arguments or flags after git commit, and you'll get the idea really quick.

"Flags" includes -m, and if you're using said flag your commit messages are probably not detailed enough. When editing your commit message, you see what files are being committed. That way, you will never accidentally commit a file you didn't mean to, because you will have ample time to see exactly what you are committing.

Global .gitignore

If you are a soydev on MacOS or Windows (Mac users are by far the worst offenders), you probably have committed .DS_Store or desktop.ini once in your lives. First off, that is a sign your workflow is stupid and you don't know what you're doing: you should never git add --a; git commit -a unless you absolutely know what is being committed.¹ But even then, it's good to have some safety against ever committing garbage like .DS_Store: you will never have a legitimate file named .DS_Store, so it's perfectly fine to just gitignore it.

But instead of adding it to the .gitignore of each project, you can instead add it to your personal global .gitignore. This means that, across all local repositories, you will never be able to add any files in your personal gitignore to the staging area.

You can read this GitHub Gist, it's quite comprehensive. If you follow the links in the comments you will also find the default locations for the git config/ignore files. I will tell you explicitly that for Linux, you want to create the files ~/.config/git/ignore and ~/.config/git/config. If the directory ~/.config/git doesn't exist (it probably won't), make that too.

What I recommend is you make a git repository in your dotfiles, and then clone it in ~/.config. (The repository should then be ~/.config/git.) This is not so feasible in GitHub because you don't have nesting (groups and subgroups in GitLab), but I still think you should make separate repositories: your git config files have nothing to do with your vimrc or whatever else you have.

Here is where my advice becomes more personal taste. I think your project .gitignore should only contain things that everyone will see. That's why my TeX projects don't have a .gitignore; my build files all go inside a build directory, because I use my own custom script called dlatexmk. And OS-specific files like .DS_Store definitely should not be in a .gitignore.²³

pull = fetch + merge/rebase

A lot of beginners don't know what git pull does and just assume it magically updates their repository.

Even though your local branch might be named master and push to a remote branch named master, your local and remote branches are two totally separate things. You just synchronize the two via --- you guessed it --- git push and git pull.

Just google git pull = fetch + merge to get a more complete picture, dozens of guides on the internet exist about this already.

If you forget to pull before making some changes, committing, and trying to push, you're typically left in a quite precarious situation. You can either try to merge and get a messy commit you never wanted, or (more frequently) give up, re-clone the repo, and copy your changes there. But if, say, the remote had file A changed and you changed file B, then your commits are independent and present no conflicts. In this case, wouldn't it be nice if, say, you could make it as if you were working on the latest commit from the remote?

And that's exactly what rebasing does: it applies your changes on top of the remote branch's. Thus git pull --rebase will do the job.

Staging part of a file

When I only wanted to commit part of my code I would usually manually delete it, commit my file, then undo. Turns out you don't have to do this: git add -p FILE gives you a way to do this interactively.

It will give you options [y, n, q, a, d, e, ?] for each hunk, and sometimes s as well. Most options are quite obvious, but the two I find most useful are e and s. The e option lets you edit the patch line-by-line, so you can remove lines starting with + or - to change what you stage.

rebase -i

If you haven't pushed to master (it's OK if you've pushed to feature/dev branches) and realize that a commit is stupid or you forgot to add something to a commit or whatever, you can use git rebase -i to drop, squash, reword, or reorder your changes.

If you have pushed to remote, you will need to perform git rebase -i HEAD~n where n is however many commits back you want to go. So git rebase -i HEAD~5, for instance.

At this point the remote will reject your changes, because your local branch and remote branch have divergent histories. You can do git push -f to force Git to accept your changes. Needless to say, this will rewrite history and is an awful idea if anyone relies on your branch history at all. (Which is why dev branches are good.)

Fortunately, this is not how creating things works. Let's say that Alice writes crosswords and sudoku, while Bob only writes sudoku. Even if we were to argue that Alice is a better puzzle constructor than Bob is, because Alice can do strictly more things, that doesn't mean Alice's sudoku puzzles are better than Bob's. Or even that their sudokus will cover the same ideas, mind you.

In the group quiz example, Bob is practically useless because Alice will do everything that Bob could've. Will, not can. The difference is that, in the real world, a great puzzle setter certainly could construct something as good as your puzzle --- but they didn't. Because there are so many things to create (such as puzzles) and discover (such as math theorems), it is actually very unlikely for someone else to come up with and execute on your idea, provided it is sufficiently novel.

So yeah, there are probably people who are better than you at making whatever it is you make. But the upshot is even if there are people better than you, you can still provide value that they don't. Because if both you and Alice can contribute something, then chances are, you won't care so much which one of you two is better at it.

Nor does a "big takeaway" work. If your essay is long enough to have some sort of overarching conclusion to draw everything together, you shouldn't be saving your main points for the end anyway. Otherwise you'll leave the reader scratching their head while you're refusing to tell them what your examples mean, and by the time they get to the end they'll have forgotten. People can only remember so many things at once.

End your essays with advice. It won't work for everyone who's reading, but it should be relevant for people who're facing what you're describing. Try to make it positive advice as well: describe what your readers should do instead of listing a subset of what they shouldn't.

What should your readers do after reading your essay? How should they change? In a way, this is a "big takeaway" --- with the extra restriction that it's specific and actionable, even if not immediately so.

That's the common conception. But even if there's a relationship between the homework and tests, the two factors are totally orthogonal. Or at least, what they represent is orthogonal.

The amount of homework there is in a class represents its required effort. How likely you are to pass the class after putting in this required effort is the risk of taking the class.

There is a lot of debate over how hard some classes are, like AP Statistics. But it turns out the disagreement is more narrow than its difficulty; it's based on the risk of taking the class. No one, not even the people who perform terribly in the class, feels a particular need to sink dozens of hours a week into the class. You hit a barrier really quickly and returns begin to diminish. Rather, what differs from person to person is the risk of failing the tests; some people have less innate ability than others or less experience doing math, and so their risk is higher.

What's important is that these two components, risk and required effort, don't change much for a person. Sure, there is wide variance across different people, and these traits are not totally immutable. But over the span of a year, and maybe even all four years of high school, you do not often see significant variance. Particularly because the people in charge and the students themselves ignore that these two factors are what really matter, and instead waste time with a tutor without ever decreasing the inherent risk in harder classes or the amount of effort they require. For these traits change globally; it's a matter of habits and values, not of specific knowledge.

When people ask if a class is worth taking they are really asking about the potential reward. That is, what lies at the end if you manage to succeed? Maybe you learn something, or more frequently, it looks nice on your college applications. Implicitly they are also considering the cost of failure: tanking their grade point average. And really, that's all the factors that you need to make a decision: required effort, cost of failure, risk, and potential gains.¹

Given two sets X and Y, a function if f: X → Y is just a subset of the Cartesian product X × Y such that for each x ∈ X, there is exactly one y ∈ Y such that (x, y) ∈ f.

As you expect, we write f(x) to denote said unique y.

So what happens when we attempt to define a function? Essentially, we are associating each x ∈ X with some subset of elements in Y. For example, in the function f : ℕ → ℕ where f(x) = 2x, we are associating x with {2x}. For example, we just associate 1 with 2, 2 with 4, and so on.

When we attempt to define a function f, we would like to have f associate each x with exactly one value in y.

Most of the time we write a function as f(x) = … and in this case there are no problems. If it is some formula of x, this is always fine. Perhaps it is a piecewise formula, where you first have to check whether x is even or odd. No matter. This will always be fine.

Now let me give you a very quick bad definition of a function. Suppose that we are attempting to map the positive rationals to positive integers. Suppose you say, "we associate the rational p/q (where p, q are integers) with the integer p + q." News flash: we associate 2/3 with 2 + 3 = 5 and 4/6 with 4 + 6 = 10. Since 2/3 = 4/6, we have associated multiple elements with the same rational.

Now what is the actual issue? When we define a function, we are associating a representation of an element in X with some value in Y. What we are really doing when we say "for all x ∈ X, f(x) = 2x" is that we are representing each element of X as x. The "for all x" quantifier guarantees that each element in X is accessed exactly once with the representation x, and then we associate each x with 2x.

But when we look at f(p/q) = p + q, what we are doing is associating a value to every fractional representation of a number. And we have already seen that a rational number has many fractional representations, such as 2/3 = 4/6. Of course, the way we fix this is stipulate "for all relatively prime p, q, we have f(p/q) = p + q". Now this genuinely is a function, because we have ensured that we are only associating values to one representation of each rational number.

If we are earnestly trying to define a function f, we are not going to associate a representation of an element with more than one value. Because if we do, the subset of X × Y we are describing automatically is not a function anymore.

Now, we do not need ascribe every representation of an element in X with a value in Y. Ideally, we would like to ascribe exactly one representation of an element with a value, which is why there is no need to check that functions of the form "for all x ∈ X, f(x) = …" are well-defined. There is nothing to check! It is obviously a function.

But what of functions where you genuinely have no choice but to associate a value to multiple representations of an element? In this case, I personally say we check for representation invariance of the function, or that such a function is representation invariant.

As an example, take modular arithmetic. We would like to be able to define 13 + 10 modulo 12 the same way that 1 + 10 is defined modulo 12, and it turns out that 13 + 10 ≡ 1 + 10 modulo 12. Likewise, 13 ⋅ 10 ≡ 1 ⋅ 10 modulo 12. So it is okay to ascribe a value to multiple representations when defining multiplication, because they are all consistent.

In general, what we need to check here is representation invariance. In other words, if a ≡ b and c ≡ d, we need to make sure that ac ≡ bd. (All these equivalences are modulo n.) And it is easy to check that this is indeed true, so multiplication can be properly defined in modular arithmetic.

To summarize,

• a function is a subset of X × Y,
• we associate values in Y to representations of values in X when we attempt to define a function,
• ideally for each element in X, we only ascribe values to one representation of the element,
• but sometimes we genuinely need to ascribe a value to multiple representations of the element, in which case we must check that the ascribed value is the same across all representations we access.

And the last bullet point is the only time you need to check that a function is "well-defined".

Future

We have already started constructing the second MAT, and we made a lot of progress very recently (yesterday, actually). If you're interested in testsolving, please email me at dchen@mathadvance.org.¹

Because contest production is confidential (I will not discuss any particulars of the next contest), there is not much I can say. But there are a few things I still want to note.

The score distribution was much lower than we anticipated or desired. There are a couple of things we are doing moving forward.

First off, we are making the first problem in each round more approachable. This year's P1 was perfectly fine, but P4 and P7 tripped some people up. P4 required knowledge of Ptolemy's, and P7 required knowing the formula for ϕ. I think the first three problems were perfect introductions for an audience we did not have. So next time, problem 1 will be about as hard and problems 4 and 7 will be a lot easier. The rest of the contest will also go down in difficulty, particularly problem 8.

We've reviewed a lot of geometry problems, and the verdict on many of them was "this problem is good, but does not feel like a MAT problem." Generally this is due to difficulty or subject matter (think more complicated geometric structures and theorems). But there are many easier problems as well that did not make the cut for MAT, but would feel right on place in a geometry contest. So after the 2022 MAT (which is projected to happen around New Year's), the next contest we are planning to host is a geometry contest.

The JMCs are going a lot less smoothly than the MAT. But I am also not as involved in the production, since MAT production is a lot easier and more fun. And I also should focus on getting the MAST backend up so I can put applications up, because at this rate decisions are going to come out after the AMCs.

Fortunately next time around, we're going to have a year between test production instead of half a year. Between Year 1 and Year 2 we had six months because we made the first contest at July rather than January. We have around six months this time around because the AMCs this time are on November. And we have never been utilizing all six months, because Math Advance does a lot of things other than JMCs. So perhaps JMCs will not be such a strain on us next time.

Fun

I actually added a brief note on fun to the contest report because I thought it was important enough to include. But I'll expand some more. (I didn't include this bit in the MAT contest report because it has nothing to do with the MAT. Most of my thoughts here are on established contests like the AMCs and HMMT.)

I think a lot of contests lately have not been fun to take. Out of all the problems, there were very few that I was drawn to do. And I think this is a result of the test construction processes. They are quite cold. There is so much focus placed on whether each problem is the right difficulty or the right subject. But past a certain point you are just approximating, and your opinions just guesses. So much weight is placed on these guesses, and nobody asks: does anyone actually feel like doing these problems?

Many people say the AMCs are becoming less approachable. I do not really agree with this assessment. I think the problems have been about as fun in 2021 as they were in 2016 or 2018. (Not to say they are/were fun, or that they aren't/weren't fun.) As for difficulty, I think that is just rounding error. Maybe on average the first ten problems have gotten somewhat harder. But the goal shouldn't be to get more people to show up, do ten problems, and not care about the contest after. It should be to make the first ten problems interesting enough so students want to actually go past that and try the next fifteen. And come on, we can do a lot better than we are doing now.

I think HMMT suffers from the same issue. Certainly the problems are hard, and that contributes to low accessibility. But oh my Lord, look at some of those problem statements. Half of the algebra/number theory section looks absolutely unappetizing. I don't want to read it, let alone solve it. And this is the issue that comes up when you just judge problems after you solve them. You are considering them from an omniscient perspective, not the perspective of an uninformed contestant. This is why I frequently talk about my initial impression of the problem. That way, people have a train of thought they can follow and know more about how the problem felt.

If a problem looks a little gross but the solution is very interesting, is it worth including on the contest? Answers vary, but for us it has almost always been no. We have enough problems that we can choose the ones with pretty statements and solutions. Even if this isn't the case, I think a lot of problems are not so resistant to a little bit of rephrasing.

You want to package the problem neatly. It's a bit like food that looks or smells gross but actually tastes good. Sure, it might be enjoyable in the end, but what if you could excise that initial revulsion? Contests like the HMMT, and more notoriously, BMT/SMT/CHMMC, could stand to follow this principle more often. Their problems often have great ideas and terrible packaging. And when there are prettier problems to pick from, there's little reason to do the ones that are ugly.

From a more pragmatic perspective, it doesn't matter if an ugly problem has a beautiful solution, because no one will want to do the ugly problem anyways.

We have laid the groundwork for propositional logic with the last two entries. As one might expect, propositional formulae are a uniquely readable inductive structure (this is why we bothered to define these two concepts).

I am going to assume you have a working idea of what a boolean expression (think computer science) is. For instance, if I give you the expression

(a && b) || !c

you should be able to tell me exactly which triples a, b, c satisfy (i.e. make true) this boolean expression.

The notation for computer science and formal logic differ. Instead of &&, ||, !, we use ∧, ∨, ¬. And we have another symbol: ⟹ (read as "implies"). Semantically a ⟹ b is equivalent to (¬a) ∨ b. This meaning is connected to what it does in the rest of mathematics: "b follows from a", meaning that if a is true then b better be true. But it doesn't matter what happens when a is false (since implication only concerns what happens when a is true). If you work through it, you will quickly realize that implications are satisfied if either a is false or b is true, which is precisely when (¬a) ∨ b is satisfied.

Formal treatment

Definition (Propositional Alphabet) A propositional alphabet consists of

• a set of propositional symbols P,
• the connectives ∧, ∨, ⟹,¬,
• constants ⊤, ⊥,
• and parentheses ( , ).

For our convenience we will refer to the set of propositional symbols and constants (i.e. P ∪ {⊤, ⊥}) as P^*.

Technically a propositional alphabet consists of P as well as the connectives, constants, and parentheses, but P uniquely identifies it so we will refer to the alphabet just as P.

A pedagogical note: we have defined these technical terms "propositional symbols", "connectives", etc. They are intentionally evocative, so you may be inclined to search for the differences in function between each of these sets. However, so far we have made no real distinction between these sets (besides that P can be arbitrary while the other sets are fixed). This is about to change.

Definition (Propositional Formulae) The set of propositional formulae of the alphabet P, which we will refer to as X, is the closure of the inductive structure (P^*, {∧, ∨, ⟹, ¬}), where

• ∧ is a function taking a, b ∈ X to (a ∧ b),
• ∨ is a function taking a, b ∈ X to (a ∨ b),
• ⟹ is a function taking a, b ∈ X to (a ⟹ b),
• and ¬ is a function taking a ∈ X to (¬a).

Note that the functions ∧, ∨, and ⟹ behave very similarly. Indeed, with only a syntactic structure, they are virtually indistinguishable. (We define all these symbols so that we may perform distinct operations on them related to their semantics.) For convenience we will let ⋆ be a stand-in for any of ∧, ∨, ⟹ and their associated functions ∧, ∨, ⟹. So we may instead just say "⋆ is a function taking a, b ∈ X to (a ⋆ b)".

Pedagogical note: It is important to recall that these propositional formulae are just strings consisting of the characters in the alphabet P. Note that we have bolded the logical connectives in the definition as ∧, etc are not the symbols in P. They are functions that are heavily related to the symbol ∧, etc.

Another pedagogical note: The propositional formula ((a ∧ b) ∨ (¬c)) is surely evocative of the boolean formula (a && b) || !c. But unlike the latter, the former doesn't mean anything yet. (Much later, when we introduce semantics and proof rules, we will be able to attach meaning to these formulae too in an unsurprising way.)

Final note: The parentheses matter!! They establish unique readability, which in turn gives us an unambiguous notion of semantics. (This is something worth proving yourself.)

Because propositional formulae are uniquely readable, we can define functions on X (the set of propositions) by induction. Here is an example.

Definition (Rank)

The rank r : X → ℕ of a proposition is defined inductively as

• r(p) = 0 for p ∈ P^*,
• r((a ⋆ b)) = max (r(a), r(b)) + 1 for a, b ∈ X,
• and r((¬a)) = r(a) + 1 for a ∈ X.

Intuitively, the rank of a formula is a metric for how complicated it is. More precisely, it is the smallest n such that X_n contains a, where X_n is defined as it was for the proof of the closure of an inductive structure.

Exercise. Prove that every propositional formula has an even number of parentheses. (Hint: recursively describe a function that returns the number of parentheses in a formula.)

A bigger picture

This is about the time some may start asking "what on earth is happening here, and why should I care?" (At least I asked myself these questions at this time.) The rules and system we are setting up are beginning to seem somewhat arbitrary, as if we have capriciously chosen the symbols in our propositional alphabet. This is not entirely false, so I will aim to explain at a broad level why this is OK.

One may verily ask, "why these particular symbols in our propositional language?" (For instance, ⟹ is semantically redundant since it can be expressed with ∧, ∨, ¬. Actually because of De Morgan's, we may remove ∧ as well.) It turns out that the specifics of our proof system (the propositional language and the rules for manipulating formulae which we will later introduce) are not particularly important.

What is important? We need something expressive enough to eventually describe set theory (spoiler, propositional logic is not enough: we need quantifiers). We will architect a set of "proof rules" with which we can "manipulate" certain formulae into other formulae. Introducing semantics (i.e. actually attaching truth values and meaning to our propositional formulae) will give us a notion of soundness and completeness. In very broad terms, a logical system being sound means that "if you can syntactically prove something, the semantic implication will always hold". And completeness means "if some semantic implication holds, you can syntactically prove it". Both are very non-obvious traits of a logical system, and being able to equate syntax and semantics gives us powerful tools to study propositional formulae.

Furthermore one may ask, "will we start exclusively using this logical system to prove things about this logical system?" And the answer is no. There would be problems of circularity (e.g. Godel's Incompleteness Theorems), and on a more philosophical note, there are always mathematical concepts we will have to take for granted. (Specifically, we cannot "rigorously" define set membership.) So we will have two fairly distinct notions of a proof: one is the standard English proofs that you and I are used to, and another will be the syntactical "proofs" that we will soon begin to develop. The latter happens to be a fantastic model of the former (after all, that is one of the reasons we care about formal logic), but we will be using plain English proofs to develop our formal logical system.

Of course, we are not introducing this logical system just for fun. Eventually we will reach the point where we can more easily study some aspects of plain English proofs by modeling them as syntactical proofs (e.g. soundness/completeness and Godel's Theorems). This is the main point, and we are ju building up to it.

One common semi-misconception is that smart people will teach you things. Well, yes and no. They will talk about complicated ideas they understand, and you will not know a good chunk of them in advance. But come on. A five minute conversation could not possibly transmit all of the nuances of, say, the Spectral Theorem, or the requisite background in inner-product spaces, let alone its applications with positive operators, square roots, or SVD.

But being around smart people primes you to learn these things. You might be a curious individual learning about something like group theory, and you probably would not randomly encounter the Spectral Theorem before properly learning linear algebra, or the construction of the reals via Dedekind cuts before real analysis on your own. But if someone is telling you about orthonormal bases of eigenvectors and relating it to diagonal matrices, and afterwards you take linear algebra, it is hard not to go "Oh! So that's the Spectral Theorem. I get it now!" And even if you would've encountered the Spectral Theorem by looking around on your own (which is very possible --- this is why it is good to stay curious and do undirected reading on your own), there are many other things that you would not have seen.

You probably will not directly learn a lot just by hanging around smart people. You still have to stay curious, follow up on conversations, and read on your own. But if we consider intelligence to roughly be the ability to learn and solve problems quickly, then being around smart people actually makes you smarter. That on its own is not good for much. But if you also work hard, it just might provide the boost you need to learn --- and do --- great things.

Not that I'll have all the details down: many times I will adjust the breakin so it flows more smoothly or because it subtly breaks in some way. Often I don't even determine important aspects of the puzzle like, "how many rows and columns will it have?" or "where will the breakin be?" Nevertheless, the entire idea of the breakin is in my head.

As an example, take my latest puzzle River. The main idea is that Renbans of length 4 or greater must have at least one of 46. This is about how far I was able to get with the breakin in my head:

I've mostly begun doing this out of necessity: there are a lot of situations where I don't have access to setting software (sleeping, dinner with family friends, graduation, etc) and I get bored easily, so I start thinking of puzzles in my head. But I do think that constructing breakins in your head does ensure they are good: it ensures that there is actually some idea behind the breakin, rather than just being a series of details to whittle down. Because if your "breakin" is just a bunch of uninteresting details, you won't be able to remember all that in your head.

P.S. If you spot me spaced out, it's probably because I'm constructing a puzzle in my head.

As you may have noticed, this website and everything related (puzz, Math Advance) have been down for a week or so. Here is the story, roughly, of why that happened.

