Inventing a new puzzle variant

For posterity’s sake, I want to record how I set a new puzzle variant (both coming up with the variant, and actually verifying that a good puzzle can come out of it by setting one). The main examples I have in mind are Alphabet Soup, Fillomino Chaos Construction, and Fillomino Chaos Construction 3. Spoilers ahead!

Everything beyond Step 1 should be generally applicable for puzzle setting.

Step 1: The Variant

Come up with something thematically consistent.

You want your objective and clue types to interact in a natural way.

• In the Fillomino Chaos Construction series, fillomino and chaos construction are both region-building variants, so it is only natural for your regions to be built out of smaller fillominoes. Naturally you need clues which interact with the fillominoes in some way (which naturally gives rise to the circle clue).

  So why the square clue? Because it creates a natural tension with the circle clues. The larger a fillomino, the fewer other fillominoes can be in the same region (for both arithmetic, i.e. “only 9 cells in a region”, and geometric reasons). In particular, this allows cells clued with a circle and a square to be very powerful, which is a tool I’ve used in both Fillomino Chaos Construction 1 and 3.

• For Alphabet Soup, I came up with the idea of clue placement (placing them in the correct regions). To keep the presentation appealing and thematic, I chose to only use one type of clue (thermos) and had a theme for their shapes (alphabet letters).

More generally, there are two ways for a variant puzzle to be a distinct variant: 1. a distinct objective, 2. and/or distinct cluetypes.

For example, Alphabet Soup only had a distinct objective (place the clues) without having a new cluetype (it used standard thermos), while Fillomino Chaos Construction had both a distinct objective (subdivide regions into fillominoes) and new clues (circles/squares/etc). Any standard variant (Killer Cages, Renban, etc) is an example of same objective, new clues.

Under this framework, a puzzle like Alphabet Soup is actually pretty rare. Whenever you have a new objective, you usually need it to interact with the original puzzle genre. For example, any Chaos Construction puzzle needs some clues which interact with the region-building. Otherwise, what is stopping you from making the regions 9 rows? But Alphabet Soup (place the clues in the correct regions) will interact with the base puzzle just by virtue of giving digits (it breaks the symmetry and gives you a chance of resolving into a unique puzzle).

What makes a good variant anyway? The rules (objectives and cluetypes) must be strong enough to force interesting deductions, but not so strong that you can hardly place a clue without breaking the puzzle (i.e. leaving yourself with zero solutions). In general, the way you set a puzzle is by using clues just strong enough to whittle down the possibility space in an interesting way, but not so strong that you whittle it down to zero.

However, you will be surprised at how wide the range of clue strengths which can set a puzzle are. Alphabet Soup is very high-powered (either your thermos are doing very little, due to how much freedom you have in placing them, or they must be very restrictive and hence can easily lend itself to zero solutions), whereas something like Jesper’s Fillomino Skyscrapers are somewhat weak: how much can a skyscraper clue tell you, particularly when the fillominoes can be massive?¹ (In fact, the break-in combines several weak clues which just barely give you something to work with.)

In any case, if you want to know whether your puzzle variant is feasible, the first thing you have to check is whether you can set an interesting break-in. Speaking of which…

Step 2: Break-in

When you set a break-in, you want your clues to both be strong and flexible.

1. Your clues should be strong: particularly because it is a new variant, you can get away with having clues that are, in retrospect, fairly straightforward and forcing. In Alphabet Soup, the first observation is that the digit 1 appears in the middle row of exactly three boxes. It is not some deep insight, it is a basic consequence of the genre. Something similar holds for the Fillomino Chaos Construction puzzles.

2. Your clues should be flexible: often times, when you are solving something like a Killer Cages puzzle, you aren’t immediately using the precise sum of a cage. Rather, we only use the fact that the cage is “large”. As a concrete example, the break-in of Fillomino Chaos Construction 3 utilizes a 20 cage in the bottom right corner. But the opening logic only uses that the total is too large for it to be a 2-cage, and likewise (by nature of how the cage and arrow operate), too large to be a 3-cage.

   I did not set the cage as a 20-cage. Instead, I wrote a range of digits that would force the deduction.² To a more experienced setter, this kind of clue may seem obvious. But I record this technique because the underlying principle is important: separate the mechanics of your break-in from the core ideas. Not only does this buy you more flexibility, but it also gives you a roadmap if the puzzle breaks. I find this holds throughout the construction process, not just the beginning.