When I went home for break, I bought a Raspberry Pi and set up a server back home. Then I moved all the websites and email onto there, with no clear backup plan for what I'd do if everything went bust. Not everything was fully migrated, but I decided it was good enough. Satisfied, I shut down my virtual Ubuntu server that had been running for several years and called it a day.

Well, everything went bust.

A few days after I leave for college, I realize my emails are suspiciously empty. That was the first sign that the installation was somewhat botched, and upon investigating, the problem is with the mailcow docker process. Okay, maybe mailcow just doesn't work on Raspberry Pis. So I spin up a test server, and it seems like Alpine is actually the issue. So I can't install mailcow on the Raspberry Pi. That leaves me two options:

• Set up another VPS just to handle mail,
• or mystically figure out how to install Postfix (grr I wish the documentation were better).¹

I am a busy college student. Life has many demands. So after trying the second route for a bit, I decide that getting my email working couldn't be that important. It could wait...

But then everything, and I mean everything, came crashing down.

I notice this when I check on my puzzle website. It's down. Hmm, odd. My first suspicion is that the connection in our house died, and that indeed was the culprit. I couldn't SSH into the server, and nor could my family. This posed a big problem: I stupidly left the Pi plugged in somewhere where no monitor could realistically be moved to it. Because Raspberry Pis are pieces of shit, they don't have a power switch. And no one can SSH into it. This means, effectively, that no one can see what is going on in the screen.

So my brother has to blindly type out commands that I dictate (none of which he really understands) to try and attempt to either reconnect to the internet or shut the Pi down (so it can safely be moved to a monitor). Neither works.

Exasperated, I decide to take the risk and instruct him to unplug the Raspberry Pi. I read online that it usually would not corrupt the data on the SD card. Apparently it did. The damn thing would not boot, or so I was informed. Either way, being a coast away meant that I couldn't look at it myself. So I needed to set up something in Pittsburgh.

I think about setting up an Orange Pi. But then I realize, a) our room doesn't have an Ethernet port (at least I have not been able to find it in a semester and counting), b) I don't have a monitor or a USB keyboard, and c) it would be a massive pain in the ass. So I go back to where I started: a virtual cloud server.

Lessons

• Always have direct access to your hardware.
• Don't host somewhere with shoddy wifi.
• Plan for and simulate failures. What happens if the Wi-Fi is out? What can you do?

All these seem so obvious in retrospect. It's the kind of thing that they tell everybody running a server anywhere. Yet it seems, for better or worse, that I have to learn the hard way: reliability above all else.

What now?

Only dennisc.net and puzz.dennisc.net are up as of right now. Here is what I plan to do:

• Get email working. This is the top priority.
• Get mathadvance.org back up. This isn't really too hard to do, but it's also not that important.
• Get mat.mathadvance.org back up. Past contests are on there, so this is quite important, but also trying to host a Node JS application seems quite difficult. (There are some really big dependency hell issues, mostly because Node 16.7.0 -> Node 17+ contained a breaking change that fixed SSL of all things, and I have not been able to figure them out yet.) So it's high priority, but also realistically high effort.
• Get the rest of the random stuff up, like git.dennisc.net and mast.mathadvance.org (which effectively have no content).

It's likely that I pay for this cloud server for the foreseeable future. (In the time between installing my old server and the new one, the minimum price went down from $10/month to $6/month.) But when I move into a real house, it is also quite possible that I give the whole Raspberry Pi thing another go. As annoying as system administration is, it can be rewarding to have a whole homegrown setup as opposed to outsourcing the server somewhere far away. And with more free time in the summer, it is quite likely I will have the motivation to try it again.

As an AI language model, ChatGPT is designed to generate human-like responses to written prompts. However, this does not mean that it poses a threat to humanities classes. On the contrary, people should not be writing like chatbots, and AI language models such as ChatGPT can serve as useful tools for students and educators.

One of the primary goals of humanities classes is to teach students how to express their thoughts and ideas in writing. While AI language models such as ChatGPT can generate grammatically correct and coherent sentences, they lack the nuance and creativity that humans bring to their writing. Writing is a form of expression that reflects an individual's unique perspective, experiences, and voice. Chatbots, on the other hand, are designed to mimic human language without having an individual perspective.

Furthermore, the ability to write well is a valuable skill that is highly sought after in the professional world. Employers are looking for individuals who can communicate effectively in writing and express their ideas in a clear and concise manner. Relying solely on AI language models to write on our behalf would not only hinder our ability to develop this skill but also undermine the value of human creativity and individuality.

It's important to note that AI language models such as ChatGPT are not meant to replace humans in the writing process. Instead, they can serve as useful tools to assist individuals in their writing. Just as computers cannot generate good Sudoku puzzles on their own, they can serve as an assistant to those who enjoy playing the game. Similarly, AI language models can help students and educators in generating ideas, suggesting sentence structures, and correcting grammatical errors. This, in turn, can save time and increase productivity.

Moreover, AI language models can be used to analyze and evaluate written works. By inputting essays or articles into the system, students and educators can receive feedback on the clarity, coherence, and organization of their writing. This feedback can help them improve their writing skills and develop their own unique style.

In conclusion, ChatGPT is not a threat to humanities classes. People should not be writing like chatbots anyway, as writing is a form of expression that reflects an individual's unique perspective, experiences, and voice. Instead, AI language models such as ChatGPT can serve as useful tools to assist individuals in their writing, just as computers can assist Sudoku players. By using AI language models as an assistant rather than a replacement, students and educators can improve their writing skills while maintaining the value of human creativity and individuality.

Yeah, ChatGPT kind of sucks. Its weaknesses?

• Way too verbose
• Its tone is always the same; it doesn't know how to emphasize anything
• It lacks domain-specific knowledge, particularly about soft skills (e.g. creativity, intent)¹

ChatGPT as of now is just a toy, one that happens to have a modest level of utility for smaller/less high-stakes situations. And yes, maybe I could've gotten it to write like me if I engineered a prompt to do that, but that would've literally taken longer than writing it myself.

Compare this to basically any site with a big, flashy marketing homepage. Think GitHub, GitLab, or Prisma --- they all feel the same. Sure, there are lots of differences between the sites, but those mostly lie in the details.¹ These websites do look nice and all, but in a sea of complicated web design a simple website can be a nice break from it all.

Why do all complex websites look the same? There's two reasons for that feeling. The first is that even if there are 10⁵ unique complex website designs versus 10³ simple ones, everyone is going for complex these days. The second is that people actually pay a lot of attention to complexity: it's one of the first things people notice, especially if they don't understand the nuances of web design. Think about art: if you don't really know art all that well, you probably mentally categorize paintings into three kinds: the normal ones, the super minimalist ones, and then the plain weird ones (think cubism).

Of course, there's another reason websites like mine feel so rare on the internet: because a lot of "simple websites" literally do not have any CSS styles whatsoever, so you get the ass-ugly default browser theme. Most the other people who design websites like these do it a la perfectmotherfuckingwebsite to make a point. So on personal and business websites, you either get people applying a lot of styling or none whatsoever.²

Yes, it may seem like your CSS isn't doing anything and you haven't really scratched your design itch if it has very few classes and declarations. But you know, not adding a bunch of stuff is a design decision too, and it's not one people take very often. There are very few websites with more than a page's worth of content that look like this, so whatever you make is bound to be unique. Plus, this may just be my personal taste, but I think websites with a tasteful amount of styling are the ones that look the best.

For the API, I'm going to use Rust, period. I'm probably going to keep my Tide + Postgres + Redis stack that I'm using for MAT-web. (This means no wacky filesystem shenanigans that I described in the previous post.) I also am going to try adding GraphQL, since I think it'll be useful for the frontend.

I don't know about the frontend, which indicates that I really ought to learn how to do it. Maybe SvelteKit? It seems promising. Anyway, I'm going to set out a preliminary goal of 100KB or less per page request.¹ If SvelteKit doesn't pan out, I might just server-side generate all of my pages, since

• the frontend does not have to scale
• but it has to load fast for people using it.

I will have to test on some old hardware with a throttled connection, just to make sure

So when I call this Git service "small", it doesn't mean that the codebase necessarily will be small. Heck, look at CGit --- it has 10390 lines.²

With regards to authentication, I want to make things simple AND force people to do things the right way. To authenticate with the frontend, you will login with your credentials (duh). The only way I will allow authentication with Git is through an SSH key, because that is the only way you should be authenticating in Git to begin with. Username-password based solutions are hacky, stupid, and insecure.

Redirects are now low priority: in fact, I'll probably ship them out only after I start dogfooding (i.e. host the development of Glee on my Glee instance).

CLI Tool

The Glee CLI is going to be written in Rust too, since I want to leverage clap. All the standard stuff, like making directories, creating a new repo, etc will be in it. However, the ergonomics are going to lean heavily on the filesystem model that Glee is using. This means that instead of glee-cli group create PATH, we will use glee-cli mkdir PATH. And of course, authentication will be based on SSH keys.

Of course, you will need some way of specifying a remote. There are two ways to do this: either have the user build from source and have the remote be specified compile-time (probably the worse option by far), or read the remote from a config file at runtime (probably what I will do).

A big feature that isn't in any other CLI I want to add is fscopy. Basically, if you run

glee-cli fscopy SOURCE TARGET

it should clone SOURCE from the remote to TARGET on your filesystem, using the default branches. The reason this is a big dealis because fscopy, unlike git clone, would be recursive by default.³ This is definitely possible in GitLab (which is the only other Git hosting service that this would make sense for), but because GitLab doesn't lean into the "Git hosting service as filesystem" mentality that I will, I only thought of this while making the preliminary designs for Glee (i.e. writing this post).

There are other operations that fit better when you consider the fs model. For instance, in GLab there is no way to change the permissions of a repo/group, and this is likely because it's hard to design. But for us, glee-cli chmod -r PATH --view=user --push=admin will do the job. (I intend for chmod to have an -r flag just because there are times you want to change permissions recursively and times you just want to change the default permissions for a directory. The difference between chmod and fscopy is that one is a write operation while the other is just a read operation.)

By the way, here I used the name glee-cli because it's unambiguous what it does; it acts as the CLI for glee. But to make typing it easier, I might change it to glee or gcli when I actually write the CLI.

Patches

Just like in Sourcehut, patches are expected to be sent by email. If you're hosting software that's big enough to need a coordinated way to send patches, it's probably a better idea to use something like Sourcehut.

For purposes totally unrelated to Glee, I've wanted to get a mailing list service for a while. Of course, reinventing the wheel here would be silly, so I will probably try lists.sr.ht (obviously it will be self-hosted). If nothing I find works well for me, I will write a mailing list myself, but I really would rather not resort to that.

Licensing

AGPL-v3, duh. Yes, my previous stuff either uses MIT or BSD-3. But that's because its potential use cases were so limited that there really was no risk of exploitation, commercial or not. But since "Git hosting service" is quite broad, even though few people will use this specific design, any improvements better find their way back to me. There is no legitimate reason for a Git hosting service not to be open source.

This will likely be my last big project

Given what's currently on my plate and the time I have before college, this will probably be the last big project I have time to do on my own, at least as a hobbyist. I don't even know if I want to work as a programmer as my day-job. Here's what's coming up before:

• Whatever maintenance of mapm is necessarily
• Maybe mapm-gui
• mat-web
• Maybe a MAST-web rewrite
• My geometry book (probably will happen in parallel --- not a software engineering project, but I will likely focus on that first)

I don't really want to do any of these besides mat-web and the geometry book, so I will probably get started on Glee soon. Making a new Git service will likely be a significant undertaking, even if I'm not offering it as a SaaS. I'm fine with this becoming my flagship software product for the next five years or so.

Most people in AoPS use AoPS' olympiad.asy module to draw their diagrams. It provides functionality such as circumcenter(A,B,C), rightangle(A,B,C), and so on --- it provides methods for common olympiad geometry constructions.

But olympiad.asy is not a standard Asymptote module, it is an external library. And there already is a module in the standard library that does what olympiad.asy does, and arguably better. It is geometry.asy.

As an analogy, consider using an external library in your programming language of choice for string replacement rather than using the standard library. It doesn't matter if the external library is written well, because the standard library exists and it works well too. Why rely on an external dependency when you don't need to?

By the same reasoning, you should avoid using olympiad.asy. Its functionality can be completely replicated by the standard geometry.asy module, which you have installed with Asymptote whether you know it or not.

Drawing a right triangle in geometry.asy

Unfortunately geometry.asy's perpendicular function is not the most intuitive. The function (simplified to only the arguments we care about) looks like this:

perpendicular(pair vertex, pair direction, pair offset)

The vertex is where the right angle will be drawn. The offset determines one of the non-base vertices of the "right triangle" (only the direction is relevant, not the magnitude), and the direction determines the angle bisector relative to the line formed by vertex and vertex + offset.

A really simple example:

import geometry;

perpendicular((0,0), NE, (1,0));

It will generate the right angle above (without any of the extra labels, lines, dots, etc).

When you're measuring the height of a plant, the plant could care less. At least, until you start selecting for or against height. But testing students will affect what you are measuring, because people are sentient. So no test can be a truly accurate measurement of achievement, because the very existence of said test distorts whatever measure of achievement you are using. It's like Heisenberg's uncertainty principle; by observing it, it changes.¹

If the existence of a test affects how you learn, we should demand that it has a positive effect on learning. Especially if you still have things to learn down the road. And there is always something more to learn.

Preparing for the tests we take is also incredibly time consuming. Let's say that in junior year, you spend two hours a week preparing for the SAT, from August to March. That comes out to be about 60 hours. If a kid wasted that much time on video games his parents would be mad. But when it comes down to it video games teach you more than the SAT.

Sixty hours is a lot of time. It was enough for me to learn Linux and become fairly proficient in it. I took less time to write and ship a functional first version of mapm --- library, command-line interface, and documentation. Not every chunk of sixty hours will be this valuable. But multiply a lot of tests to study for and the entire population of the United States, and you get a sense of how much productivity has been lost.

It's true that whatever high school students do mostly amounts to chump change. But just as we give kids allowances to promote responsible spending, we also have to promote responsible learning.

Some tests have been able to do this. I think any major STEM contest (especially the AMCs) does this well. You can't just "study to the test", or more accurately, the only effective way to study is to genuinely learn. For the AMCs, you don't agonize over the specific format or types of questions,² you just work on problem-solving and learning new things.

In all fairness you can't reliably "hack" the SATs anyway. Evidence of a significant benefit from prep programs is limited at best. But the problem is, if the SAT is an IQ test that also determines your future, two things are true: you have extremely high incentive to study for the SATs, and you can't get better at the SATs.

So what should a good test look like? Pretty much the opposite of the SATs. It should not be too high-stakes, because even an initially positive distortive effect taken to an extreme becomes harmful, it should not be something worth specifically studying for, because it should measure things that align with skills people find worth developing, and it should also be possible (though not guaranteed!) to improve significantly with hard work. And perhaps most importantly, it should be fun.

Most of the time when parents email me I don't really know how to respond. And it's because I've only had the perspective of a student, which means that any answer I give will be tailored for students. Since students won't be reading emails sent to their parents, it honestly feels like a waste of time.

Let me put it this way. I receive about a dozen emails a month asking for math instruction ("Can I apply for MAST" included). I'd say about 2/3 of it comes from students and the other 1/3 come from parents. So far every student has gotten a helpful answer from me, and as far as I can tell, no parent has gained anything from emailing me.

So if you are a parent, what should you do? I think the answer differs depending on how old your child is and who you're asking, but here are a couple of pointers.

• If you are forcing your kid to do math contests (especially if it is for college), just stop. I have a lot of harsh things that I want to say to these people, but I will leave such comments to the imagination.
• If your kid has just started math contests and might not be as willing to cold contact people, then cc your child in the email. The reason parent emails are a waste of my time is because the kids don't read them.
• If your kid is really young, i.e. too young to send an email themselves, unless they are an AIME qualifier I don't think I'll be of much help anyway. If they are an AIME qualifier, look at MAST.
• If your kid is old enough to send an email, then please have them do it themselves. Typically I can glean a lot more information from students than I can from parents.

I wrote this essay not to discourage people from emailing me --- there are certainly many useful emails parents can send me! But I do hope that my future emails become more productive. Every email I've received from parents are of the form "my kid is in Xth grade and wants to prepare for contests A,B,C". Really, the only difference X makes is whether the kid is old enough to send an email on their own, and the contests they want to prepare for invariably are the AMC 10 and maybe AIME or MATHCOUNTS. It's very hard for me to give any guidance, or even to tell whether my classes would be appropriate, with such little information.

So in summary: have students email me if possible, cc them otherwise, make sure I have a direct avenue to communicate with them, and give me specifics. If you write and send your email in a way that makes it easy for me to help, then there is no reason for me not to.

Sudoku

Sudoku puzzles have been computer-generateable for a long time now, so the role of computers in creating puzzles is pretty well-established. And what setters have found is that computers are pretty good assistants¹, but they are terrible at independently setting puzzles.

That's because computers have no intent. They can't actually design a break-in or a finish or anything in between. They just throw shit at the wall until they set something that technically works, and these kinds of puzzles are terrible.

You can easily tell the difference between a computer-generated puzzle and a handcrafted one. Computer in no way have replaced puzzle constructors.

Writing

So why is ChatGPT such an existential threat to the humanities? I freely admit that there are substantial differences: language is much easier to codify than logic, and you do have some control over the output with prompts (in other words, you can inject some intention).

But at its core, the role of computers in the two are the same. If you're setting puzzles like a computer, then stop setting puzzles like a computer. And if you're writing like a chatbot, then stop writing like a chatbot.

I predict that a lot of English-adjacent humanities, particularly literary criticism, will be existentially threatened, because what they do is just putting nonsense words down on paper. And people have already managed to do this before ChatGPT.

Ultimately, I think ChatGPT might pose a threat to some humanities classes, but I regard this as a good thing. Because if a class is literally making you write like a robot and no one can tell the difference, maybe you're better off actually writing it with a robot.

How blessed I was. As a college student now, let me tell you: there isn't a single class I haven't fallen asleep in. And this isn't because the professors are boring.¹ All of my professors during the spring of my freshman year were downright fantastic, and yet I could not keep the lights on in my head.

And I didn't even find CMU unreasonably hard. I wasn't staying up many nights to finish assignments. I was staying up some nights for really stupid reasons, like destroying some scrubs in Smash. But even on the nights where I would sleep at a reasonable time, I'd just be perpetually exhausted.

In fact, this has been a concerning trend in my life. In elementary school I would never want to take a nap because I just had so much energy. High school, I'd wake up tired some days, but I'd be more or less at full power whenever I needed to. This year, I don't think I've been fully sharp for more than 30 days. This is terrifying. Sure, after this year I can tell you all about the universal property of the free group. But despite this, I now feel like a useless moron most of the time because my brain is too exhausted to do anything beyond the bare minimum.

Back in the day, I'd take plenty of long walks with no distractions and think about something meaningful. Or just sit down, breathing in the fresh air, with the only thing on my mind being "how do I make this puzzle work?" Now in college (especially during Tuesdays last semester when I had eight hours of classes) I just shuffle between classes, wishing for the day to end so I can just go and play Smash with my friends, goddamnit.

Despite the bad parts of high school (and college is by far an improvement), at least I spent most of my waking hours fairly sharp. Yes, every day of high school was 8 hours of classes, which was absurdly exhausting too, but I was just bored. My brain wasn't completely fried, and when I got back from track practice, more often than not I'd be itching to work on something.

Now I am at an internship, and it is so exhausting that I don't think I spend a single day being sentient for more than two hours. And many days the amount of time I spend being sentient is a big fat zero.

At least in my internship, I have two saving graces: it is a startup, so the internship program is much less bullshit² than it would be at big tech,³ and I am going to be back in school at the end of the summer, which helps to stave off a lot of identity crises.

It is no coincidence that in my senior year of high school, despite college apps, I was able to publish nearly fifty puzzles. After college started, I have only published five. A similar trend has followed in my writing, though thankfully to a much less alarming extent.⁴ And it isn't because school takes more time --- if anything it is the opposite. And I actually enjoyed most of my homework, rather than wanting to skewer my brain with a screwdriver while doing it like I did in high school. So the rigors of college, however high they might be, do not entirely explain this change.

I can only imagine that --- at least in the average case --- things will not get much better in the typical software engineering job. If left unchecked, my life is heading towards a trajectory where I become more lifeless as time goes on, until one day a Dementor comes and eats my soul wholesale.

Dreams

In the months that follow you bend to the work, because it feels like hope, like grace – and because you know in your lying cheater's heart that sometimes a start is all we ever get.

Of course, I have no such plans to ossify. But it has been important for me to note that I have been more and more exhausted as time has gone on, and that this trend does not seem likely to reverse itself on its own.

In some sense I have been living my life by the greedy algorithm. I skipped the introductory math courses because it would make my academic experience more interesting. I took on a ridiculous courseload because each individual class seemed interesting to me. And the hard part is, many of these were good decisions, at least when taken individually. But putting it all together, I have spent my life becoming unnecessarily good at math and computer science. Do I really envision doing this for the rest of my life? Or will the greedy algorithm just lead me to a local maximum, stuck in the middle of a valley and surrounded by alternative possibilities far more worth living?

Potentially math. There's a real case made for being a professor, though from what I can see the road there is quite painful. But I couldn't envision programming becoming anything more than a hobby: if I'm going to program something, I'm going to take the time to do it well, but I cannot be assed to spend time writing shitty code I don't give two shits about for eight hours a day, five days a week, forty weeks a year for the rest of my life. In the long term, this rules out most big tech. Because if I have to write another line of React or import magic from Python again, I might actually go insane.

But in the ideal world, the world where I have a trillion dollars and the skies in Pittsburgh are clear every day, I'd want to read, write, and maybe even draw stories. I can't say for certain, but gun to my head, that would be a more appealing future than doing math --- especially if there comes a point when math literally stops making sense for me. At the very least, it is about equally as important to me as math. Yet I have spent half of my waking hours in college doing math, and yet not a single story has been written in the past three years (let alone been published anywhere, including this website).

Why such the disparity? Because I'm much better at math than writing stories, and people are much more likely to pay me exorbitant sums to be good at math than to write. That's the problem with following the greedy algorithm: at any point in time, specializing even further in the thing I am good at is the only move that makes sense.

But in the grand scheme of things, I am terrible at writing stories and just slightly less terrible at math. There is still time to decide to never listen to another word of math again, learn enough CS to graduate, get a part-time software engineering job and spend the rest of my life in a Starbucks with some pen and paper. Ditto for a thousand other things I want to try but am too scared to take seriously.

Yet I am still in a tech internship building badass robots⁵ and still enrolled for a graduate class where I will learn how to show that CH is independent of ZFC. It's not for practical reasons: there is no universe where model theory makes me rich or even employed. But even with the many other things I want to do in my life, model theory is goddamn cool. Math is cool. There is no way I'm ever completely giving up a subject where you can look at a quintic polynomial and conclude it is unsolvable, and then turn around and prove the Fundamental Theorem of Algebra using the same tools (Galois Theory). Life would be a lot less complicated if math was just this terrible, disgusting subject, but it just is not.

Even computer science is cool. Functional paradigms are cool, Rust is cool, systems and comparch are cool (but massive pains in the ass). And I really do want to learn more about each of these subjects, even though going to college for computer science is the reason I'm so exhausted.

But the simple arithmetic of time means if I want to spend time learning how to write, draw, and push around comically large weights in the gym, something has to give. It's not going to be sleep, I'm already barely functioning as it is. It's not going to be socialization or video games because that's the only thing that unfries my brain. I do need to read less manga, but that won't be enough. By sheer process of elimination... it's got to be math or computer science.⁶ As unfortunate as it is, I really cannot justify learning category theory or the Curry-Howard Isomorphism if it comes at the expense of writing the things I want to write. If it comes at the expense of not taking myself seriously in any domain besides mathematics.

Have you considered not spending three hours a day solving logic puzzles---

Hahahahaa that's a funny one.

Listen, I'm fully aware someone's going to say "you're just a college sophomore. Why are you thinking so hard?" You're right. I haven't even truly experienced the life of the working man. But at some point in my life, I will be wondering what went wrong and start digging through the past like a complicated stacktrace. And the stacktrace certainly will not have started now, this summer, or even this year. It may have started when my stupid little brain decided to prioritize getting better at multiplying over anything else. But I can't do anything about that now, can I?

What I can do is at least think about how I'm going to steer off this course of being constantly exhausted and only giving half-caring about my work. I would rather try and adjust my life right now as the naive little idiot I am, instead of waiting until I am a jaded old idiot wondering why I want to drive a skewer through my eyeballs every time I see someone commit .DS_Store to a goddamn Git repo. I may not have lived for very long, but it doesn't take a genius with significant life experience to figure out that churning out CRUD APIs with a lifetime of precisely 0 seconds on prod is not the ideal life.

And unfortunately, though the things I am learning are very cool, the habit of "focusing on the right things" that studying computer science is building is very not cool. One day, it will feel deeply wrong to do computer science, and that is when it will make sense for me to immerse myself in computer science. But for now, I need to decouple it from my increasing risk aversion and the future where I sit in front of a screen and write ultimately inconsequential code for eight hours a day for the rest of my life. Even if that inconsequential code makes me a lot of money.

Puzzles

The silly puzzle man strikes again

This post is sectioned into three sections. The first is on my concerns about where my life is headed. Then where I want my life to head. Now I will pat myself on the back for not completely blowing it. If you have any understanding of me or this blog --- an understanding that takes all of eight seconds to achieve --- you know that I spend half my waking hours thinking about logic puzzles. Well, I am glad to report that nothing can stand between me and my puzzles, and I'm goddamn proud of it.

That's not to say my puzzle hobby came out unscathed. A year ago I had the stamina to write Alphabet Soup and my brain was fed puzzles zany enough to construct my Fillomino puzzles. I like to think the last few puzzles I've published are really good too, but you don't have to be very observant to realize I've been exclusively using Killer Cages and Arrows for the past half year. I no longer have the chutzpah to cook up a new variant that'll take an entire week to resolve, and I think both me and the world are worse off for it. Because if I don't wrap up a puzzle neatly, I get to wake up to a million other responsibilities that need to be taken care of before I finish the puzzle. The takeaway is that I need to make time for myself where things just stop piling onto me, or more realistically, where I at least get to pretend they don't exist for half a week.

I'll admit it. Right now I'm completely out of ideas. Nothing I've really solved recently has inspired me, which is no fault of those incredible puzzles. The last puzzle I posted was the result of a stupid idea I had while lifting weights, and I just got really damn lucky that it worked out somehow. But you know, I just set a puzzle less than a week ago. I think it makes sense to be a little optimistic right now, even if I have no clue how I'm going to set the next one.

But what do puzzles have to do with writing stories---

Everything. I've written stories before, though looking back I'd like to beat the life out of each one of them with a fireplace poker. And I write puzzles, of course, that's at least an eighth of my personality. And even though one seems like a purely logical endeavor, while the other is emotional, I believe they speak to the same part of the soul. The part that hopes the ideas you want to tell fit together, the part that screams in relief when it all just works. The part that reaches for an easy conclusion, and the part that says Wait. You can do better. There is a good way to finish this puzzle, to finish this story, and it's somewhere in you, so you better start looking goddamnit.

So maybe I'm not completely screwed. I believe that the part of my soul that writes puzzles can be rekindled to help me write stories again, and to nurture that part of my soul I will proceed to spend the next 3 hours doing nothing but puzzles---

Takeaways

There are none.

I dunno, life is really complicated. If I knew what I was doing, if I could coherently tell you what I wanted to do in the future, it wouldn't have taken me three thousand words to do it.

There is a story I want to write that is still fresh in my head. Only time will tell if it will be good, and above all, whether I have the courage and stamina to go through with it. I want to end this on a hopeful note, so I'll say this: please look forward to it sometime in the future.

This is the kind of thing good writers want to avoid. A good writer can know that a sentence feels wrong, not because of some specific grammatical rule, but because it just seems nasty. Where other writers might think, "I have expressed my idea, this is fine", a good writer will only be satisfied with "I have expressed my idea and it sounds right."

A good writer also doesn't follow arbitrary rules just because some authority figure told them to. They have a good instinct for what conventions they should naturally adopt. For instance, it makes perfect sense for me to use the personal pronoun when signposting. When English teachers tell you not to use the word "we", they're wrong: even English papers use "we". That's because it sounds right.

If you want to write well, develop good hygiene. Learn to be icked out when phrases or sentences are awkward.¹ Be aware of how your sentences sound. And above all, when your writing starts getting nasty, clean it up. Make it a compulsion. If you can't tolerate bad writing, you won't do it. It doesn't get any simpler than that.

It seems counterintuitive, but writing about things you don't feel passionately about can actually make you better at writing --- and it might turn out to be better writing to boot. It's the difference between acknowledging something as true and ardently believing that it must be. With the former, you are just finding the dots and connecting them, and with the latter, you can get easily blindsided because you are putting yourself out there.

Not every essay on something you're passionate about will turn out bad. But not every good essay has to be on something you're passionate about. The reflective side is just as powerful as the passionate, and it's okay to sometimes be a little disconnected from your words. That's the power of writing: you can say things you never even would've thought about otherwise.

Usually I begin with a surprise. Not the sort of surprise that contradicts the reader's intuition, but rather a surprise that confirms and clarifies it. It's something that I expect the audience to know on some level but the general public does not. It's almost like letting them in on a secret: you've established a common ground with your audience, and nobody else knows about it.

Hooks get people interested when it tells them your writing is worth their time. And a surprise is a sure sign that what you have to say is worth their while. It means you've told them something that they don't know, and more importantly, surprise means it's interesting. But that's not why I start with surprises. It's not merely cold calculation that shapes my essays. In fact, it mostly isn't cold calculation.

It's really hard to bullshit your way through a piece of writing. Imagine what your favorite book would look like if the author didn't care. It'd be really easy to tell. Maybe not on its own, but the difference is night and day if you compare it with something the author wanted to say. That's what shapes my writing. What do I think is worth saying, and how do I say it?

Typically I write things in the order I think of them, at least roughly. It seems to work in the middle, for the most part. Maybe in the end I have a conclusion whose points I didn't clarify. Sometimes this means abandoning my old draft and writing the essay I really meant to. Other times I figure out there are a couple more points to add.¹ But the beginning is always the scariest, because I have no clue where I want to go.

When I start writing it's always because I have more ideas than I can remember. Going in, it seems that I have enough to write about. But my ideas start out with a lot of breadth and not a lot of depth. In fact, my private writing notes look a lot like this too. I have a long list of things I want to write about and most of them are half-formed ideas. This may be the threshold for being a good writer: having depth be a larger focus than breadth.²

I think depth is a lot more interesting to read than breadth. It certainly is more fun to write with depth. So I do a depth-first search when it comes to figuring out what I want to write about. But I don't pick my first node at random. I look for what'll most likely lead me down a rabbithole. I look for surprises.

What a smart audience wants is not a random factoid, or clickbait, or even a specific topic. They just want to see that there's something to explore, that the rabbithole doesn't end abruptly. This sounds remarkably similar to what makes something worth writing. Indeed, what the reader wants and what the writer wants converges the smarter the audience gets. So you just need to convince the reader your essay was interesting enough to be worth writing.

I thought it was interesting that the beginning of an essay was so hard to write. But why is it interesting? Because it's unusual. It's not the same kind of hard to start a conversation, write a report, or send an email, because at least there you have an idea what you want to do. For an essay you have to convince yourself it's worth writing, because nothing will happen to you if you don't write it. And you have to convince the reader it's worth reading, because nothing will happen to them if they don't read it.

The hook is taking what made you start writing and putting it into words. When you're finished that reason might have changed. But that's fine. An essay is supposed to be a train of thought anyways, not merely a list of points supporting a predetermined conclusion.

Here is the naive answer. Suppose P(M) and P(N) are the performances of models M and N, respectively. (Maybe P(M) is the amount of money that M makes you.) Then the value of model N is P(N) − P(M) if P(N) > P(M), and otherwise N is worthless to you. Basically, the value of N is how much better N does than M.

But this is very much the wrong answer. The value of N is based on the best model you can get by combining M and N. Let me illustrate this for you with a concrete example.

Suppose I have a model M to predict yes/no events. It is correct only 2/3 of the time. Suppose you and your friend have two models N₁ and N₂ which do the same. Individually, they are no better than guessing. However, the models have an interesting relation: The majority vote of the models is always correct. Simply put, if at least two of the models think the answer is "yes", then it is "yes". Likewise for "no".

1. correct, correct, incorrect
2. correct, incorrect, correct
3. incorrect, correct, correct

Do you see how by acquiring N₁ and N₂, models which individually do just as well as M, combined with M do much better?

Now this is a bit of a contrived example. But the point remains the same. This is quite literally a mathematical example of the power of teamwork.

What's the upshot? A lot of people implicitly hold the naive view that because they are worse than someone else in their group or team, they hold no value in it.¹ Two years ago, I wrote about how I don't believe this is the case for creative fields. Two years later, I am glad to report that this is this is the case even for fields with objectively right answers. That group test? Unless Alice is literally perfect, the A-student Alice and B-student Bob do better than Alice could alone, even though Alice gets better scores than Bob every test.

On a related note, there are some people who believe that generative AI will render humans completely useless. Again, this is not the case. Even if a moron with generative AI can produce code faster than a skilled software engineer working alone,² that skilled software engineer will be leagues more useful with generative AI.

Background

Several people were fired for being gay or transgender. Both the plaintiffs (the people fired) nor the defendants (the employers who had fired them) agreed that these firings were discrimination on the basis of sexual orientation. Title VII of the 1964 Civil Rights Act states:

It shall be an unlawful employment practice for an employer ... to fail or refuse to hire or to discharge any individual, or otherwise to discriminate against any individual with respect to his compensation, terms, conditions, or privileges of employment, because of such individual's race, color, religion, sex, or national origin

Notice that sexual orientation and gender expression do not appear there (despite many efforts to amend Title VII to include it). Both parties agreed that Title VII applied to the employers. The question just is: if sexual orientation is not explicitly protected in Title VII, then is it protected all the same?

Majority opinion, by Neil Gorsuch

The 6-3 majority (liberals + Gorsuch and Roberts; at this point Ginsburg was still on the court) ruled that Title VII prohibits discrimination on the basis of sexual orientation and gender expression, because you cannot discriminate on the basis of sexual orientation/gender expression without discriminating on the basis of sex. Rather than being a case of constitutional law, this is a case of statutory interpretation: SCOTUS ruled based on the text they had and not any constitutional considerations.

Gorsuch starts with a brief historical note on the Civil Rights Act of 1964: no one intended for Title VII to protect sexual orientation or gender expression. But,

Those who adopted the Civil Rights Act might not have anticipated their work would lead to this particular result. Likely, they weren't thinking about many of the Act's consequences that have become apparent over the years, including its prohibition against discrimination on the basis of motherhood or its ban on the sexual harassment of male employees. But the limits of the drafters' imagination supply no reason to ignore the law's demands. When the express terms of a statute give us one answer and extratextual considerations suggest another, it’s no contest. Only the written word is the law, and all persons are entitled to its benefit.

Then he establishes the background of the case. Some gay and transgender people were fired after the companies discovered they were gay/transgender. District courts had different rulings. The background is not incredibly complicated or central, because everyone agrees what the facts are: the employees were fired for being gay/transgender. The entire case hinges on whether Title VII protects against this discrimination.

One major detail about the established facts:

The only statutorily protected characteristic at issue in today’s cases is “sex”—and that is also the primary term in Title VII whose meaning the parties dispute. Appealing to roughly contemporaneous dictionaries, the employers say that, as used here, the term “sex” in 1964 referred to “status as either male or female [as] determined by reproductive biology.” The employees counter by submitting that, even in 1964, the term bore a broader scope, capturing more than anatomy and reaching at least some norms concerning gen- der identity and sexual orientation. But because nothing in our approach to these cases turns on the outcome of the par- ties’ debate, and because the employees concede the point for argument’s sake, we proceed on the assumption that “sex” signified what the employers suggest, referring only to biological distinctions between male and female.

Now for the meat.

But-for causation

The word "because" implies but-for causation. If X not being true would've prevented Y from happening, then X is a but-for cause of Y. In Gorsuch's words,

A but-for test directs us to change one thing at a time and see if the outcome changes. If it does, we have found a but-for cause.

Then he follows up with an example:

So, for example, if a car accident occurred both because the defendant ran a red light and because the plaintiff failed to signal his turn at the intersection, we might call each a but-for cause of the collision.

Notably, it is important that Congress chose to use "because of", and not "solely because of" or any stricter standard.

Discrimination

Gorsuch spends a while rigorously defining discrimination, but for our purposes the major point is this:

Discrimination is illegal on an individual basis, not just a group basis. This is important because

The consequences of the law’s focus on individuals rather than groups are anything but academic. Suppose an employer fires a woman for refusing his sexual advances. It's no defense for the employer to note that, while he treated that individual woman worse than he would have treated a man, he gives preferential treatment to female employees overall. The employer is liable for treating this woman worse in part because of her sex.

Therefore, firing both lesbians and gays doesn't mean the homophobia "balances itself out", provided we establish that firing people for being attracted to the same sex is sex discrimination. (We establish that next.)

Why sexual orientation/gender identity discrimination is sex discrimination

Now here is the main point: take some man who is attracted to men but is otherwise inoffensive to the employer. Now change his gender, and we have a woman attracted to men. The employer has no reason to fire this hypothetical woman now, and so the man's sex is a but-for cause. This is, plain and simple, what "discriminating because of sex" means.

Similarly, take someone born as a man but who presents as a woman who is fired by an employer because of this. Someone born as a woman who presents at a woman would not have been fired. Sex is, again, a but-for cause.

Gorsuch then buttresses this with SCOTUS precedent: the "simple test", i.e. this but-for analysis, has held water for a number of unexpected applications of Title VII. This is but the latest.

Common refrains

Gorsuch addresses two common refrains in his opinion. I'll let his own words do the trick here.

Refrain 1: Sexual discrimination must be willful. How can it be willful if the employer doesn't even know the sex of an applicant?

Sup- pose an employer asked homosexual or transgender appli- cants to tick a box on its application form. The employer then had someone else redact any information that could be used to discern sex. The resulting applications would dis- close which individuals are homosexual or transgender without revealing whether they also happen to be men or women.

Doesn’t that possibility indicate that the em- ployer’s discrimination against homosexual or transgender persons cannot be sex discrimination?

No, it doesn’t. Even in this example, the individual ap- plicant’s sex still weighs as a factor in the employer’s deci- sion. Change the hypothetical ever so slightly and its flaws become apparent. Suppose an employer’s application form offered a single box to check if the applicant is either black or Catholic. If the employer refuses to hire anyone who checks that box, would we conclude the employer has com- plied with Title VII, so long as it studiously avoids learning any particular applicant’s race or religion? Of course not: By intentionally setting out a rule that makes hiring turn on race or religion, the employer violates the law, whatever he might know or not know about individual applicants.

Refrain 2: In the but-for analysis, you changed a gay man into a straight woman! Two things changed at once! Now you can't prove it was sex discrimination rather than sexual orientation discrimination.

When we apply the simple test to Mr. Bostock—asking whether Mr. Bostock, a man attracted to other men, would have been fired had he been a woman— we don’t just change his sex. Along the way, we change his sexual orientation too (from homosexual to heterosexual). If the aim is to isolate whether a plaintiff’s sex caused the dismissal, the employers stress, we must hold sexual orien- tation constant—meaning we need to change both his sex and the sex to which he is attracted. So for Mr. Bostock, the question should be whether he would’ve been fired if he were a woman attracted to women. And because his em- ployer would have been as quick to fire a lesbian as it was a gay man, the employers conclude, no Title VII violation has occurred.

In response, Gorsuch makes the following analogy:

Consider an employer eager to revive the workplace gender roles of the 1950s. He enforces a policy that he will hire only men as mechanics and only women as secretaries. When a quali- field woman applies for a mechanic position and is denied, the “simple test” immediately spots the discrimination: A qualified man would have been given the job, so sex was a but-for cause of the employer’s refusal to hire. But like the employers before us today, this employer would say not so fast. By comparing the woman who applied to be a me- chanic to a man who applied to be a mechanic, we’ve quietly changed two things: the applicant’s sex and her trait of fail- ing to conform to 1950s gender roles. The “simple test” thus overlooks that it is really the applicant’s bucking of 1950s gender roles, not her sex, doing the work. So we need to hold that second trait constant: Instead of comparing the disappointed female applicant to a man who applied for the same position, the employer would say, we should compare her to a man who applied to be a secretary. And because that jobseeker would be refused too, this must not be sex discrimination.

And he basically concludes with, "the conclusion in this hypothetical is ridiculous, the simple test has worked consistently for every other instance of sex discrimination where sex was a tangential cause, there is no reason for it not to hold here."

Conclusion

I've left out a lot of details in order to summarize the majority opinion, and you should read the full text on your own time. This is a remarkably simple case, so the majority opinion stands at a slim (by SCOTUS standards) 37 pages. This section has mostly been summary instead of analysis because the majority opinion is just correct. There's nothing left out, and there's nothing wrong. The dissents, on the other hand...

Alito's dissent

Joined by Thomas.

The dissent essentially goes like this: "No one expected Title VII to cover sexual orientation and gender identity so it doesn't". Never mind that interracial marriage wasn't intended to be covered by the 14th Amendment either. Apparently legislative intent overrides the clear plaintext of the statute. Purposivism as a legal framework is pretty sketchy anyway, so if that's the best Alito can muster, the dissent holds little water.

Alito appeals to originalism:

our duty is to interpret statutory terms to "mean what they conveyed to reasonable people at the time they were written."

Yet any reasonable person would conclude that but-for causation is enough, more than enough, to satisfy "because". It does not matter whether these but-for causes were anticipated by the legislature or not. (Gorsuch's majority opinion notes many examples such as motherhood where it not being expected to be covered by Title VII, and gender not being the primary cause, are irrelevant.)

Alito then spends some time saying, "sex was not construed to cover sexual orientation or genedr identity":

Determined searching has not found a single dictionary from that time that defined "sex" to mean sexual orientation, gender identity, or "transgender status."

Nobody contests this. He even concedes this a page later. Now he gets to his main point:

If an employer takes an employment action solely because of the sexual orientation or gender identity of an employee or applicant, has that employer necessarily discriminated because of biological sex?

Uh, yeah? The majority opinion literally covers this. Discriminating against a man who is attracted to men, when the employer would've had no issue with a woman being attracted to men, is sex discrimination.

I will not belabor the point: Alito's dissent makes no real counterargument to any of the majority opinion's arguments. He instead dances around the fact that you cannot define sexual orientation and gender identity without sex (and therefore cannot discriminate against either without discriminating against sex).

However, this quote is so awful that I had to just include it. Just one more...

Contrary to the Court’s contention, discrimination because of sexual orientation or gender identity does not in and of itself entail discrimination because of sex. An employer can have a policy that says: "We do not hire gays, lesbians, or transgender individuals." And an employer can implement this policy without paying any attention to or even knowing the biological sex of gay, lesbian, and transgender applicants. In fact, at the time of the enactment of Title VII, the United States military had a blanket policy of refusing to enlist gays or lesbians, and under this policy for years thereafter, applicants for enlistment were required to complete a form that asked whether they were "homosexual."

Yikes. I mean, wow. This is impressive. Alito is appealing to authority, and the best part is, he's not even appealing to the right authority. If the Court was bound by the executive's interpretation of laws¹, then what exactly is the Court supposed to do? Affirm illegal/unconstitutional executive actions because the executive said "trust me bro"?

If an employer discriminates against individual applicants or employees without even knowing whether they are male or female, it is impossible to argue that the employer intentionally discriminated be- cause of sex.

Counterexample: An employer has a policy that men who like Barbies and women who like Hot Wheels cannot work at their institution. They apply this policy without knowing whether they are discriminating against a man who likes Barbies or a woman who likes Hot Wheels. So apparently it's "impossible" for this to be discrimination? Alito blathers on about this point for quite a while, ignoring the fact that it is very possible for this to occur. Helpfully, the majority opinion already provides numerous other examples.