The break-in you set can either be local or global. Local break-ins, such as those in region building puzzles (like my Fillomino Chaos Constructions), are easier to set because there are fewer degrees of freedom and global implications you need to juggle. With global break-ins and constraints (such as Alphabet Soup, but more obviously, Circle Counter puzzles), trying to set clues in the future interfere far more with clues you set in the past. Furthermore, it is far harder to make small adjustments to the break-in, because you have to ensure whatever changes you make preserve your global invariant that your break-in utilizes.

Step 3: Follow Through with Subtlety

When you have finished your break-in, you will have your strong clues interact in predictable ways. You will realize that weakening them or adding additional weak clues will create additional, less obvious interactions.

What do I mean by a weak clue? I mean a clue which only immediately does very little to further resolve the immediate setup you have. In reality, no clue is truly “weak”, and the only reason it will appear weak at the moment is because there is no pressure forcing it to resolve to anything. To make an analogy, it is a clue with low kinetic energy and high potential energy; it is a ball on the edge of a cliff.

As a concrete example, consider the 29 cage in Fillomino Chaos Construction 3. It originally was meant to block a 3-fillomino on the bottom from expanding horizontally, and it does that quite effectively.³ Notice how the puzzle does not utilize the full strength of the clue: the clue tells us “this cage is a big fillomino at the following location”, but we only use very little of it (that it blocks another fillomino). Eventually we will have to come back for it, and in many cases, it is the latent complexity in the puzzle that dictates when we have to revisit our weak clues, not us. After you have added enough of these weak clues, the pressure they collectively exert on the puzzle will cause a cascading reaction. I have gone from thinking a puzzle is about 30% done to having the weak clues I have placed solve about 70% of the puzzle.

A lot of the subtlety you see in my puzzles is emergent complexity, and in order to spot it and take advantage of it, I have to thoroughly solve my puzzles as I go. In some sense, there is a lot of luck involved.

1. First, you have to have a good sense for what kinds of weak clues will combine in interesting ways in the future. This is taste.
2. Then, as you keep setting the puzzle, something interesting must emerge that you can use. This is luck.
3. You then have to be able to spot and utilize these interesting setups to your benefit. This is discernment and skill.

The better you are at 1 and 3, the less luck you need for 2. As I have gotten better at setting, I have been able to spot areas where even small amounts of pressure are building (3 and a little bit of 2) and continually apply more pressure there (1), until there really is some opportunity for the dam to break (3).

As an example, consider the 38 clue in Fillomino Chaos Construction 3. Before I had placed it, I realized that the weak clues I was building had carved out a lot of regions. My instinct (3) told me that lots of regions were being carved out, many of which were likely mutually exclusive. So I checked and this was the case. I then realized (1) if I could carve out more regions, I could potentially create some interesting logic involving region counting. So I placed the arrow in r5c5, and ended up with 9 regions which must be distinct (2). Furthermore, none of these regions had room for an additional large fillomino. I then decided to take advantage of this (3): I specified that a large fillomino must exist with the 38 clue, and since it cannot be new, it must fit into one of the existing large fillominos.

There is more to it. I realized that to have any hope of finishing the puzzle, I must resolve the 7-fillomino in r1c8. Up until now I hadn’t committed to placing the large cage anywhere, just recognized it as an interesting possibility to explore. With this extra goal in mind, I now knew where to place it.

Here is the way I like to think about weak clues. When you place them down, you are putting yourself in debt (there is unresolved strength in the puzzle). As you keep borrowing, you must be constantly on the lookout for a way to pay off this debt. Be conscious of what you owe so that you may spot the opportunities to pay them off. Problems like non-uniqueness (who doesn’t love the infamous lone Kropki disambiguator?) arise when you do not give yourself a way to address these debts. But at the same time, you must spend money to make money, so you must borrow.

Endgame

After this grand cascade has occurred, there is very little of the puzzle left to set. You now just need to find the right configuration of clues to wrap the puzzle up in a satisfying way. At this stage, you should be confident that a solution exists, either by checking with computer software or (more likely) constructing one yourself.

Setting the endgame can be subtle and difficult, but because your possibility space is so constrained, you literally can brute force. When you are trying clues, you look for

1. the resolution to be interesting,
2. and not to have an overwhelmingly strong or boring clue that resolves a small disambiguity.⁴

For example, it is very bad to be left with a single 8/9 disambiguation. A good example is the disambiguation in Fillomino Chaos Construction, where a given 9 resolves the entire 589 disambiguation through a fun coloring argument.