I'm not going to analyze every quote from this dissent, because it is quite long and everything is just wrong in the exact same way. Summary: the dissent argues about things that do not matter and it does not feel like Alito even read the majority opinion.

Kavanaugh's dissent

If Alito's dissent was aggressively wrong, Kavanaugh's is wrong in a more measured sense. For some reason, I'm a lot more sympathetic to this dissent.

Initially, he makes much the same arguments as Alito does: appealing to the history of the 1964 Civil Rights Act, and failed attempts to amend Title VI/VII to include sexual orientation and gender identity. He also appeals to separation of powers, essentially saying "it's not our job to make laws, just to interpret them".

However, his jurisprudence soon diverges from Alito's: where Alito claims that no reading of Title VII extends to sexual orientation and gender identity. However, Kavanaugh makes a significant concession:

For the sake of argument, I will assume that firing someone because of their sexual orientation may, as a very literal matter, entail making a distinction based on sex. But to prevail in this case with their literalist approach... The plaintiffs must establish that courts... adhere to literal meaning rather than ordinary meaning... Or alternatively, the plaintiffs must establish that the ordinary meaning of "discriminate because of sex"... encompasses sexual orientation discrimination.

(Note: All parties agreed that the ordinary meaning of "sex" does not encompass sexual orientation. Therefore I only focus on the bolded portion.)

This is a very reasonable framework. Where Alito has his head in the sand and does not admit there is any reading of Title VII that agrees with the majority opinion, Kavanaugh understands that one exists. Instead, his primary focus is, "should we prefer the literal approach over the ordinary one?" This is a much more decisive question, and honestly, many of the people celebrating the literalist approach because of the result it brought would likely bemoan such a literalist approach in another case (e.g. a 4th Amendment case).

Kavanaugh's conclusion is where I diverge from him. The "reasonable interpretation" does not mean you pick some random passerby off the streets from 1964 and ask them. Even if the average person does not know "because" means but-for causation, the "reasonable person" would have to understand this. Given that, Kavanaugh fails to explain why discriminating on the basis of gender paired with whether they applied to be a mechanic or secretary, while unexpected to the (truly) average person, is prohibited under Title VII, yet discriminating based on sexual orientation somehow is permissible. It is not the majority's job to show that sexual orientation is somehow special when being included in Title VII; rather, it is the dissent's job to show that it is somehow special and should not be included. The dissent fails to do this, and this is why I disagree with Kavanaugh's conclusion.

Regardless, I am somewhat sympathetic to his claim that few really expected Title VII to apply to sexual orientation, and other (subtly different) cases have been decided differently by the Court. That is why I think Title VII should be amended to explicitly include it, so everyone knows how far it extends: all the way (more on this later; note how, for instance, Bostock makes no mention of bisexual people).

One last point: Kavanaugh mentions that Clinton's 1998 Executive Order explicitly mentioned sexual orientation, and somehow this reveals that Clinton thought sex discrimination didn't cover sexual orientation. This is wrong, because even if you think sexual orientation is covered in a literal sense, there's no harm in making doubly sure by explicitly spelling it out. Also, explicitly banning sexual orientation discrimination is good policy: it prevents the need for cases like Bostock v. Clayton County in the first place. A years-long litigation process would not need to ensue: the (prospective) employee can simply say, "Look, they said they fired me because I'm transgender and the law literally says it's illegal right here." Even if Kavanaugh is correct in assuming that Clinton thought sex and sexual orientation as distinct concepts, Clinton could've been wrong. For much the same reason as Alito's appeal to the US military's policies, Kavanaugh's appeals to authority and history hold little water.

Title VII should be amended

The ruling is correct. However, this ruling is somewhat narrow and may not actually amount to a blanket ban on discrimination on the basis of sexual orientation. Consider this: a man is fired for being attracted to both men and women. Sex is not a but-for cause here, because if that man were a woman, they would also be fired for being attracted to both men and women. (Gorsuch's hypothetical, where a person walks in with a woman to a company gala, would probably still protect a bisexual woman from being fired because they are dating a woman, but makes no promises as to their general attraction.) Perhaps this would still violate Title VII under the eyes of this SCOTUS, but the best way to make sure is to amend the law.

We saw what happened when the legislature and the public took abortion rights for granted because of Roe v. Wade. I'm not suggesting that such a similar thing might happen here, but with civil rights you do not fuck around and find out. Especially when SCOTUS has been overturning lots of precedents, there is no reason not to buttress a narrow protection with a proper, comprehensive amendment to Title VII, rather than relying on a ruling whose wider consequences may not be immediately clear.² Regardless of your personal beliefs, if you have any respect for the premise of Amendment 14 and Title VII, then you should understand that immutable characteristics such as sexual orientation ought to be protected just the same as race, etc.

The clauses in Title VI and VII about unlawful discrimination are identical. Just as this ruling applies to Title VII, it also applies to Title VI. Title VI should be similarly amended for the same reasons. Apparently I'm not the only one who thinks so: attempts to pass the Equality Act have occurred every two years, and Biden issued an executive order that extended "sex" to "sexual orientation and gender identity".

• MATHCOUNTS?
• The AMC 8/10/12?
• AIME?
• USAMO/IMO?
• College contests like HMMT?

If your answers are anything like mine, you'll have noticed a pattern: contests with less problems are more memorable.

On the surface this is a really obvious observation. If there are less problems, it's easier to remember each individual problem. But why does this matter?

The primary purpose of math contests is education. You want good, thoughtful problems on contests, and you want to learn as much as you can for each problem. This is why, besides "for fun" events like Number Sense, you don't really want filler problems to appear on contests.

The second purpose of math contests is for math socialization. You conveniently bring everyone together to do the same set of problems, thereby giving them something to talk about: said problems. But especially with MATHCOUNTS and HMMT, there are so many problems that the problems one contestant wants to discuss can easily be forgotten by other contestants. And this is literally seconds after the contest ends!

So here is the second way to tell if your contest format sucks: if you have more problems in one contest than you can remember, the format sucks. If you have so many problems that a contestant cannot reasonably discuss them all, the format sucks. Put a small number of problems in your contests so everyone can learn from every problem.

Diligence

Picture this. You'd like to learn how to sing, but you know you don't sound very good. So you don't really make a big deal out of it.

Maybe one day, when you are feeling exceptionally brave or ambitious, you will sign up for lessons. You will feel really good about yourself when you go to your one or two hours of lessons every week --- so good that you don't feel much of a compulsion to sing outside of it.

Sure, you'll have improved, and it's a lot better than nothing. But you will never be as good as the people who sing in the shower.

My parents were always out at work — surprisingly, I didn’t really hang out with friends too much back then. My friends all lived quite far from me. My sister would be studying. I’d have nothing to do. I’d watch TV for three hours, and by the end I’d be bored, so I’d start playing the piano, until my parents literally said, “We need to sleep now! Can you be quiet?”

(Source: Rolling Stone Interview with Rosé)

In my university, the drama majors and theater kids have to practice for a long time, getting out late in the night. Every time I see them leave, they are always singing.

I doubt they are doing it for the extra ten or fifteen minutes of practice which will make them marginally better. Instead, they're singing because they just love singing that much. They can't imagine not singing. It's a good opportunity too - no one else is around anyways, so why not just go for it?

What happens as a result? They accumulate the practice that you don't. They are singing while taking a walk, waiting at the bus stop, or taking a shower. That probably adds up to hours every day. If you consciously had to take three hours out of every day to sing, you would get sick of it. The only reason they are able to sustain this is because it is almost subconscious. It is just the natural thing to do.

Details

My sense of rhythm is not particularly good. For the longest time, I believed it was because I wasn't diligent enough to practice my rhythm. But one day, my friend showed me how he could tap out an eighth-note/triplet polyrhythm,¹ and the way he talked about it, he made it seem like the most entertaining thing in the world to try. Eventually it dawned on me: he's passionate about rhythm itself, not just broadly passionate about the instrument he plays.

And I realized the reason I couldn't do polyrhythms wasn't because I didn't have the discipline to drill them. It's because when I'm waiting for food at a restaurant, I am not tapping my legs with my hands while counting "1, 2, 3, 4, 5, 6, ..." It's because I am not tapping my feet in that rhythm at the bus stop. It's because, whenever I'm bored and take out my phone to check my email, I don't feel the compulsion to find some rhythm exercises on YouTube instead.

I might have a passion for singing as a whole, but for the longest time, I dismissed details as just that: things you need to diligently drill in order to become good, pain points you just have to get over. No. You have to be passionate about the details to be any good at them.

• If we divide an interval in half, then one of these intervals must have an infinite number of points in the sequence. (Else the union of the intervals contain a finite number of points.)
• We can take the interval with infinite points, keep doing this, and converge to the limit of the lower bounds of the intervals.

The natural thing for a professor or textbook to do afterwards is assign exercises that either directly use Bolzano-Weierstrass or look very similar. And to be clear, this is good: if we're teaching an idea, we better use it. But I would additionally assign Konig's Tree Lemma: on the surface it looks fairly different, but the underlying ideas in the proof are quite similar.¹

When I read the proof of Konig's Tree Lemma, I understood it very quickly because I had seen the proof of Bolzano-Weierstrass. So it stands to reason the converse might be true: learning the proof of Konig helps you better understand the proof of Bolzano-Weierstrass too. Explicitly, we ought to assign a couple more exercises using the ideas of theorems rather than their results.

There is some subtlety here. For one, we need to avoid being overly broad with what we assign. The idea that "a finite partition of an infinite set must have an infinite part" is generally useful, but it is also known as the Pigeonhole Principle. Konig is a suitable exercise because it recursively uses this method to generate an infinite "path", just as Bolzano-Weierstrass does. It is this high degree of similarity that keeps it from being just another randomly assigned exercise.

This is hard to do well, and I'm skeptical we could afford to do this for every theorem we proved. But I think we could stand to do it more often.

What do I mean by direction and magnitude? Perhaps a couple of examples would best illustrate the distinction. "The algebra 1 professor this semester does not teach well, take algebra next semester and analysis 1 this semester instead" is directional advice. It substitutes one thing with another thing of a similar difficulty. On the other hand, if you see someone working 60 hours a week, "take some breaks and stop working so hard" is advice on magnitude. It removes (or possibly adds) difficulty.

These components are not precisely orthogonal. But the overlap is small enough that this is still a useful distinction to make. And both forms of advice can be useful! "Stop working 60 hours a week" is still good advice, full stop.

But by and large --- especially for people who already have things figured out --- advice on magnitude is at best useless. Unless you have known someone for very long, chances are they know their limits much better than you do.

So when someone is trying to do something ambitious, the least helpful thing you could do is say "don't try to be ambitious". They're not going to listen. They aren't looking for advice on magnitude. They are looking for advice on direction.

It also works the other way: if someone is spending 50 hours a week and still struggling, the least helpful thing you could say is "well just try harder". What do you mean, try harder? They're already going to every lecture, every office hours, and making extensive use of resources like Piazza. There is no possible way they could try harder.¹ This is why many of the people being generally unhelpful to the people who are ahead also cannot really give good advice to the people who are struggling.

However, where the correct² magnitude of someone's efforts is largely determined by internal factors, the correct direction of someone's effort is more affected by external factors. For instance, the CMU math department does not have a lot of opportunities for algebra. So if you want to study math at CMU, it makes more sense to direct your energy towards another field or cross-register at Pitt. Either way, something in the direction of your efforts should change because of this external factor.

Whereas the internal factors affecting someone are best understood by themselves, the external factors are a different story. Say you are an upperclassman giving some freshman advice on what courses to register for. You don't know nearly as well as they do what they can handle, but you do know much better what the courses are about. Closing this knowledge gap is probably the most helpful thing a piece of advice can do, and I think it should be the explicit aim of said advice.

Of course, this isn't to say "course X is generally regarded to be hard" is bad advice to give. But maybe your gut reaction to "I want to take course X" anyway shouldn't be to issue a blanket statement discouraging them. Just my two cents.

So what are the main differences? Here's a list in roughly descending order of significance:

• package manager (apt, dnf, pacman, etc...)
• libc implementation (for almost every distro it's glibc)
• init system
• kernel version
• default desktop environment installed and default theme (important for overall "market share of Linux", which I don't really give a rat's ass about, but totally unimportant for an individual user)

Obviously this post will be opinionated, otherwise it'd be useless.¹ So for transparency, my allegiances:

• apk
• musl libc (but some people need glibc)
• runit, but openrc is also quite good
• I typically run a very recent kernel.
• sway (but please don't install shit with my distro, I'd rather login from a tty and manually run sway)

I have three machines --- a desktop running on Bedrock, an old laptop on Alpine Linux, and a new laptop running Void Linux.

I'll be going through the distros in roughly descending order of popularity. A couple of distros I know exist but decided to omit due to lack of knowledge are Gentoo, Fedora, and RHEL/CentOS. Since most systems use glibc/systemd (and the purism of your libc and init system probably don't matter much to you), I will make no comment unless a distro uses something other than glibc/systemd.

This post isn't really meant for Linux beginners, by the way, although I think Linux beginners can glean some useful information here.

Debian-based (e.g. Ubuntu)

Even though they certainly have a place in the Linux world, I really don't like Debian-based distributions (mostly Ubuntu, which I will primarily focus on for the rest of this section). That said, if you use Ubuntu, you are by no means a "bad Linux user" --- ease of installation should be a pride point and there's nothing cool about pre-archinstall Arch being super hard to install.

Debian's package manager (technically frontend for the real package manager dpkg, but come on, who actually uses dpkg for anything besides .deb files) apt is one of the more intuitive package managers, at least on the surface. But this is only on the surface. Dependency management is hell and there are a bajillion different binaries you need to actually manage your system. It's also super slow and super out of date. It is literally an order of magnitude slower than pacman, and texlive/pandoc, two fairly significant programs, are years behind (texlive is on 2020 and the stable pandoc package is using a version from 2019). Of course, you can mitigate this somewhat by going on an unstable branch, but even then there is no way to get anything super recent. For programs like pandoc, there are subtle breaking differences between one version and the next, and all things consider you really want to be on latest.

(Yes, I know Debian recommends aptitude, but the fact that exactly 0 Ubuntu/Mint users know it exists is kind of my point. There are just way too many frontends for dpkg and no one can make heads or tails out of it!)

That said, it is almost always not a big deal to have a Debian server, because most critical infrastructure is up to date enough. You really shouldn't even have Pandoc installed on your server, because the way you are supposed to update a website is by building and then sending a tarball with your files to prod... same for TeX Live, really. However, I'll probably stop using Debian when I clean-install a new server because I see very little technical advantage in continuing to use it. Desktop though? Yeah... no. Have fun adding PPAs for shit while everyone else is using the AUR or xbps-src.

Also re Ubuntu snaps: LOL, let's hate on Ubuntu for a bit.

• It's closed source. (Yeah, parts of it are open source, but you cannot make a useful Snap app store because you don't have the closed-source backend.)
• It's slow as FUCK, and not just for installations --- apparently Firefox takes 15 seconds to start if you're calling the Snap. (This is likely an exaggeration and not collected using time.)
• Chromium and Firefox are snaps by default. If you run apt install chromium you get a Snap. Linux is all about transparency. Even if you have nothing against snaps, there is literally no reason for apt to use anything other than a .deb file from either the Debian repositories or a close downstream derivative.

At this point Ubuntu is literally pulling the wool over your eyes much like Windows (but not as bad). Sure, everything is technically documented, but everything's so opaque by merit of "a bunch of crap no one wants is shoved in your face." So yeah, for new users: I don't really recommend Ubuntu. Ubuntu also asks you to update your software a bajillion times. Mint patches out the Ubuntu-specific bullshit, so it seems to be a better choice for now, but I really recommend archinstall and setting up Sway on Arch if you have a weekend to spare instead.

Arch

Arch's package manager pacman, unlike apt, is fast, recent, and totally unintuitive. Where in Debian you run apt install or in Alpine you run apk add, in Arch you run... pacman -S. And then to update, instead of running apt update && apt upgrade you run pacman -Syu. The options are literally alphabet soup, and you could certainly understand pacman with some effort, but when you compare to the good package managers which don't require any time to understand pacman leaves something to be desired. Also pacman is kind of slow, but my standards for speed are based on xbps/apk now.

On the bright side literally anything you could ever want is in the AUR. On the not-so-bright side, as nice as some AUR wrappers are, I just want to manage my dependencies with pacman and I can't. (Well, unless you use the Chaotic AUR --- but that's a huge mess too for reasons I won't get into right now.)

Arch is also very DIY, just like pretty much every not-Debian distro. You have to set up Xorg/Wayland on your own (though a lot of Systemd-related stuff is just installed and set up for you). Networking is also annoying, just like on every other distro.

Alpine

Alpine is my favorite distro. It doesn't do everything, but for the people who'll be satisfied with its capabilities it is the best in its class by far. And its limitations aren't because of technical flaws with Alpine, but flaws with the programs you want to run. (Namely, companies who release proprietary binaries only build them against glibc, so if you use musl libc, you're screwed.)

The main sell of Alpine for me is the apk package manager, particularly how small and neatly divided each of the packages are. For instance, package foo usually does not include docs, and the docs are contained in foo-docs. This is because if you're on a server you don't need the docs, you should be reading the manpages on your own system. Granted, docs don't really take up that much room, so it's not the biggest deal. For your development environment, you can and should just install the docs metapackage to get the documentation for each program you install.

apk is super intuitive too. If you want to find the package that contains file foo, simply run apk search foo. apk is also much faster than any other package manager, including pacman and xbps. It's expressive enough to do anything you'd ever want and I don't know why you'd ever use another package manager at this point.

Since Alpine uses OpenRC, not systemd, and barely has anything configured by default, you may have to do some basic hardware configuration yourself. For instance, in every other distro

• the brightness value on boot is taken from whatever it was on shutdown
• wifi autoconnects pretty easily
• laptops hibernate on close lid

yet with Alpine, I had to set all these up on my own.

Alpine does have limitations, but there are ways to get around it. You can use a chroot of a different operating system (or have Bedrock handle it for you) or a universal package manager like Flatpak, which bundles it dependencies as well as the binary. You can choose to use different software --- and I don't mean to go all FSF "you must never under any circumstances use proprietary software" here. There are genuinely good interfaces that still maintain compatibility with the proprietary backend. For instance, 6cord is a good Discord terminal client that just works and rclone lets you treat your Dropbox folder as a remote filesystem. Steam not working is a bit of a pain point, but I grind Spider Solitaire in KPatience all day anyway so it's OK for me. In fact if you want to focus, Alpine is a really good distro for a work environment.

Adelie Linux also seems very similar to Alpine Linux (I haven't personally used it myself). It uses the same package manager (apk), the same init system (OpenRC), but a different userland (Adelie uses GNU while Alpine uses BusyBox). I get why Adelie uses GNU coreutils --- BusyBox does have some problems, like sort not seeing whitespace as a separator by default when you pass in the -t flag.² (I do like BusyBox a lot though! Full disclosure, I use BusyBox on most my systems.) The gcompat library, a glibc compatibility layer, is also developed for Adelie, so that is one of the many reasons Adelie might be worth considering. (gcompat also can be installed on Alpine, though I've had zero success in getting it to work.)

Void

Void is similar to Alpine but it uses runit (IMO a better init system since it makes you manually symlink stuff, which is basically what happens under the hood with systemctl enable but without any black magic). The package manager is very similar, but unlike Alpine, Void also lets you build a vast array of packages from source with xbps-src. Everything under the sun can be found there (even PolyMC, a Minecraft launcher!) though it does feel a bit clumsy compared to the AUR.

Void also uses glibc unlike Alpine, so pretty much everything is compatible with Void. (You can choose to use musl libc if you want, but to me that eliminates 90% of the benefit of using Void over Alpine.) For 99% of people this is the ideal desktop/laptop distro.

Bedrock

You know how you can make a chroot of one distro in another (usually done for compatibility purposes)? Bedrock is like that but on steroids. You can mix and match coreutils, kernel version, package managers, etc.

However, making a chroot directory for another distro is a reversible operation, while converting to Bedrock is not. Thus I only suggest it if you are totally out of options otherwise. (I use it on my desktop because Gnome/Ubuntu just works with NVIDIA monitors while everything else is totally pure agony, though if I were to set it up all over again, I wouldn't.)

Verdict

For servers, use Alpine, hands down. The proprietary programs you're "missing" for Alpine are mostly for personal use, you shouldn't be running a server with proprietary programs anyway. (That said, if you want to run Alpine in production you probably should also run it on your dev machine.)

Personal computers are a bit more complicated. If you want to lean more heavily into gaming with Steam use Arch or Void. I personally think that if you're going to install more "random" packages³, Arch is the better choice, mostly because I hate xbps-src. I run Alpine on my "work" laptop, but Dropbox, Discord (I use 6cord, a terminal client for Discord instead on Alpine), and Steam are all major PITAs.

There's also an argument for installing Bedrock on top of a sane base like Alpine Linux. It's like chrooting but you don't actually have to care about building packages yourself, or setting up the environment. It is a bit bulkier though since you pretty much have most of a distro's base installed, minus the init parts.

Final note: If you are a soon-to-be new Linux user planning to install a "user-friendly" distro (usually a Debian-based distro, and said Debian-based distro is almost always Ubuntu), don't let my criticism of Ubuntu stop you. For all the flaws of Ubuntu, it is still bona fide Linux and a thousand times better than Windows/Mac.

• people don't understand what "testsolving" means,
• or they are too irresponsible to write their contest themselves.

If I suspect you are the second type of person, I will just flat-out refuse to help you with your contest. But I suspect there are a lot of hard-working people who may unintentionally be taking advantage of their "testsolvers" because of misunderstandings.

How do you write a contest?

• Write problems. At some point, possibly after arranging them into a test but no later, you must write full solutions for each of them.
• Arrange problems into a test. The people who do this can be the people who write problems, but this is not always the case.
• Verify feel. Are the problems appropriately easy/hard for the intended target audience? Is the "intended" target audience who you will actually get? Are the subjects distributed reasonably? Are the problems fun to do? Iterate until the answer to all these question is yes.
• Verify correctness. Make sure every problem is well-defined, answers are what you think they are, and the solutions are clear and logically sound.

Particularly for verifying correctness, it is imperative that you have full solutions written. Verifying correctness does not mean finding the answers to your problems. It does not mean verifying your answer key, either. It means you verify that absolutely everything is correct.

You aren't ready to ship a contest until you're ready to release everything right now, meaning the test itself and the solution manual. It maybe even means preparing some advertising materials, or the means of testing (testing venues for in-person, testing portal for online).

So when do testsolvers come into play? At earliest it is at the third step, verifying feel. Personally I make a rule for myself not to involve testsolvers until I think I'm ready to ship a contest. (More precisely, by "I" I mean Math Advance.) Testsolvers are just there to make sure that the contest is ready. They are the final correctness verification.

One-line summary: If you expect your contest to change after receiving feedback from someone, they are not a testsolver, they are helping you write your contest.

But I need help on my contest!

Fair enough. Testsolving is not the only role non problem writers can take, and it's perfectly normal to ask for help, even from outsiders to your group. Just don't tell people you want them to "testsolve" if you're giving them an obviously flawed product to look at. Testsolving means that you have made your very best efforts to create a final version of your contest, and that you are very sorry if anything has to change at all.

Obviously you won't be able to catch everything, and you may need slight assistance or want feedback on earlier steps, such as "what about this new arrangement of problems?" or "is the feel better now?" It is okay to ask for these things. But remember, the job of a testsolver is not to help you write your contest. They are just the final verifiers. Inevitably your mistakes will be revealed, but the thing about mistakes is you're supposed to put some effort into avoiding them.

Don't ask me to testsolve until...

...you have all your solutions written. No exceptions. I don't want to read a test half-filled with problems that make zero sense. I mean, you should be careful enough that you don't even put problems on your test unless you know they are actually comprehensible. Everyone in Math Advance can do it, it's really not that hard.

But if you know that you've been cursed with the inability to properly express your mathematical ideas, then one way to make sure it doesn't affect me is by writing solutions. It'll be really hard to write clear solutions if the problem isn't clearly stated, so there's that.

If you often get the feedback "I don't understand this problem", one thing I haven't tried but recommend is having at least one solution per problem by someone who isn't the problem author. That way if a problem is unclear, it becomes apparent quickly.

I'm not ready for testsolving, can you still help?

Do not ask me for help unless you are using mapm. If my time is not valuable enough for you to use a totally new tool for creating your contests, that is totally fair. But my time is worth that much to me, and I am sick of reading disorganized, carelessly written Overleaf documents that are not even in sync with each other.

For those of you who are willing to use mapm :) If you want me to help you write your test, the answer is probably still no. I just don't write enough problems to do that, period. If you want help with arranging your test, maybe, with the caveat that I will not take a leading role. I am pretty opinionated and vocal though, so you will definitely get lots of feedback.

If you are willing to use mapm but don't know how or want me to help you set it up, I'd love to do so.

If you have any questions about contest-writing, feel free to email me. If it's a good question, I might just update this article to include it!

Update (2022-07-27): Please ask your testsolvers to look through your contest before you publicly announce it! If your contest is significantly flawed, in 99% of cases the "testsolver" will end up taking more responsibilities (because anyone who volunteers their free time to testsolve a contest is probably really nice). You don't want to force a time crunch on them that they never signed up for.

The alternative is that the testsolver goes "fuck it, I don't care anymore" and BSes through the rest of the test or just stops commenting. This happens a lot with contest creators who are either unnecessarily stubborn or fail to plan ahead, and you don't want this to happen to you either.

If you don't know what Gemini is, it's an internet protocol based on simplicity. It totally separates content (server responsibility) from style (client responsibility). So you can read my website however you want: through a terminal, a GUI, through light mode, etc.

I recommend Amfora for terminal enthusiasts and Lagrange for everyone else; at least, those are the two I have looked at. I am sure other Gemini clients are also amazing, I just haven't had the chance to look at them yet.

Reproduced below is my first status update. I don't think I will mirror this content again, I want to keep some content gemini-exclusive.

I've got a ton of things on my notepad's to-do list that I'd like to keep in a more permanent medium. (Some of them I have recently checked off.) So here they are, roughly separated by topic.

Writing

I've had a couple of thoughts about pedagogy that I've never really gotten around to writing. One is that group lessons are probably more valuable than individual lessons, despite the latter costing far more per hour. I think this is only true in some limited circumstances: you have to be teaching smart people, they have to be self-motivated, and they have to be sufficiently extroverted. People tell you the main value in schools like MIT isn't really in the professors or the classes or whatever, but in the peer group. Doesn't the same hold true in general?

Recently I also finished porting over the ARU-Telescoping unit for MAST, and the solutions all follow the same template:

Algebraic expression A can be rewritten as B, which obviously telescopes to value C.

There's no padding where I explain the motivation of the solution. There is motivation, obviously, but I have omitted it. I've considered adding it before, but in the end decided not to.

Whereas in my geometry textbook, I sometimes explain in great detail where each step comes from, even writing out obvious statements like

Angle ABC and angle ABD are congruent where B, C, and D are all collinear.

Of course, this is in the context of a chain of equalities to show that some angle is equal to another, so said statement clarifies the chain of equalities a bit more.

Just like how some books and movies deliver a message more effectively by implying it, rather than outright stating it, I think some non-fiction texts might also do the same. Brevity can be good, and so can omitting details, since it can be used as a tool to magnify the most important parts of the text.

So there is something that makes it right to omit motivation for telescoping and makes it wrong to omit it for geometry. What is this difference? I don't know what it is, and if you have any ideas I'd be glad to hear them.

Glee, the (hypothetical) Git server

=> https://www.dennisc.net/writing/tech/glee-design Design for a small-scale self-hosted Git service

Recently in my investigations with Gemini, I've heard whispers in the wind that authentication and primitive read/write access might be possible with Gemini. I don't have any ideas on how the details would work, but if all the pieces line up I am interested in writing a Gemini frontend for Glee in addition to (or possibly in replacement of) a web frontend.

The graphical frontend would likely just be for read access anyway, I don't care about giving people a GUI to set permissions or whatever. That stuff should be done with a command line that works around the API (even if the graphical read-only frontend is on Gemini, the API exposed for writing to permissions files would probably be on HTTP). But if too large a subset of functionality can't be easily created with Gemini, I think I will just not use Gemini for Glee, period.

Math

I've been telling myself that I should look at the CAMO 2022 for a while, and maybe now is the time to do it, what with it being two days away from USAMO.

=> http://cmc.ericshen.net/CMC-2022/CAMO-2022.pdf CAMO 2022

I also need to finish solutions up for the GQU-Triangles MAST unit. That's sort of been blocking me from porting other over stuff.

Also, sometime after the USAMO, I would like to give a lecture on my ARU-Telescoping unit to the MAST kids. (I would've done it before the USAMO had I not qualified.)

latex-lint

I have a small project called latex-lint which I use with Vim. Currently all it does is indent through latexindent + move punctuation outside of math mode. Eventually I want to add functionality to do this:

Git diff replacing double dollar signs in LaTeX -$$1+1=2$$ +\[1+1=2\]

=> https://gitlab.com/chennisden/dotfiles/latex-lint

mapm

I've been telling myself for a long time that I need a GUI of mapm to really make adoption widespread across Math Advance. But it seems that even with just a CLI, adoption is pretty high across the active problem writers and test compilers. Many thanks to everyone who has begun using it, particularly Prajith Velicheti, Aprameya Tripathy, and Kelin Zhu, who have done a lot of work migrating problems to mapm, providing installation manuals and helping people through the process, and using it for their own projects. (I am somewhat deliberately obfuscating who each description is attributed to.)

Of course, I have also been writing my own manuals and maintaining the documentation for the project. Though mapm is still in an alpha stage right now, I think it is far better than any other publicly available alternative. I encourage you give it a try if you want to compile your own math contests. The (informal) spec is super short (just like Gemini) and very extensible.

=> https://gitlab.com/mapm/zero-to-mapm From Zero to mapm => https://mapm.mathadvance.org Comprehensive mapm documentation

Just like how I recently wrote a "Why latexmk" article on what niche the latexmk script fills, I should do the same for mapm. A lot of open source projects have this great, technical description of what it does, and no two sentence summary on why you would use it. mapm is among that number and hopefully I can change that soon.

=> /latexmk Why latexmk, mirrored on Gemini => https://dennisc.net/writing/tech/latexmk Why latexmk, on the web

As for the GUI: I wonder how much I need to write this utility, versus just wanting to write it to fully take advantage of our tags and buckets system. It is still my next major programming project, though I can already see I am inventing new things to do (like programming the build script for this gemlog) instead of actually writing the GUI.

If you are interested in using mapm, join us at Zulip. Unfortunately you are required to make an account, but unless I've screwed up, anyone should be able to join.

=> https://mapm.zulipchat.com mapm Zulip organization

Linux (mostly Sway)

I had planned on switching to Sway on my desktop, but it uses Nvidia drivers. (I know, sue me, --my-next-driver-wont-be-nvidia.) Nouveau kind of really sucks and I don't care enough to get it to work properly. But my laptop, which is a Lenovo IdeaPad, is basically the Linux gods' gift to humanity. So I do have Sway on my laptop.

Previously my twm was i3, which is largely compatible with Sway, but there's a couple of little Sway-exclusive things (like the holding tray for windows) that I hadn't really configured yet. Plus, I'm using Polybar for desktop but that doesn't exactly work on Sway as far as I know (at least not without X, which sort of defeats the point).

Also I'm really surprised people recommend Arch so much over Alpine. I mean, I'm one of them, just because Arch is so much more popular that chances of you fixing a problem are much higher (everyone and their mom has a pacman post, the only thing it lags behind in converage is Ubuntu). But besides bleeding edge + more package coverage, I think Alpine is just better in a lot of ways. Although no AUR is a dealbreaker for a lot of people, I'm already using Bedrock, so really no reason not to have my base running Alpine. I won't deal with that now, but hopefully in a few months I look back on it and take care of that then.

I hope you will consider installing Gemini and checking out my gemlog.²

With that said, here is a litmus test to see if your contest format sucks: can you explain it in no more than twenty words?

• AMC 8: 25 MC in 75 minutes. (5 words)
• AMC 10/12: 25 MC in 75 minutes with 5 options. 6 points for correct, 1.5 for blank, 0 for incorrect (18 words)
• AIME: 15 questions with integer answers 0-999, 3 hours, 1 point for correct 0 points for wrong (16 words)
• USA(J)MO/IMO: 2 days of 3 questions, 4.5 hours each. Proof-based, responses graded from 0 to 7. (15 words)
• MAT: 3 sets of 3 problems with positive integer answers, 30 minutes each, plus a set of tiebreakers broken on time. (20 words)¹
• HMMT Nov: 10 questions 50 minutes for general and theme. 10 questions in 60 minutes for team. 12 sets of 3 for Guts, last set is estimation. (25 words)²
• HMMT Feb: 10 questions 50 minutes for 3 subject rounds. 60 minutes for team, proof-based. 12 sets of 4 for Guts, last set is estimation. (23 words)

In all of these cases I didn't even have to try particularly hard to get it under 20 (except HMMT). I suspect I could've gotten most of these under 15 if I word-golfed hard enough. If you can't get your format under 20 words, then you should change it before it comes to bite you into the ass in the form of a poorly designed test.

It doesn't matter what it is. Even a press release handled well has a moment that you can't help but revisit. A police interview has that moment of catharsis that some of us love to watch over and over.

Hitchcock famously said that "suspense is when the spectator knows more than the characters." And indeed, anything worth reading --- or watching --- must have that same feeling. Not necessarily of suspense, mind you, but just the feeling that it's worth seeing the piece through even when you know its conclusion. After all, there are very few mystery novels that are stunning the first time around and absolutely boring the next.

That's not to say that reading something for the first time has no value --- obviously it does. So a good book, essay, movie, or game needs to be interesting enough to capture your interest the first time, even if you don't know or understand everything that's happened.

But there's a separate sort of value you get from rereading something. I feel that this isn't given as much attention as it really deserves. Even when it comes to documentation, how often something is reread tells you a lot. It tends to either be important or have some fundamental concept it's easy to get confused on.

Consider the mount command: is it # mount /dev/sda2 /mnt or is it # mount /mnt /dev/sda2? Turns out it's the former. And it has to be the former, because otherwise you can't run mount with one argument. And this makes mounting USBs without user intervention so much easier to handle because now you don't have to worry about where to mount it.

As a writer, I am good enough to make all my essays worth reading the first time around. But I don't think that's enough. In fact, I have several essays that I haven't published --- both on my old site and this one --- that I think were worth reading once. But that's the problem: it's only worth reading once.

The day before publishing this essay I made a couple of user interface changes on this site. One of them was removing the boxes with the links in the writing subpages, but not the main one. Why? Because no one will ever reread the descriptions.¹ I might even remove it for the main writing page, because the sections are honestly pretty self-explanatory.

This is a new idea for me as well. Only for the last few weeks have I been writing with a focus on rereadability. The essays on my old site were littered with hyperlinks everywhere. I didn't notice at the time, but that was a good sign --- it meant that at least I was interested enough in my writing to reread it and reference it. And the reason to aggressively prune writing? To increase the proportion that's worth rereading. I've even started trying to minimize the number of footnotes in my essays, because often people don't find them worth reading even once, let alone rereading.²

Rereadability is the one word summary for what makes a good hook. I don't know how rereadble the rest of this essay is, but I hope at least the hook is.³

• both CS
• UT Austin Turing actually has a higher mean salary than CMU (caveat: methodology for comparison was questionable)
• money isn't a major issue

Either way, I don't think I would've made a wrong decision. But I'm only ~40% sure that I made the better decision, for a couple of reasons:

• parents played a big influence in where I'm going. In some hypothetical world where my choices were MIT vs. Harvard, I'd be like "ew no I hate Harvard I am definitely going to MIT", and the strength of my preference would be enough to override any hypothetical parental desires. If my choice was made totally independent of their influence, I would almost certainly have chosen UT Austin, but I was barely more sure that UT was the better choice. Well, at least my mom will get to enjoy her Asian parent WeChat group.
• apparently CMU is really fucking small and that sucks. In some sense, it's cool that people density is higher, so you can just hang out with people and not be lost looking for another human soul, but I'd also be able to explore campus and still find something new even in my fourth year. With a school only four times the physical size of my high school, that's just not very likely. At least Pittsburgh is pretty big, so I can just step off campus and go downtown.
• CMU allegedly induces depression. Obviously this is overblown, but where there's smoke there's fire.

But hey, there's lots to be excited about too in CMU. For better or worse, there's no turning back at this point, so might as well be optimistic and make the most out of it. Some part of me is thinking of putting in a transfer application to MIT, but by the time I get there I'll probably really enjoy CMU and be too lazy to fill it out.

Also one of my friends from school is going to CMU, so that's cool too.

UCL Rejection

I couldn't really fit this anywhere, but I felt that college talk was the best place to put it. I got rejected from a British university I applied to a long time ago, University College London. At this point I didn't even care enough to have a reaction either way, and almost decided not to check. (Which I did to Irvine, and I honestly might not have known they accepted me if they didn't email me about their honors program.)

Roommate rizz

So a lot of people have asked to room with me, which is pretty cool because I'll have a roommate for sure and there's a group of people I'll definitely be able to talk with during orientation (though I want to meet new people, especially outside of SCS!)

But of course, roommate rizz doesn't count unless you get rizz rizz. In high school it's always been some combination of "it's complicated", "circumstances", and "waited too long to take any actual initiative." But things will never be the same when I get to college. This will be my life goal that I value even more than that sweet sweet Jane Street internship 🫡

(To be clear, I'm mostly joking, but I should try and be more extroverted. Right now I'm this weird mix of extrovert and introvert, where I'm not afraid to put myself out there and have eyes on me, but at the same time I might find it difficult to talk to specific people I want to talk to because of circumstances or any other litany of excuses. Also at WARP I discovered that I get really tired after doing stuff with other people and need to hole myself in my room for a few hours afterwards, so consistency rather than intensity seems to be the correct approach here.)

And for no reason whatsoever, I feel like the Tepper kids are probably the most sociable/normal/etc. Which, I guess, is part of the job description...

Track and Field

My track season ended with a whimper.

Three weeks ago at Dublin, I nearly fell asleep during warmups and got a terrible time on the mile.

Two weeks ago at the Amador Valley Spring Fling, I hit a decent PR and thought, "maybe I can bring this momentum forward and close my last season with a good-ish time."

A week ago at Granada, I made a few tactical errors in the race and could only match my PR.

And Thursday, I had a terrible cough and severe allergies. It wasn't as bad as Dublin, but it was noticeably slower than my PR and I felt like I was dying the entire race (moreso than usual).

So rather than closing the season with a big PR, or even modest gains, it was just very meh for reasons entirely out of my control. And I ended up not being able to run the fun events I wanted to because the pollen was making me bawl my eyes out and I could barely see the track, let alone run on it.

So how do I feel? Well, I'd probably care more if I was good at distance running. It kinda feels the same way as my 3 on USAMO last year --- pretty embarrassing, but I'm not sure I was really invested in the result at that point. I enjoyed the process, it taught me some pretty valuable lessons, and I'm a better person because of running. Or less fat, at least.¹ Very much like college applications, I've come out through the other side and nothing really terrible happened, but at the same time, nothing great really happened either, and I'm still a little regretful about the whole thing.

I do plan to continue running in college. I don't think I'll train as intensely as before at college, though I'll probably still show up for summer training when I'm in town this year. It's very possible that I've run my last 1600m race already, or at least one of my last three if some one-off miracle occurs and there's a track meet convenient enough that I can register for. I might try some actual distance races though like 5Ks, 10Ks, etc.

AP Chemistry

Man, AP Chemistry was (is?) such a wild ride. To this day I have never done a practice MCQ, but based on my performances on FRQs I think I could probably get a 5. However I do lack some problem-solving experience, which is expected since I started self-studying literally less than two weeks ago.

Before I actually made a serious attempt, I was like "I have three weeks left I'm screwed, why don't I just give up." But then I tried for a bit, learned pKa/buffers/thermochem/the other stuff I didn't know pretty quickly, and now I actually think that a 5 is not only within reach, but honestly should be pretty easy. I'd told myself that you could learn the entirety of AP Calc BC in 2 months, the entirety of AP Stat in 2 weeks, etc, and I also wrote about how most people suck at any given thing, yet I didn't really internalize either of these things until now.

Right now, I just need to memorize VSEPR and maybe review thermochemistry. That should be enough to get me a 5, although now I'm studying chemistry, it actually seems pretty interesting to me again :)

Multivariable Calculus

My relationship with vector calculus has been a real doozy. First I thought, "huh the multidimensional analogs are pretty natural generalizations." Then I got to the four fundamental theorems of vector calculus: line integrals, Green's, Stokes, Divergence, and then I went "what the heck is this physics nonsense, this makes absolutely no sense and I still don't really know what an integral is why didn't I take real analysis," but now I'm learning that there's something deeper behind these four theorems.

Honestly I've been less engaged with the material than I hoped I'd be, and without a final I really might just check out mentally after the last unit test. Which is a bit sad, considering multivariable calculus is still math even if it's gross applied math with the applications (physics) neutered. But learning real analysis then vector analysis seems to be a better approach, skipping the non-rigorous calculation nonsense portion, and also discrete math is good to learn too. The opportunity cost of evaluating irrelevant integrals is just too high to justify actively doing it.

But we do have a final project, and it's something I've been thinking about doing for a long time. I've always wanted to write an article about numerical computation, history and present day included. Logs were invented so people could multiply big² numbers easily, but they did it first with trig tables. And the reason they did it was to perform calculations with astronomical data, which is how they derived Kepler's Laws. I think this is really cool stuff and want to share it with you guys too, and this has finally given me an excuse to do exactly that. I'm going to write the article, then actually care about meeting the project requirements, which are not too demanding anyway and won't compromise the article in any fashion.

Also, have you ever wondered how your computer calculates (well really, approximates) sin 1? It's not just Taylor Series, there's a lot that goes into it from the little I've seen. (This is going to be the present-day portion.)

A trend

I've noticed that most of the time, I really don't share a lot of good news on my blog, and when I do there's always a tone of solemnity. This isn't too surprising, because nothing purely good and noteworthy has happened to me recently (life is rapidly getting more complicated), and even if it did, it probably wouldn't be interesting enough to write about.

I mean, since my life is about to be consumed by CS, it's very likely that my website turns into a math/CS blog for real. I don't think I've written about math without using it as an example for a philosophical framework --- when I write about math now, it's usually in-depth enough to be an article of part of a book --- and my CS posts are along the lines of, "how to use Git", "how to make a non-shitty website", etc.

While my CS posts have actual meat, unlike the "beginner-friendly" crap most article mills put out, it's often written as a high-level, cursory overview in order to remain accessible. But when CS occupies a lot of my headspace, I'll likely have ideas (maybe even rehashes of lectures) in my head that I'll just speedily put down as a post on this website. So the ratio of "personal things that happen" that I share is likely to decrease.³

I think as you grow older, you start to realize that Noteworthy Good Things don't really happen to people that often. Life is usually complicated, and the only Noteworthy Good Things people generally share are marriages and the birth of a kid.⁴ Maybe if you're a celebrity or write books this list is longer, but generally life feels quiet because you learned to ride the waves. It's a little sad to think that, now I've mentally checked out of high school, I'll never get to feel the same highs I did when I first qualified for USAJMO or ran the 1600m under 6:00 in my first real track meet.

There are these things called vectors. They don't exist in isolation. Rather, they're defined as parts of vector spaces. A vector space has objects called vectors u, v and scalars k such that u + v is a vector and kv is a scalar. Vector addition and scalar multiplication behave as you'd expect: it commutes, associates, distributes, etc. Don't think too hard about what vectors are, just what they do.

The canonical example of a vector is a list of real or complex numbers. These vectors exist in ℝⁿ or ℂⁿ. We will be focusing on these vectors soon, but for now we work with generic vector spaces.

We say a set of vectors {v_i} is linearly independent if the only solution to ∑a_iv_i = 0 is a_i = 0 for all a_i. We say a set of vectors spans a vector space V if for any vector v ∈ V, there is some solution to ∑a_iv_i = v. If {v_i} is linearly independent and spans V, then we say {v_i} is a basis of V.

All bases of a vector have the same length (i.e. the same number of vectors). Therefore we define the dimension of a vector space to be the length of the basis.

A subspace U of a vector space V is a vector space whose elements are all in V and whose addition/multiplication operations are identical. That is, if u + v = w under U, then u + v = w under V as well, etc.

Linear Maps

A linear map T is a function V → W such that T(u + v) = Tu + Tv and T(av) = aTv for all vectors u, v and all scalars a. (V and W are vector spaces.)

The most boring map is I, the identity map. It is the function V → V such that Tv = v.

The null space of a linear map is the set of vectors v such that Tv = 0. As the term implies, the null space is a vector space itself. The range of a linear map is the set of vectors v such that there exists some vector u satisfying Tu = v. It is also a vector space. In other words, the range is the set of outputs possible from T. (Take note that null T ⊆ V and range T ⊆ W.) The Rank-Nullity Theorem states dim null T + dim range T = dim V.

Also, linear maps can be injective, surjective, and invertible. T is injective if a = b ⇔ Ta = Tb. (T injective is equivalent to null T = {0}.) T is surjective if range T = W. T is bijective if it is both injective and surjective. And T is invertible if TT⁻¹ = T⁻¹T = I. Obviously, T must be of the form V → V, i.e. V = W, for T to be invertible.

A linear endomorphism is a map of the form V → V. In other words, the domain and codomain are identical; this is the definition of an endomorphism. In an endomorphism, injectivity, surjectivity, and invertibility are all identical.

Now we discuss invariant subspaces on linear endomorphisms. A subspace U is invariant under a transformation T if Tu ∈ U for all u ∈ U. You can decompose a vector space into invariant subspaces.

The most useful kind of invariant subspace is that with dimension 1. These subspaces are essentially equivalent to eigenvectors. We say v is an eigenvector if Tv = λv for some scalar v. Also, we call λ the associated eigenvalue. If v is an eigenvector, then {av} is an invariant subspace with dimension 1.

A set of eigenvectors in distinct dim 1 invariant subspaces is linearly independent.

All complex vector spaces have an eigenvalue. All real vector spaces with odd dimension have an eigenvalue. The former is essentially equivalent to the Fundamental Theorem of Algebra, and the latter is equivalent to the fact that all real polynomials with odd degree have a root.

Inner-Product Spaces

Up until now any arbitrary vector space sufficed: the scalars could come from any field. Now we must specifically deal with either the real or complex vector spaces ℝⁿ and ℂⁿ.

An inner product ⟨u, v⟩ satisfies the following:

• ⟨v, v⟩ ≥ 0, with equality iff v = 0
• ⟨u + v, w⟩ = ⟨u, w⟩ + ⟨v, w⟩
• ⟨av, w⟩ = a⟨v, w⟩
• $\langle v, w\rangle = \overline{\langle w, v\rangle}$

(To clarify, $\overline{z}$ is the complex conjugate of z.)

These are our axioms, but one can also deduce that ⟨u, v + w⟩ = ⟨u, v⟩ + ⟨u, w⟩ and $\langle v, aw\rangle = \overline{a}\langle v, w\rangle$.

The standard example of an inner product is the dot product. Note that the complex dot product is not quite what you'd expect; $\langle (w_i), (z_i)\rangle = \sum w_i\overline{z_i}$. The motivation is so that $\langle z, z\rangle = \sqrt{\sum |z_i|^2}$.

Also, the norm ||v|| of v is $\sqrt{\langle v, v\rangle}$.

Some (in)equalities you might expect from dealing with the dot product:

• Pythagorean Theorem: If ⟨u, v⟩ = 0, then ||u + v||² = ||u||² + ||v||².
• Cauchy-Schwarz: |⟨u, v⟩| ≤ ||u||||v||, with equality when one of u, v is a multiple of the other.
• Triangle Inequality: ||u + v|| ≤ ||u||+||v||, with equality when one of u, v is a non-negative multiple of the other.

With the inner product defined, we can construct orthonormal bases of a vector space. In an orthonormal basis, ⟨e_i, e_j⟩ is 1 if i = j and 0 if i ≠ j. An orthonormal basis always exists, and can be explicitly constructed given some basis with the Gram-Schmidt Procedure.

Spectral Theorem

A linear functional is a map from Fⁿ → F. For all linear functionals ϕ, there is a unique vector v such that ϕ(u) = ⟨u, v⟩ for all u ∈ V.

Fix some vector w ∈ V. For a linear map T: V → W, observe ⟨Tv, w⟩ is a linear functional from Fⁿ → F, so there exists some T^⋆w ∈ V such that ⟨Tv, w⟩ = ⟨v, T^⋆w⟩. Each w is associated with a unique T^⋆w, so we can consider T^⋆ a function from W → V, and furthermore, T^⋆ is a linear map.

Say now that T is a linear endomorphism, i.e. T: V → V. Then we say T is self-adjoint if T = T^⋆ and T is normal if TT^⋆ = T^⋆T. Obviously all self-adjoint operators are normal.

The Spectral Theorem states that T has a diagonal matrix iff

• In a complex vector space: T is normal
• In a real vector space: T is self-adjoint

I've been trying really hard to avoid matrices, but bringing them up is very natural here. To use terms we have defined before, "has a diagonal matrix" is equivalent to "has an orthonormal basis of eigenvectors".

Conclusion

So that's linear algebra! A lot of detail has been left out, and we haven't even touched matrices, but it's still quite doable right?

For existing students: you will be getting a tarball of all the finished units as well as a separate tarball of all unfinished units.

A public explanation of why Math Advance has decided to place MAST on hiatus and what this means for its future is in order. This is said explanation.

We still stand by the claim that MAST's materials are some of the most thought-out and comprehensive lecture notes in the math contest community. However, our standards are not just "offer better material than everyone else" --- they are offering material that is as close to perfect as we can get. This even applies to the code that runs MAST and its content. Part of what makes Math Advance what it is now is our unrelenting strive for perfection, and these last few months have shown that we do not currently have the infrastructure or manpower to meet these high standards. Most math organizations (and organizations in general) would be content to keep offering a program like MAST under these conditions, and perhaps we would do more good than harm if we did so. But my personal philosophy as Math Advance's director is that it is okay to sacrifice the scope of our organization if it means the things we keep doing are as close to perfect as possible.

Currently we have our hands fill with the contest side of things: further developing mapm and its ecosystem (many thanks to EvilMuffinHa for agreeing to help out!) as well as rewriting the MAT website and its contest portal to align closer to my web development ethos. Personally, I am also working on a Git hosting service because I'm very picky about my infrastructure. In short, my manpower is limited and my priorities have shifted.

Teaching has, in part, been a selfish endeavor for me. Of course I enjoy seeing my students improve, but I also enjoy making my content as good as possible for its own sake. Since it's not currently my main interest (outside of my draft of an introductory competition geometry book), I hope you'll forgive me if I don't pursue it as frequently. Plus, my teaching philosophy has dramatically changed, so I need some time to collect myself and reorganize my materials in a way that makes sense to me.¹

As for the future of MAST? I don't think we're going to see that exact progrma back as is, though I am inclined to keep the name (so that way we can keep the logo too). However, I do hope to continue teaching with Math Advance in the future! It's more likely these materials will be used for shorter, more intensive math camps during the holidays. Or I might use them as bases for private tutoring. We'll see what happens.

I'm hopeful for the future. Perhaps this hiatus of MAST will not only represent the closing of a certain chapter for Math Advance but also the beginning of new opportunities. Putting MAST on hiatus, in a way, is like returning it to the staging area of a Git repository. Of course, any staged changes there may never make their way to remote regardless of how much work has been put into it, and just the same, there's the real possibility that nothing more will come out of the work we've put into MAST.

However it goes, I'm grateful to everyone for their support.

The original email sent to MAST students:

I emailed this to the staff a while ago, and after careful deliberation I have decided that it is best to indefinitely suspend the MAST program.

Besides the reasons I have laid out in the original email, one more stands: we are in tech debt. The MAT website is a complete disaster as it stands, though I will not elaborate until after a rewrite is complete for security reasons. I'm also writing Git server software to improve Math Advance's infrastructure, and Math Advance is undergoing aggressive expansions on the contest side --- we plan to hold a Winter MAT (more details on this later).

The long and short of it is, I don't have the time/energy to spend on MAST, and frankly it is not my first priority anymore. When Math Advance's tech stack becomes more stable, MAST may come back in some shape or form!

Forwarded message from Dennis Chen on Thu Jul 7, 2022 at 8:26 PM:

There's a good chance I end up forwarding this verbatim to the students, so I'm going to directly address them. And to the students, if you end up seeing this, note that this was originally a staff email --- so many of the concerns I bring up will be pertinent to them and only them.

MAST is waning, and though the lecture spree did redeem the end of Season 4, it is obviously not something sustainable at that level. We've been on minimum maintenance mode for maybe 10 months at this point, and it's about time we put a name to the decay that's been going on. Let's call a spade a spade: MAST has fallen in a state of disrepair because there is no one willing to work on it, including myself.

Many students may not know the difference between MAST and Math Advance, or that the latter even exists. So, to clarify: Math Advance is the parent organization of MAST, and we are also the organizers of the MAT (mat.mathadvance.org). Math Advance is still alive and well, but our interest (and our start) has always been with writing math contests, which is why we have recently put much more attention into it.

Part of this "lack of interest" is largely selfish. I've personally been more involved in writing programs that do useful things for Math Advance and the math contest community at large. The culmination of my last 6 months of work is in programs like mapm (mapm.mathadvance.org), the Math Advance website (mathadvance.org), the MAST website (mast.mathadvance.org), the design for the MAT contest PDFs, and so on. However, this shift to programming has largely whittled away at my familiarity with the math contest scene since I haven't been practicing or teaching as often.

I still am doing technical writing, but most of those efforts have been focused on a geometry book for beginners to contest math (mostly at the middle school level). As such, I am left with little time and energy to work on MAST materials, and my familiarity with them has honestly slipped past the point where I can recover. I still faintly remember the philosophies I put into Perspectives, but that's about it.

At the same time, I recognize that a lot of people have gotten value from MAST. I don't want to take too much credit --- but I have noticed that with the new batch of students, MAST has played a role in their improvements, however small it may have been. Many USAJMO qualifications, multiple MATHCOUNTS Nationals qualifiers (and some from really difficult states, too) --- I don't have the time to name you all in this email, but I'm proud of every one of you. My only hope is that you pay back the kindness, generosity, and patience you've encountered in math contests in some shape or form, even if it's not to the math community. (You're all really talented, and Math Advance would appreciate seeing some of that talent here :)

Beyond just winning titles in math contests, I'm glad to see that MAST has also formed a community. We've competed in a ton of math contests together under the Math Advance banner, played games together in voice chat, and had some of the silliest conversations. None of this is going away. If any staff are still willing next year, we will continue to organize teams for college math contests.

So here's what's going to happen: I probably will not host another round of applications, or actively give units to people anymore. Feedback on submissions definitely will not be happening, not that it has been consistently happening anyway. I will still be answering questions in the MAST Discord, whether they are about our handouts/contests or not.

What I will do is probably send a tarball with all of the MAST materials, both what has been migrated onto the website and the units which have not. There is no official guarantee for quality for the units which have not been vetted, but anecdotally speaking, most of them have some sort of value, even if they aren't quite up to par. This way I do not have to respond to any unit requests, which further reduces the workload I get from this program. (It's not like I really decline any unit requests anyway.)

If I end up doing this, I request that you do not distribute any of our materials to the wider math contest community. You guys have been really good about this, so this is more just to let you know that the redistribution policy is the same. I want to keep the option of using these materials for some other purpose open, possibly including the revival of MAST in some shape or form.

Thanks to everyone who has ever contributed material to our program. Thanks especially to Prajith, who has expanded our workable content base by writing solutions for many units. Your work will not go to waste --- though MAST may not exist the same way it did two years ago, I do have plans for these materials sometime in the future! One of Math Advance's primary goals is to teach, and I fully intend to continue doing that in some shape or form.

Finally, thanks to everyone who's donated the suggested $40 to Math Advance (no one chose to donate a different amount). Though the program you donated to, MAST, may not be around anymore, we will still use these funds to support Math Advance's greater mission of supporting math education. A couple of things that donations will help cover:

--- Prizes for math contests (which we will be hosting more of, as MAST's hiatus will provide us with more focused energy in this regard) --- Server costs & domain costs --- Registration fees for college contests

I've learned a lot from teaching MAST for the last 2 years. I hope you learned just as much participating. Thank you to all the students and staff for supporting me for so long --- it's been a blast.

As the title suggests, I spent a good chunk of time in California watching Alice in Borderland (which, by the way, is much better than Squid Game). A brief rundown of my summer:

• Went to China
• Actually did stuff in very hot places
• Got my wisdom teeth extracted in a place with reasonable weather
• Did not want to live (and did nothing) for two weeks
• Got back and got (mildly) sick
• Did not want to live (and saw nobody) for 4 or 5 days
• Saw people

I don't think a listicle of things done makes for interesting reading, but I'm at college now and barely remember any of it.

Vaguely, the way I remembered feeling was "ok well I just wanna get this over with and like be on my own". I didn't really have much to do, although at least on the last few days I started seeing people (my boy Sword Looting went on a trail run with me). But yeah it is nice to just, maybe have problematic decisions like staying up till 1 or weird shit like showering twice a day (I will never do the opposite) or even technically irresponsible but more agentic things like skipping shitty O-week events with friends. (I hate the word "agentic" with a passion but it actually worked perfectly here.)

Also I just decided "fuck it I ain't seeing anyone" on Thursday, the day before my flight. Probably a good decision so my social battery doesn't go to 0 before I even go to college.

Day 0+1 (Locked Out)

Friday 4:00 AM I woke up for a 7:00 AM flight from Oakland. I don't know why we left so early.

Anyway, as a result of being lazy last night and my mom being convinced we had no time for me to do some last minute packing (she was wrong!) I was left without

• my Switch
• Secret Hitler
• Ethernet cables
• my passport(!!!)

Anyway the point is I am an idiot. And apparently now I am coming back for Fall Break because I really need some of these things (my passport) and really want some others (my Switch and Secret Hitler if I don't just cave and order it on Amazon).

After arriving we did some shopping at Costco, and I was too lazy to actually go and find a restaurant so we just ate Costco pizza and went to the hotel. The hotel was literally in the middle of nowhere, and to make things worse the people next to us were blasting TV really loudly at 1 AM, which made it hard to sleep. My movein time was 9:45 AM and I should've woken up around 8:30, but I got woken up at 7:30 ish (I am guessing here) by the fucking TV and was like "ok I never even fell asleep I can go back to bed".

Then I wake up at 9:55. At first I think my dad is playing a prank on me; no way I slept that long. But no, it is 9:55 AM and I am definitely late for my move-in time. "Well, crap", was not the first thing --- or even one of the things --- that I thought, because I knew (read: assumed) move-in times didn't really matter. I was right.

After moving in I kick my dad out because I like having friends. Then we eventually osmosised into a large ass group and it was really funny when I ran into him again because I forgot to take my blazer out of some luggage (and so I would've accidentally shipped it back oops). And I won't lie I forgot the rest of what happened but I vaguely think it involved the gym a little? Also I met up with a lot of people that day, so I didn't do the two things I wanted to:

• Sleep before 11 PM
• Study linalg

The one thing I do remember, though, was getting locked out of our dorm for 2 hours and sitting outside because our door key swiper thing ran out of battery?? And then watching Axler videos with my roommate while my other roommate slept through the loud-ass knocking on the door. Eventually the RA contacted campus police who opened our door and contacted maintenance to fix the lock for us. So we just permanently left it open before then.

Also I still owe people money and should pay them back eventually instead of being a bum.

Day 2+3 (Bonding over O-Week by hating O-week)

One of my friends asked (rhetorically) "how do you even make friends during O-Week this shit is so boring". (Because it is. Pro tip if you go to CMU next year, skip almost everything because it sucks, go explore Pitt with your friends instead!!) Or maybe I asked it. Anyway, someone else answered "by shitting on O-week events with the people next to us". Honestly, he's not wrong.

The actual O-week events are by far the worst part of O-week. The president's welcoming address was boring and made me play Clash Royale for the first time in literal months (and my roommates too). The SCS speech was like, "bro why are you flexing 50% yield and stop I don't care about the details let me leeeeave." The floor event was like "OK let us now memorize names" (I don't care). And the Day 3 (everything before was Day 2) house meeting was like, "ok and now the RAs will socialize while you sit here and do nothing for two hours".

The worst part isn't that these O-week events are nominally mandatory. It is that everyone else goes there so you can't actually plan to hang out then, although thank jesus it is beginning to change. At least I had a few friends who were sane and didn't go to more house meetings from 3 to 5 or some bullshit?? And when you get there it's literally kindergarten. The lunch tables are too reminiscent of high school, and not in a good way. (Although that's just the atmosphere; the people I eat lunch with are amazing.) And we literally had to "fill time" so we could get "properly dismissed for lunch". Like bro I'm not some high school freshman. I can vote so please just let me stand up and leave.

We also had a math placement test. I didn't ace but did pretty well (as far as I know, maybe the best?), so it was ok. Better than I expected and worse than I hoped for.

I complain a lot here, but by and large the positives outweigh the negatives by far, especially since I skipped a lot of events (mostly to recharge before placement test, and because the placement test overlapped). I like CMU and I like the people here. But the most novel thing these two days is the O-week events, and also the least confidential, so it is what I focus on here.

Also my gym ratting + two showers a day regime is going well. I will probably stay consistent because I still study a LOT. Especially now the placement test is over.

Day 4

I dunno if it was post-test nerves or something, but I could not sleep the entire night. So around 6 AM, I just gave up and went outside (and now I am absolutely exhausted). When I got back I just started doing work and puzzles, which somehow requires the least brainpower compared to actually socializing and stuff. (Goes to show that I'm still secretly an introvert despite how hard I try to gaslight myself otherwise.)

Then we tried to register for whitewater rafting in the outdoors event:

Try to register for event

Reg is open from Aug 22, 8 AM - Aug 22, 1 AM

???

Number 1 CS school for real. Now we just have to camp the website??

The link then got replaced under our noses and we only realized when it was too late, so my roommates + our adjacent friend group didn't get our first choice (water rafting). Plus my second choice (paintballing) disappeared too, so I went with my third choice, caving. But at that point you can't really get everyone to compromise on a third choice (most peoples' second choice was paintballing too), so we all got split up.

Also I went to the floor meeting (for a few minutes anyway, right as it ended) this afternoon. That was pretty out of character for me, but I got a free shirt from it so hooray?

At some point I figured I needed to clean our dorm, do laundry, and buy eggs (for breakfast). I would've done that but got too busy socializing (I need to be more responsible) and was going to do it at midnight but things happened.

Day 5+6

After two days of not sleeping and a bunch of things going on, I just crashed.

Every time I went to a live-in camp, I would be super extroverted for the first few days, but at some point I'd just go "OK I'm done" and basically hole myself in my room working on my own stuff. (It could be worse; I could hole myself in my room doing nothing.) And these two days are when I went, "I'm done".

Day 5 wasn't too bad --- at least, I didn't realize what I'd done until Day 6 (which is when I am typing this section up). I did sleep a lot, eat breakfast in my room (instead of with other people --- my roommates were either asleep or gone?), skip the pool event I signed up for, and cut my gym session short because I was so tired, but it wasn't all going downhill. Yesterday I took the initiative to actually go grocery shopping and went to the Google field trip (which wasn't quite what I expected or hoped for it to be).

I did nearly fall asleep on the Google trip because it was a lot of "sit down and listen to panelists" and I was super sleep deprived.

Now for Day 6... I slept at 10 PM last night and woke up pretty late (8 AM ish)? I decide to skip CMU breakfast because CMU breakfast is dogshit. Unfortunately, I am not particularly experienced at cooking stuff nor particularly intelligent, and by the time I got back from grocery shopping on Day 5, the aforementioned Google trip was happening very soon. So in my haste to pack everything into the mini-fridge, I accidentally put the eggs in the freezer instead of cooling.

The eggs were fucked.

So instead of having something nice for breakfast, I was basically just subsisting on the bread and fruits I bought a while back. I had to chuck the eggs out as well. It seems small --- and honestly, it really is --- but I became irrationally upset for a few hours. I say became, but right now I'm still holed in my room playing BTD Battles 2. Orientation events might be a bad use of my time, but the difference between me skipping earlier and skipping now is that in the former case, I actively had better things to do. I was hanging out with people or at least going shopping, but now I am just a bum playing video games instead of socializing with other people.

The worst thing is, right now I am waiting on external events --- whether it be lunch, a (hopefully good) MMS result, or other people --- rather than relying on my own initiative to change anything or improve my mood. I know what I need to do, but I just can't be assed to do it. This feeling of learned helplessness is pretty awful.

Ok but whatever who cares right? Apparently I did "quite well" on MMS and need to "discuss options", which is distinctly different from "oh OK you can just choose to take 242 or not". Which means WE GETTING OUT AND WE GETTING CHOICE WOOOO

My roommate also got the same email so there'll be someone I know in my classes maybe. I dunno though because our choice on what to do next might differ.

Day 7

Yeah, basically everything went as perfectly as it could have. I'm taking 0 math classes and 2 CS classes this fall which is nice, because most people take 3 core classes. So I can just chill this semester.

Also Carnegie Cup happened. We did puzzle hunt, the counselors running it dipped, so we dipped back to my room for a while too. The logic puzzle round was nice and I solo-solved it in ~10 minutes, the rest was between difficult to impossible for me.

A thought I had near the end of the night: I think there's a point where it's better to meet new people, and then past that point it's better to get to know the people you just met. I think we crossed that point by Day 4. Orientation should not be nearly this long, and it really should've just been speeches (I guess, because they're kind of mandatory) and the fun things like Pittsburgh Connect. (More on that later!)

Day 8

I figured out what my second gen-ed will be today: 80-180 (Nature of Language). I know a couple of people in that class, and technically I'm on the waitlist but my advisor told me there were 9 open spots, so I'll basically be guaranteed to get in the class.

More importantly, today was Pittsburgh Connect. The idea is you either touch grass (go outside) or see cultural stuff (eg museums) to create, well, a connection with Pittsburgh. I chose to go caving and honestly it was so fun. And if you're in disbelief, yes, I actually went inside a cave. Like, a real one. I was surprised when I saw the details too.

We went to the upper two levels of the Laurel Caverns, and the actual caving was about 90 minutes long. (The bus ride was 2 hours each way.) I got really dirty on the trip, and I also got to experience genuine pitch black and near-silence in a cave. The highlight was the spring water though. Holy shit it tasted so good. Like genuinely, that shit changes lives. I swear I might've almost gotten high on just how good that water was.

I also started reading up on a little bit of Haskell, since I got into 15-150 (the functional programming class). I already know some big-picture things about Haskell and FP because I've just been reading about these ideas on and off, but it didn't really register that I'd be actually learning about FP this soon until yesterday. I'd heard about people on LinkedIn who got an Amazon/Jane Street/etc internship who took 150 in the fall but until now I thought those people were a myth and me doing the same thing would be a pipe dream. Apparently not.

Day 9 (last day)

Man. I was so tired that I slept at 11 and when I decided it was time to wake up, it was already 9:30. Today the goal is to actually become adjusted for classes. Not that I can do that in one day, but I'll be doing what I can.

Today I decided to figure out my schedule (i.e. what I'd be doing every day) and also dealt with a bit of MAT logistics. (Sorry it's taken so long, guys!) And, surprising even myself, I decided to attend the final floor meeting because it's the last one.

Then I got dinner with some people and we went grocery shopping but there were no eggs. And so I didn't get anything, because the apples in Scotty's Market are $2.59. Not per pound. EACH. That is pretty fucked up. Scotty, you are one disaster of a mascot dog. Get your act together.

...and so it begins.

So I've had a couple of days of classes already (literally a couple: exactly two).

First thing's first: CMU stopped serving their shitty orientation food and put us on the meal plan proper. Actually, this happened a few days before (Day 7 if I remember correctly?) but I never thought it was really worth mentioning. The food is pretty good, I really like Rev Noodles. Meal blocks are still a scam though and there's no way I finish them all.

Speaking of eating, I've basically only had time to eat two meals a day. Part of this is because of scheduling shittiness; sometimes I only have enough time to eat brunch and dinner. But another part (and honestly most of it) is self-inflicted. More on that later.

My actual classes were mostly OK. Imperative was a little slow but surely it'll catch up. Functional programming was pretty interesting and I liked the first lecture better, but I almost collapsed during the first lecture. (More on this later, and this is related to my two meals a day situation too.) I don't want to write anything mean about my writing class so the less said the better. Hopefully Nature of Language shapes up to be interesting.

This is what my schedule looks like for this quarter:

This is very chill. As I've said before, testing out means I had no math classes. But also taking zero math classes is one surefire path to brain rot, so I decided to sit in on a couple of classes and asked a friend to join me. Well, something awakened in his mind, and now I am sitting on three classes. This is my fake schedule (that reflects what I am actually doing):

If I have to drop something I'll drop Complex Analysis first. (I didn't go on Monday and I was already fucked up from four back-to-backs.) Because Complex is more standard than Formal Logic (e.g. online resources and friends who know it are more common), it's also something I can afford to put off for a semester.

My fake classes are by far the most interesting. They aren't large ass lecture halls and the professors can actually individually see people (wow).

In Linear Algebra (which is covered with abstract fields and has multilinear transformations hopefully) we just went over fields in the first lecture. Mostly just review, but it was funny when Prof. Schimmerling was making an effort to assume as little prereq knowledge as possible, got to 𝔽_p, and just thought for a second and gave up (i.e. he said "the field of integers modulo p"). I also am not sure whether he knows we're not actually registered students or not.

In Basic Logic (remains to be seen whether it will turn out to be truly basic... my guess is no) we covered "the view from 3000 feet up" and got a general overview of the difference between syntax and semantics, Godel's (in)completeness theorem, why the Peano axioms cause incompleteness, and what the heck induction actually is. He also gave some supplemental textbook recs for the course, which we later learned are on the syllabus. (The course uses no text.) And now we are on the Canvas and Piazza, which is pretty nice.

And this is why I get two meals a day some days... (and probably why I nearly fell asleep in FP lecture). Oh well!

Also, if you think my fake schedule is fucked up...

Yeah I think I really awakened something.

Anyway, I'm still hanging out with friends and having fun. Life is fine even if I technically have 80 units on my fake schedule.

Thanks so much for organizing this contest! Usually math contests are way too fast, leaving no time to think about the problems or check your answers.

Thus, I really enjoy slower-paced contests like MAT where I can enjoy the problems.

I feel the same way, which is why I designed the MAT the way I did. For posterity and for the sake of my future self, I want to record my thoughts on what makes a contest format and (briefly) what makes a contest's problems good.

Contest formats, as I wrote about in a slightly facetious manner before, should be simple so as to not confuse the contestants or distract from the main thing the contest is about: math. For this reason, I personally don't like the convoluted weighting scheme contests like HMMT use. In fact, I don't think problems should be weighted at all, and every significant US contest (well, that's just the AMC series and maybe MATHCOUNTS) agrees.

As the quote implies, contestants should have ample time to solve problems. The only complaints I see with giving lots of time for each problem is that it makes the contest take too long or makes the contest fail to differentiate between contestants. In the next two paragraphs I will address these criticisms, but I first want to expound on the benefits of having generous time limits. It makes the contest more relaxing because it allows people to work without fear of running out of time. Most of all, it gives people the chance to chew on interesting problems and struggle with them for a while, which is what math contests are supposed to promote in the first place. No one wants to sit with a test of problems they either instantly know how to solve or instantly give up on, and that is what shorter time limits force. Plus, every contestant should get a chance to see every problem, and if people run out of time halfway through the contest then what was the point of the other half?

Having longer time limits makes the contest take longer, but this can be offset by having less problems. And I think there should be less problems on most math contests. How much discussion do you reckon the average HMMT problem gets? Compare that to the MAT: Summer 2021/1, Summer 2021/3, Summer 2021/7, Summer 2021/9, Summer 2022/2, Summer 2022/3, Summer 2022/4, Summer 2022/6, Summer 2022/8, and Summer 2022/9 all have a good amount of discussion, and you can find past discussion for the problems I didn't list if you know where to look. This is 50% of all the main 9 problems across both years. I bet you every American contest (besides the AMC series) has about a 10% rate of discussion at most. Contests are largely a social experience, and if you have so many problems that contestants don't discuss because they literally forgot about doing them, you lose out on the education factor.

Maybe this makes MAT worse at distinguishing who the "best" contestants are, but top contestants really do not care that much about the competitive aspect. First off, winning an unfun contest is meaningless, and I believe math contests with less problems and more time are just more fun. But more directly addressing this criticism, which is partially correct: it doesn't really matter if a contest isn't perfect at distinguishing performances because contests are only nominally competitive. If we really wanted to spot the best 5 contestants out of 100, there'd be no reason to include problems 1, 4, or 7 in the contest. We could just have 9 really hard problems and put them together. But we also want the other 95 contestants to learn something and enjoy the contest, and the top 5 contestants probably enjoyed solving the easier problems on the contest too.

For a contest to be fun, so must its problems. This means every problem must be satisfying to solve --- it can't be too tedious, and there actually has to be some sort of idea behind it so the contestant gets something out of the problem.¹ I think pretty much every MAT problem is a good example of this: in particular, Summer 2021/2, Summer 2021/9, and Summer 2022/TB3 are worth looking at. (You can find past copies of the MAT at mat.mathadvance.org.) Beyond this, I won't talk too much about problem-writing philosophy; if you want more, see Ankan's thoughts.

Both these sets of instructions technically convey the same information. But not all information is made equal. Telling you to turn left is much more helpful, because it tells you the point of what you are doing.

Mathematics is much the same. Often, I will describe a mathematical argument as being "morally (in)correct". My friends think this is a strange thing to say. Surely mathematical arguments are either correct or incorrect. What on earth does "morally" mean?

Mathematical arguments can be seen as a series of directions. We start from a certain point (a collection of assumptions), and we arrive at a destination (a conclusion). And it very much can be the case that one set of directions is far clearer than another.

Let me give you a fairly advanced example. Consider Sylow's Theorem. It is perfectly possible to prove it without using the words "orbit" or "stabilizer" once. But when you do so, you will inevitably end up saying "turn 117^∘ counterclockwise" instead of "turn left" a bunch of times. Here, these left turns are just the theory of group actions (which include orbits and stabilizers).

Definitions and theorems are much the same. You should not just treat them as lists of causes and lists of effects.

For instance, I could tell you that the implicit function theorem says that given real/complex Banach spaces X, Y, and Z...

But your eyes would probably glaze over that. At the very least, mine did the first time I was learning about it. Why should you care?

But now let me tell you the following. Consider a function like f(x, y) = x² + 2y² − 2xy + 5x + 7y − 20.

Given some value for x, we can find a value for y such that the equation f(x, y) = 0. It turns out that we can do even better; there is some function F taking values of x into values of y such that this equation is satisfied (more precisely, f(x, F(x)) = 0). And because f behaves very nicely, so does the function F. In fact, if f is k times continuously differentiable, then so is F.

Now we have a reason to care about the Implicit Function Theorem. Equations like f(x, y) = 0 can be solved, and the function that carves out the solutions is just as nice as f itself.

Hygiene

This is why we care about mathematical hygiene. Picture an unnecessary proof by contradiction that goes like this:

Assume for the sake of contradiction that X is false. Insert direct proof that X is true. Contradiction.

This isn't bad because it makes the argument longer; it barely makes a difference to the length. Rather, it obscures the point of the argument. The point of this argument is not that we may derive a contradiction by assuming "not X"; the point lies in the direct proof of X. But using a proof by contradiction makes it seem otherwise.

Orthogonality

Being morally correct and technically correct are separate things. Of course, they are correlated: if you are egregiously wrong, then you are likely not morally correct either, and a good portion of correct arguments are morally correct.

Though it may be rare, you can be morally correct without being technically correct. Usually it is some technical issue that prevents you from being fully correct. Take the existence of the algebraic closure of a field for instance.

Definition (Field Extension). We say a field K is an extension of F if F ⊆ K.

Definition (Algebraic Extension). Suppose K is a field extension of F. We say it is algebraic if every element in K is a root of some polynomial in F.

For example, ℂ is an algebraic extension of ℝ because every complex number is the root of a real polynomial.

Definition (Algebraic Closure). An algebraic extension K over a field F is an algebraic closure if K is algebraically closed.

Notice that a field is algebraically closed if and only if it has no (non-trivial) field extensions.

Here is the theorem we are trying to prove: every field F has an algebraic closure. The proof idea is simple: consider the collection of all algebraic extensions of F endowed with the order ⊆. Then Zorn's Lemma produces a maximal element $\overline{F}$, which is algebraically closed because it has no non-trivial field extensions.

This argument isn't technically correct: the collection of algebraic extensions of F is not actually a set. But you and I ought to agree this is morally correct. Telling you this will make it much easier to understand the actual proof; in fact, reading this and the proof will be much faster than just reading the proof.

There are a lot of otherwise really smart people who swear by React. And using two completely different languages and frameworks to write a website and an API that the website calls. And using component libraries that to me just seem completely useless.

I suppose I ought to include a disclaimer: it is almost certainly not the case that React is always a bad idea. In fact sometimes it may be the best idea. My point here is that often (but not necessarily always) it does turn out to be a bad idea.

React

Let's start with React. Why on earth am I writing className instead of class? Why should I write <Link> instead of <a>? So on and so forth. There are technical reasons for this. I know what they are, thank you very much. But if we are going to depart from standards and muddle up the 1-to-1 relation between "what I write" and "what is rendered", there better be really big benefits to doing so.

Using <Link> as an example, does making your website a single-page client-side-rendered application actually help anyone?

No! You are not making the next Google Docs. Most websites with a backend are some selectively rendered text, a few images, and a bunch of HTML forms. Websites with significant client interactivity are a different story and not one I know anything about. But if your website is literally just a blog (which doesn't behave any differently for different clients, by the way), you really should not be writing it in React. It's just complexity for no reason most of the time. Oh and by the way, this isn't just "React is unnecessary most of the time", it's "Javascript is unnecessary most of the time". React is much more complicated than plain Javascript, and so the bar for its inclusion ought to be much higher.

Component libraries

Component libraries are also hard to justify most of the time. Let's take one of the more popular libraries, React Bootstrap (to be clear, Bootstrap CSS itself seems fine), and look at the first component they have: the Accordion. Here is the HTML code that gets generated.

<h3 class="anchor anchorWithStickyNavbar_LWe7" id="basic-example">
    Basic Example
    <a href="#basic-example"
        class="hash-link"
        aria-label="Direct link to Basic Example"
        title="Direct link to Basic Example">
    </a>
</h3>
<div class="playgroundContainer_TGbA">
<div class="playgroundHeader_qwyd">Result</div>
<div class="playgroundPreview_bb8I">
  <div>
    <div class="bs-example">
      <div class="accordion">
        <div class="accordion-item">
          <h2 class="accordion-header">
            <button type="button" aria-expanded="true" class="accordion-button">
              Accordion Item #1
            </button>
          </h2>
          <div class="accordion-collapse collapse show" style="">
            <div class="accordion-body">
              Body text #1
            </div>
          </div>
        </div>
        <div class="accordion-item">
          <h2 class="accordion-header">
            <button type="button" aria-expanded="false" class="accordion-button collapsed">
              Accordion Item #2
            </button>
          </h2>
          <div class="accordion-collapse collapse">
            <div class="accordion-body">
              Body text #2
            </div>
          </div>
        </div>
      </div>
    </div>
  </div>
</div>

Yikes. Why is the title of a collapsible item an <h2>? This is not good from an accessibility standpoint. This collapsible item is organized under the <h3> with the title of "Basic Examples" for Pete's sake. So clearly, it and its title should semantically be a part of the <h3> header. But the HTML code indicates otherwise. And any programmatic tool which relies on the semantics of the HTML (which by the way, includes accessibility software) is not going to work well if you generate semantically incorrect HTML. And no amount of aria tags will make up for the fact that your HTML is terrible.

Also, a button? It's not the worst thing you can do, I guess. At least aria-expanded=true is properly set. But you can do so much better for a dropdown.

<details>
    <summary>
        Dropdown code
    </summary>
    <div>
        Pretend there is some code here.
    </div>
</details>

Given how simple this is, why would you ever write this code?

import Accordion from 'react-bootstrap/Accordion';

function BasicExample() {
  return (
    <Accordion defaultActiveKey="0">
      <Accordion.Item eventKey="0">
        <Accordion.Header>Accordion Item #1</Accordion.Header>
        <Accordion.Body>
          Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do
          eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad
          minim veniam, quis nostrud exercitation ullamco laboris nisi ut
          aliquip ex ea commodo consequat. Duis aute irure dolor in
          reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla
          pariatur. Excepteur sint occaecat cupidatat non proident, sunt in
          culpa qui officia deserunt mollit anim id est laborum.
        </Accordion.Body>
      </Accordion.Item>
      <Accordion.Item eventKey="1">
        <Accordion.Header>Accordion Item #2</Accordion.Header>
        <Accordion.Body>
          Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do
          eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad
          minim veniam, quis nostrud exercitation ullamco laboris nisi ut
          aliquip ex ea commodo consequat. Duis aute irure dolor in
          reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla
          pariatur. Excepteur sint occaecat cupidatat non proident, sunt in
          culpa qui officia deserunt mollit anim id est laborum.
        </Accordion.Body>
      </Accordion.Item>
    </Accordion>
  );
}

export default BasicExample;

The <details> tag is more accessible and easier to style to boot. There is literally no downside.³

And the worst thing is that frameworks like React Bootstrap pass automated tests like Google's Lighthouse with flying colors. But look for a couple of seconds and you can already see issues. (I don't know if Accordion titles are always <h2> elements, but I suspect the answer is yes. Regardless, it's not the greatest sign if the title is a header on their official website.) And React Bootstrap is doing quite well for itself, actually. Just look at MUI's accessibility issues on GitHub.

But this isn't supposed to be an expose about accessibility.⁴ The main point here is that using a custom "Accordion" component is moronic if <details> is already available to you at no cost. Accessibility just happens to be one of the big reasons it is stupid.

A potpurri of unnecessary uses

There are a lot of reasons people have to use React, component libraries, and other similar things. Let's run through them:

• "It is more accessible." Making all of your code (including your rendered HTML) more complicated and error-prone is not accessible.
• "It resizes well on different devices like phones." Only if you apply different styles based on screen width, which is much simpler to do with plain CSS and media queries.
• "It makes my website look nicer." No. Everything is reproducible with plain HTML and CSS. It all compiles down into (semantically poor) HTML and CSS. And using plain HTML and CSS ends up being simpler. It is hard to grok dozens of 2,000 line minified Bootstrap CSS files whose default values are a mystery to all of mankind. It is much easier to understand a small CSS file
• "It is necessary to connect my frontend to my backend and create a full-stack website." Factually false. There are simpler and more robust ways to do this. Rust frameworks like Actix-web, Rocket, etc are much more robust . And it is simpler because there are less layers in the client-to-server connection: browser-website-server as opposed to browser-website-api-server. And only having one middleman is important because it allows you to guarantee consistent datatypes to represent your SQL tables (or whatever your database uses).

Now I am not going to go as far as to claim React is totally useless. This is very likely untrue. But it is the case that a lot of the reasons people think React is useful are total hogwash, at least for their specific use cases.

What do you do?

So how should you write a website, particularly a full-stack website? Two years ago⁵ I teased an answer to this question:

I hope this post was helpful! A post on full-stack websites will be coming soon :)

Two years later and I will tease it a bit more. I am planning to write a total of three entries, one of which is already done:

1. I will start with an abstract, simplified explanation of the principles of client-server communication. We will start with username-password authentication, then dive into cookies, and finally we will discuss why encryption is necessary.
2. We will add more detail to the abstract client-server model. The client is usually just the browser; the server can be divided into the API and the database. Furthermore, there are many other types of clients that work. For instance, you could use the terminal to interface with a CLI, which sends a request to a programmatic API that queries the database. The important point is, anything on the server can interact however it wants with anything else on the server, and with a few exceptions, the same is true for the client.
3. We end off the series by looking at web frameworks, discussing what a "tech stack" is, and some of my recommendations for what tools to use for creating a website.⁶

I plan to write the first two entries fairly soon. The last entry will be written eventually, likely when I have finished Glee.

Without opinions all you have are a set of facts. Opinions are the backbone of a framework.

We've all seen those side-by-side factual comparisons of GitHub vs. GitLab, Postgres vs. Mongo, React vs. Angular, so on and so forth. And they are boring. It's like it was written by a person without a soul, because rarely is an actual opinion presented. And just as importantly, these comparisons are completely unhelpful.

If you search up "GitHub vs. GitLab" and click on the first comparison article you see, chances are it won't even mention the nested directory structure, let alone emphasize it as the main sell. Even GitLab cannot articulate the main feature it has --- you know, the one that GitHub will never be able to replicate --- and instead chooses to populate their page with marketing nonsense. You can even go on Reddit, which for all its flaws at least has sentient human beings with opinions, and not a single person mentions this crucial difference.

And you're telling me the first person to realize this and write about it was a high school freshman.

Opinions like these, fervently held and correct, lead to a better command of the facts. Pushing the idea that "Git hosts should store repositories like my filesystem" further led to me drafting the design of my own Git host (something I should actually finish). My geometry students learned the area formulae in twenty minutes rather than two months because I had really strong opinions on how they should be presented.

You're unlikely to fervently hold an opinion if it's uncontroversial. You might believe that Minecraft is an awesome game, but this is also deeply obvious to the rest of the world. It's not exactly the kind of thing you need to convince someone else of. So the balance of opinions you strongly hold will be controversial.

To be clear, by "controversial" I merely mean that it is not a widely held belief. It does not necessarily mean people are actively disagreeing with it. For instance, I believe that hand-crafted puzzles are incomparably better than computer-generated ones. No one who has seriously thought about logic puzzles would disagree with that statement. But the average reaction is still along the lines of "does it really matter that much?" In other words, it's not the kind of belief that people take for granted.

So if you have strong opinions and act on them, you are going to disagree with other people. And some of them will have more experience in the relevant field than you do. (Or at the very least, it will be something that a lot of experienced people have overlooked.) And I don't mean a milquetoast sense of disagreement like "Professor, I think you wrote a plus here when you meant to write a minus". I mean something far more fundamental.

Fortunately, this doesn't mean you are necessarily wrong. There are plenty of gaps in the world.¹ Sure, on the balance, people who have more experience in a field are typically more likely to be right. But all you have to do is be right once, confirm you indeed are right where everyone else is wrong, and then patch that gap to the benefit of yourself and those around you. Boom. All the sudden, having strong opinions --- even if they conflict with other people's --- no longer becomes a bad idea.

Sure. If you are young, you cannot possibly have fifty years of experience building websites, by virtue of not having fifty years of experience being alive. And to address the elephant in the room, just so neither of us ignore the fact: I am not even twenty. I write about puzzles even though I haven't been constructing for very long, I write about math even though I don't have my bachelor's degree, and I write about programming even though I've never even held a full-time programming job. I am just some kid on the internet, screaming my thoughts into the void and hoping that someone comes out better for it.

In the grand scheme of things, I don't have a single clue what I am doing. There is a very good argument to be made that I should shut up, put my head down, and just listen to the experts. On the surface this would be a commendable display of humility. But the price is not thinking for yourself, and that cost cannot possibly be worth it.

But just between the two of us: I have a sneaking suspicion that even the older people have less of a clue than we might expect. I won't know for sure until I become older and hopefully wiser, if ever, but there are a lot of signs that often the "experts²" are clueless too.

• Who thought standardized testing was a good idea? You put a bunch of guys in a room who've been doing this their whole lives, and this is the best they can come up with? Seriously? You could not find a single better way to make sure kids were learning?
• AP Computer Science is still in Java, and I hope everyone involved in this decision drops a 45-pound weight on their foot.
• Why on earth are single-page applications still a thing?
• "You don't need a gym membership to get in shape." Thanks, mister health guru. You also don't need to ever read a book, you nitwit. But you don't go around telling people they don't need to read, now do you?
• Why are all you monsters still doing NYT Sudokus. You sick freaks.

Fundamentally, most people are clueless about most things. And I'm not convinced this changes just because people get older. And it's almost certainly not the case that it changes so fast that all your opinions as a 20-something year old or whatever become entirely worthless, because everyone else has gotten it figured out already. I refuse to believe that, because there is no way that is true.

At this point, you might be guessing the takeaway is to start thinking for yourself and form your own opinions. But that'd just be preaching to the choir. Rather, the real takeaway is not to be ashamed of having strong opinions, even when they are in deep contrast with everyone else's, so long as you've done some reasonable amount of due diligence. In other words, take your own ideas seriously, even if (or perhaps especially if) no one else is saying the same thing.

It is okay to be wrong. And it is okay not to be ashamed of being wrong, much less just the possibility of being wrong. So long as you take proper care to not unduly harm or burden other people with them, holding strong opinions serves as an outlet for tremendous growth, even if you are wrong in the end. Because all it takes is being right once.

I don't think so. And I think knowing what characterizes the two actually helps you become better at both, in particular because you know what broad class of techniques you should be focusing on.

Math and computer science, broadly speaking, both ask questions about the solutions to a certain problem. "What are the objects X that satisfy some property P?"

1. Math is focused on studying the space of solutions, without necessarily providing a way to construct or explicitly understand individual instances of solutions.
2. Computer science is focused on studying procedures to construct solutions, without necessarily providing you with a broad understanding of the space of solutions.

I think this is best explained through example.

Math: Borsuk-Ulam

Assume the Earth is a perfect sphere and temperature/air pressure vary continuously across its surface. There exist two diametrically opposed points with identical temperature and air pressure.

We will prove a simpler theorem. Consider the circle S¹ of radius 1 in ℂ and a continuous function f : S¹− > ℝ. Then there is a point x ∈ S¹ such that f(x) = f(−x).

Proof: Consider the function g(x) = f(x) − f(−x). Observe g(−x) = −g(x). Pick any x; either g(x) = 0, in which case we are done, or not, in which case the Intermediate Value Theorem provides us with some y between x and −x such that g(y) = 0.

(Note the zeroes of g are precisely the points which f(x) = f(−x).)

The Borsuk-Ulam theorem gives us insight into the solution space (i.e. that it is non-empty). This is a typical example of a "non-constructive proof", where we learn that a solution exists, but the proof is no use for actually finding the solution.

Computer Science: The non-decreasing multiples of N

The source is AtCoder ABC 443 Task F.

A number is called non-decreasing if its digits don't decrease, e.g. 1123. What is the smallest non-decreasing multiple of N?

We can easily modify this solution to yield all non-decreasing multiples of N (just don't terminate) in increasing order. So we have a reasonably fast way to generate all such multiples of N.

Yet this solution provides no insight into what properties these non-decreasing multiples of N have. How many of them are there? How quickly do they grow? We have no idea.

git add .
git commit -m "backend refactor"
git push

later and the entire codebase has been upended. It's impressive, really, and I say this without a hint of irony. To immediately start executing massive changes without faffing around for weeks is genuinely a skill worth emulating.

Unfortunately, that is the only part worth emulating. Because when you drop that many changes all in one commit, any semblance of thought that went into them is completely lost to time. And it really was just a semblance of thought to begin with, because there is just no way to write code that fast while thinking deeply about what you are doing. There are no explanations, only bugs and bad UX choices.

The consequences are obvious. You wake up the next morning, everything is broken, and it takes the next few days just to fix things up. And if there are a few parts that you want to revert later for whatever reason, you have to trudge back through the codebase, figure out what does what, and manually undo the changes one by one. After all, git revert will do you no good if you are only nominally using Git in the first place.

I can't help but think: if only you took the time to properly think your changes through and make good commits, we wouldn't be in this mess in the first place.

The pitfall

But this is not just an essay about Git. Rather, it is about a common pitfall that people fall in: trying to do things quick rather than doing them right.

Picture this. You're taking a calculus class, you don't understand differentiation, and there's a test on it tomorrow. So instead of sitting down for a week, asking yourself "what is differentiation really?" and shoring up your understanding, you memorize the derivative of xⁿ and sin x. And that is enough to pass, so you rinse and repeat.

If I lay it out like this, the problem becomes obvious. If you never take the time to understand even the most basic of concepts in calculus, of course it seems impossible no matter how much effort you put in. Yet what a pernicuous trap it is.

You need to put in some effort to do anything meaningful. That is just a fact. But we extrapolate too far. In our heads, hard work looks like staring at a textbook for six hours a day and grinding derivative exercises, rather than sitting on a couch and vaguely wondering "it's a little weird that I don't actually know how we get from rational numbers to real numbers". It looks like writing thousands of lines of code in one night rather than taking a walk and asking why access control in every Git host sucks so much. But working hard and actually getting something done are not the same.

Of course you need to do exercises in order to understand calculus. And yes, I know that all my brilliant ideas for a Git hosting service won't do me any good until it exists. But you also need to ask the fundamental questions: what is really going on here? How can I make these complicated ideas as simple as possible? And how would I explain it to another person?

Category theory

The right thing to do when you don't understand something is to take a step back and ask: "What underlying idea am I actually missing? Why don't I understand this?" This much is obvious. Yet knowing this does not make you immune to falling in the pitfall. At the very least, it didn't save me.

A few months ago, my algebra professor gave a very fast presentation of category theory. If you understand the universal mapping property of the free group, you can easily guess how it went. First we discussed categories, then functors, then natural transformations, and finally adjunctions. As a homework exercise, we were asked to prove that some pair of functors were adjoint, and I knew it was a really trivial verification.

Yet I could not for the life of me understand what an adjunction was. So I look at the Hom-set definition and I am immediately greeted by more confusion. What is a Hom-set? And how on earth do you get a natural transformation between two Hom-sets, I thought you needed a functor for that? And what is a natural transformation, actually?

So I kept spinning in circles. I stared at the definition of an adjunction, digged through the terms that I didn't understand, realized I still had no clue what an adjunction is, and went back to staring at its definition. A few days later I push some symbols with the counit-unit definition and hack together a solution. In the process I gained very little understanding of what was happening.

Now fast forward to a few weeks ago, when I decide to bite the bullet and write a category theory section for my algebra primer. Now I have to step back and think: what is a category, really, and how do I explain it? How do I build up to a functor? A natural transformation? And an adjunction?

A week later and the concept of an adjunction is crystal clear to me.

In the process I realize that many of the things I wrote were pointed out by my professor in class. So really, if I had taken fifteen minutes to think "why did he talk so long about reversing this arrow, and how do I use that fact to update my mental framework of category theory", I would have understood the hom-bifunctor (and by extension adjunctions) half a year earlier.

Burnout

I have a confession to make. Officially I should be re-implementing email-password authentication for work. But I'll be honest with you, this is probably the fourth or fifth time we are doing this, and I've gotten really sick of it. (In fact, this is the reason I thought to write this essay to begin with.)

But I'm not here to complain. My point is that you and I are both smart enough to tell when we are running in circles. Re-doing something because you didn't think it through the first time is a sure-fire recipe to burnout.¹ And the worst thing about this kind of burnout is, when you feel the work you've done before is pointless, you're right.

So how do you avoid this? I confess I do not have all the answers, but here is what I think. Sitting back on a couch and thinking for half the day is nice. It is relaxing. It doesn't carry much risk of burnout. So it is tempting to conclude that you have to trade all of these comforts away to do real, hard work. I admit you have to make this sacrifice sometimes. But I am becoming increasingly convinced that most of the important work gets done in the shower, on a train, or by nature, rather than at a desk.

I've talked about this stuff before when writing about why I do math, but I wanted to write a standalone piece to emphasize the ideas I'll discuss here.

Prerequisites

Basic familiarity with sets, logic. Familiarity with "different sizes of infinity" is also helpful (for instance, that ℕ is smaller than ℝ since no function from ℕ to ℝ covers all the elements of ℝ).

What is independence?

As alluded to in the prerequisites, you may know from Cantor's Diagonalization Theorem that |ℕ| < |ℝ|. But is there some set S whose size is strictly in between, i.e. |ℕ| < |S| < ℝ|? This question is known as the Continuum Hypothesis.

You may have heard that the Continuum Hypothesis is "independent of ZFC". So what does that actually mean?

What is ZFC?

It turns out mathematics is governed by a set of rules which we call "axioms", and the language of mathematics is really the language of sets. For instance, we can encode the natural numbers as sets. And ZFC does two things:

1. It gives us certain sets to work with, such as the empty set and the set of natural numbers¹.
2. Given a collection of sets, we can perform operations on them to get new sets. For instance, given two sets A and B, we may find their union A ∪ B.

This is a very imprecise picture, but it will do good enough.

At the same time, mathematics may be considered the study of whether things are true or false. And formally, a proof is a syntactic object which starts from a list of assumptions and derive a conclusion. For instance, if we start with a ⟹ b and b ⟹ c, we may conclude a ⟹ c. And so on. The point is, we are given a list of rules which tell us how we may combine assumptions to reach conclusions. As another example, we are allowed to combine a ∧ b with a ∧ ¬b to derive a. These syntactic considerations are what we will refer to as "first-order logic".

The point is, ZFC (along with first-order logic) describes how mathematics ought to behave.

Many worlds

I'll give you the punchline up front: there are multiple worlds of mathematics satisfying the rules of ZFC, and they are genuinely distinct.

We've just established that "mathematics" is "anything satisfying the rules of ZFC." Let me give you an analogy: When we study linear algebra, we study "vector spaces", which is "anything satisfying the rules of a vector space". Importantly, we are not studying just one particular vector space: when we create definitions such as "linearly independent", or prove theorems such as "all linearly independent spanning sets of a vector space have the same size" (this is the definition of dimension), we are doing this for every vector space, because we are only using the language and axioms of vector spaces.

The same holds for mathematics. Who says that ZFC only describes one world of mathematics?

Common misconceptions

If you're anything like me, you may have watched certain YouTube videos that say "we cannot prove whether the Continuum Hypothesis is true or false." This is technically true, but it is some very misleading phrasing.

In particular, if you were to explicitly unpack your assumptions about what this means, they might look something like this:

1. There is one world of mathematics.
2. There is no proof that the Continuum Hypothesis is true. There is also no proof that it is false.
3. Furthermore, we somehow know that we can never produce such a proof.
4. So I guess we'll just never know whether the Continuum Hypothesis is true or false, and we can never know.

This is completely wrong. Here's what's actually going on: assuming there is a world of mathematics satisfying the rules of ZFC, there is some world of mathematics where the Continuum Hypothesis is true, and another world of mathematics where the Continuum Hypothesis is false.

Godel's Completeness Theorem

From now on we abbreviate the Continuum Hypothesis as CH.

So why does "CH is independent of ZFC" imply that there is a world of ZFC mathematics where CH is true, and another world where CH is false? On the surface, "there is no proof of CH or (not CH)" is a statement about proof manipulations (syntactics) while the statement about multiple worlds is about actual worlds of mathematics obeying the rules of ZFC (semantics). But as we will soon see, our system of syntactics and semantics are carefully designed to be tightly connected.

This shows itself in Godel's Completeness Theorem:

Suppose a set of rules does not lead to any contradictions; we call such a system consistent. Then there is an object satisfying those rules. (The converse also holds.)

A few other related facts:

1. If I can prove something given a set of rules, then it is true for every object satisfying those rules. (For instance, if I prove "every basis in a vector space has the same dimension", then it is actually true of every vector space.)
2. And importantly, if I can't prove a statement given a set of rules, then there is an object that doesn't satisfy that statement. (For instance, since I cannot prove "every vector space is finite-dimensional", then there exists an infinite-dimensional vector space.)

Given these rules, here is how you prove that CH is independent of ZFC. Start with some universe of ZFC. Use it to construct a universe of ZFC that satisfies CH and a universe of ZFC that does not satisfy CH. Now we can't prove CH or not CH, since there is a universe of ZFC that breaks each of these statements. Voila! We have independence.

Godel's Incompleteness Theorem

But this isn't actually the end of the story. So far, the impression I've been giving is that "there is a universe where CH is true and one where CH is false." But this isn't really the correct picture.

Instead, it would be more accurate to say the following:

There either exists no universe of mathematics satisfying ZFC, OR there exists a universe where CH holds and a universe where CH does not hold.

Furthermore, we have no idea whether there exists a universe of mathematics satisfying the rules of ZFC. Now, why would we bother studying a system of rules if we can't even show there's a universe of mathematics satisfying these rules?

The short answer: we have to. Godel's Second Incompleteness Theorem² says

No system of mathematics can prove its own consistency.

In short, we cannot prove any system of mathematics is consistent without working in a stronger system of mathematics, which we then need an even stronger system of mathematics to show the consistency of, and so forth...

This means that, at some point, we just need to take the consistency of our system for granted. We can rely on heuristic arguments such as "we've tried really hard and haven't found a contradiction" and "the rules of ZFC seem simple enough." But we can never know for sure... unless we find a contradiction. (For example, naive set theory has a contradiction in Russell's Paradox, so we know for sure it is not consistent and need to toss it out.)

Finally, we should talk about why inconsistency results must exist, regardless of which system of mathematics we work in. This is a consequence of Godel's First Incompleteness Theorem, which says

Given any system of mathematics, there exist statements that can neither be proven nor disproven.

This necessarily implies that there exists a world where the statement is true and another where the statement is false. Why? Because if there were no world where the statement were true, then it would be false in every world, which means we can disprove it by Godel's Completeness Theorem. Similarly, if there were no world where the statement is false, then it would be true in every world, which means we can prove it.

What does all of this mean? If we have a system of mathematical rules, we don't know and can't know how many worlds satisfying these rules exist (unless we derive a contradiction in these rules). However, we know that there cannot be exactly one such world.

Independence is a good thing, actually

We know that no matter how many rules we tack on, we will never get a collection of rules that carves out exactly one universe of mathematics. But we can at least try to patch up our theory. For instance, if we know CH is independent of ZFC, why not just assume that CH holds? That could make proving certain things simpler. After all, we literally have more assumptions to prove theorems with. So why don't we?

The core of mathematics is to prove things in as much generality as (reasonably possible). We could, for instance, study linear algebra only in the vector space ℝ³ and prove theorems such as "every basis of ℝ³ has the same size". That certainly would make things easier, but we would be losing a lot of insight. And sometimes we want to focus on specific vector spaces, such as finite-dimensional vector spaces. Then it makes sense to tack on an extra assumption. But if we don't need it, we really shouldn't use it.

Studying ZFC mathematics is much the same. For instance, all of real analysis still holds whether CH holds or not. So when we are proving, for example, that "ℚ is dense in ℝ", we aren't just pointing to a specific world of mathematics and saying ℚ is dense there. Rather, we are making a statement about the entire multiverse of mathematics.

Isn't that neat?

Storytime (skippable)

Previously I had a hare-brained idea to store the metadata of a repository (who has permissions to act in this repository?) as an actual file in the filepath. It suffices to say this is a terrible idea, but it would have worked just fine with my previous design.

Then I started thinking about the goals of Glee. At the start, Glee was meant to be a host that didn't need to scale, but now my goal is for Glee to scale towards large single-organization hosts. (Because otherwise it is utterly useless and you can just use cgit.) The underlying complexity, however, will still remain very low, somewhere between cgit and sourcehut. It is the goal that

• all of the design decisions can be explained in a single medium-sized document,
• understanding the design is sufficient to use the software proficiently.

But this post is not meant to be that document. I will hold off on that until I have Glee in a somewhat usable state.

Now, if we want large single organizations to use this for enterprise-ish stuff, we really don't want random people creating accounts. (I also cannot be bothered to implement a "send registration email" feature.) The logical flow is that if you are an admin of a directory or repository (which we will henceforth call a path), then you may give an email address (say dchen@dennisc.net) permissions to (say) write to the path. If dchen@dennisc.net is already associated with an account, then that account will now have permissions to write to said path.

If not, an email will be sent containing an invite link, with which dchen@dennisc.net can use to register. In the database, we will note that this invite link should also give the new account permissions to the path (say) a/b/c.git. So we will store the permission write: a/b/c.git in the associated database entry for that invite link, and when the invite link is used, that permission will be added for the new user.

Now what if a/b/c.git moves to a/e.git after the invite link is created but before it is used?

Oh.

The correct way to deal with this issue is to instead point to a unique id representing the path, one that doesn't change. Which calls for storing information about paths in the database.

Directories and Repositories

To motivate this section, I will say what I have said before in many of my previous Glee posts. Glee is about storing your Git repositories as a filesystem.

We are going to use a standard tree structure in Postgres. We will have a table Directories and Repositories. The fields of each are going to be

• id: a uuid
• public: whether the directory or repository is publicly viewable, an optional boolean (no value means "inherit")
• name: the name of the directory or repository, for example c.git
• parent: a nullable foreign key pointing to the uuid of a directory
• permissions: a list of read/write/admin with user ids, e.g. [read, b3994226-6761-456d-879c-7b18facbbd81]

Of course, the parent of the root directory / will have no parent. It will be the sole path with no parent. It also will be the sole path we cannot move.

For now I'm thinking the struct representing this unified model in Rust should be DirRepo in backend.rs. Or maybe just Path, but that is not the most ideal name because it conflicts with a filesystem path, and Path implies a complete path rather than just one step (i.e. current file plus parent).

We will index the column id in the table Directories. That way we can emulate ls for the directory b3994226-6761-456d-879c-7b18facbbd81 by simply searching for anything with a parent id of b3994226-6761-456d-879c-7b18facbbd81 and have this query be efficient.

We will have to validate that name does not contain any / characters upon any client POST request for obvious reasons.

Handling redirects

Now handling redirects is trivial, which means we will do it. We will have a table of Redirects which store

• parent
• name
• link: which ID the redirect goes to.

Let me give you a concrete example to explain how resolving redirects will work. Suppose we rename a/b/c.git ->a/d.git. As expected, we look at the entry forc.git`,

• change its name to d.git,
• change its parent to a (more accurately the id of a).

Furthemore, we create a Redirect with

• parent: whatever the id of b is (this is the parent of c.git before rename)
• name: c.git
• link: whatever the id of c.git was, i.e. the id of the new d.git.

Note that the path a/b still exists, we just moved c.git. Here is what happens when we try to navigate to a/b/c.git:

• There is a directory named a with parent /.
• There is a directory named b with parent a.
• There is no directory or file named c.git with parent b. But there is a Redirect with the name c.git with parent b that points to d.git. So now we look at d.git and get a repository.

To be clear, when we say "with parent /", we really mean that its parent is the id of the root directory, etc.

(Basically this is the idea I had with symlinks, but it solves the problem of changing identifiers because we use a static id.)

To demonstrate the robustness of this idea, suppose we now move a/b to a/e. We still want a/b/c.git to go to the correct repository. What happens?

• There is a directory named a with parent /.
• There is no directory named b with parent a. However, there is a redirect with name b and parent a that goes to e (which has path a/e).
• There is no directory or file named c.git with parent e (remember that when we do parent checks, it is with the id; saying "with parent e" is merely a shorthand). However, there is a redirect with name c.git with parent e that points to d.git. So now we get the same d.git, precisely as desired!

In fact, a/b/c.git will always redirect to that same repository until a new directory or repository is made at that same path.

And these redirects persist until they are "overwritten" by a new path at the same location. When the overwrite occurs, we will delete the redirect. The leading principle here is very simple:

The combination of name and parent id must be unique among all directories, repositories, and redirects.

That means there are no unused redirects lying around, meaning that we never have to prune redirects. So our analogy of a filesystem with repositories and directories can be extended with redirects. Now we just have two types of files: redirects and repositories, and directories.

What do we do when we try to create a new directory/repository and there is a conflict?

• If the conflict is with a redirect, simply delete the redirect.
• If the conflict is with another directory/repository, forbid the operation.

How are we concretely storing repositories?

Now it would be stupid to actually perform a filesystem move every time we do a "virtual" move in the database. The correct answer is very obvious: we store the repository with uuid b3994226-6761-456d-879c-7b18facbbd81 in the path b3994226-6761-456d-879c-7b18facbbd81.git.¹ That way when we resolve a path to a repository, we merely need to look at uuid.git in the filesystem. Furthermore, because we never move repositories in the actual filesystem and never change uuids, any bug with filepath resolution is fixable. This means we will never corrupt our data with moves, because we are never changing the underlying data; bugs will only appear due to incorrect filepath resolution.

User permissions, again

Specifically we will talk about

• how users ought to be invited
• how permissions will be managed on the frontend

because those are the only things which I have changed the design of.

A link to a special page to "manage permisisons" for a directory/repository will appear if you have admin access. We will not be modifying a raw TOML file because that is a bad idea. Here is our new approach:

When resolving whether a user is an admin, we should also determine whether they have directly been defined to be admin (i.e. in the current directory or repository) or whether they have inheritd admin from a parent directory. We will say an admin is an Inherit Admin if they have inherited and a Direct Admin if they have been directly defined as an admin.

An inherit admin will be stronger than a direct admin. So if we have determined a user is a direct admin, we also must check whether they inherit admin as well, since making someone a direct admin on top of being an inherit admin should not reduce their permissions.

A direct admin cannot delete admins in the current directory/repository. An inherit admin can delete direct admins in the current directory/repository.

Regardless of what permisisons you have (read/write/admin), the main page of the repo will tell you what permissions you have upfront. (Many other sites are awful at doing this.)

Admins, whether inherit or direct, can see both who has direct permissions on the "manage permissions" page. You may think that resolving who has inherited permissions might be complex, and you would be right, but we already do this work when determining whether to show the permissions page. Instead of just resolving permissions for one user, we will create a list of all users and the permissions they have. For example, we might say

• Alice is an admin of the repository
• Bob inherits admin from /gym
• Charles inherits write from /gym

However, it might be privileged information that Dean is an admin of /, and it would be bad if a low-level admin saw "Dean inherits admin from /". Suppose the highest parent directory we inherit admin from is /path. Then we only want to show users who inherit permissions from /path or lower. So when we traverse the tree to resolve permissions, we will be keeping track of

• the highest parent directory you inherit admin from,
• and the lowest parent directory every other user inherits their strongest permission from,

where "high" means "less deep" and "low" means "deeper".