What is a puzzle?

A puzzle is a medium of communication from the author to the solver.

I think this is the most notable part of this essay, and also the most important aspect of the definition. There should be an author that purposefully writes the puzzle, because they want to tell the solver something. The existence of this author serves as a promise that the puzzle is worth doing. Without such an author, the solver doesn't know if the "puzzle" they are seeing is worth spending time on or not, or even whether it has a solution at all or not.

One example where this author aspect is absent is in open research problems. Mathematics is a vast field with a lot of unknowns; people around the world are still chipping away and discovering new truths. An open research problem satisfies most things about being a puzzle: it hides something (the proof that it is correct), which the researcher will hopefully be able to figure out. Getting a breakthrough and solving one such problem comes with a very similar elated satisfaction as cracking a puzzle; the researcher feels happy to have figured something out.

However, an open research problem has no author. You cannot trust that what you're working on has a solution in the first place. In fact, it's entirely possible what you're trying to prove is false: then all your work is for naught, since you should have proven the opposite thing!

Mertens conjecture is one such example. It works with a function called the Möbius function μ(n) that only takes values 0, +1, or −1. This function appears erratic and random enough; in 1897, Thomas Mertens conjectured that the sum μ(1) + μ(2) + μ(3) + ... + μ(N) always remains within ±√N. It has been verified computationally up to 10¹⁶ = 10,000,000,000,000,000. Despite that, it has been proven to be false (in 1985, a gap of almost 90 years!) using a sophisticated mathematical argument. Even when numerical evidence was in our favor, turns out the statement is false anyway. There is no promise that whatever people are working on is actually solvable, or even true.

While a research problem can be fun and rewarding on its own, it is not a puzzle. The promise that there is an author helps set your expectations.

What about mathematical competitions? You have to solve some problems (in the form of proving statements), but the problems are actually set by the problem committee. Therefore, there are authors behind the problems. Moreover, they have intended solutions, so the problems are trying to convey an idea given in the solution. They sure appear to be puzzles!

I actually think mathematical competitions are not puzzles, but for a different reason. I'm not denying that they have an author; I'm denying that they are "conveying" an idea. I'll get to this again below.

A puzzle has a human author.

The emphasis here is that the author is a human (or a sapient being), in contrast to a computer. I strongly believe that a computer is unable to come up with the ideas that are required in my definition of a puzzle. A more general statement is, coming up with these ideas is a task in creativity, and I believe a computer has no concept of creativity and so cannot come up with creative things.

As an example, many bookstores carry a lot of Sudoku books. The mass-produced Sudoku books you might find at a bookstore are usually computer-generated, and a computer cannot put in any idea to those Sudoku "puzzles". They are simply activities that you can do in your spare time; a more pejorative way would be to call them "time wasters".

But wait! Who wrote these computer algorithms that generate puzzles? A human! While an unfiltered output by the computer doesn't qualify as a puzzle, a human aspect overseeing the output can help them become puzzles. A human will be able to test the "puzzles" generated by the computer, and if the puzzle turns out to carry some idea, the human will be able to spot this. The human can also tweak the output, allowing it to be improved.

I have an example, but it is only hearsay; I haven't been able to verify it. I heard that, for the game Stephen's Sausage Roll, increpare used a computer to generate a lot of puzzles, and handpicked several of them to make up the first world of the game (perhaps after some tweaking). This is exactly what I discussed above, that the human aspect is present. It shows in the game; the puzzles were still interesting to solve.

At the time of this writing, there is a lot of talk about AI-generated images. While the morality and ethicality of getting the dataset to train the AI is itself a good topic to ponder, I want to touch upon the other side of the program: its output. There have been cases of malignant people trying to pass an AI's output as their own work. This is unacceptable, not only because it's plagiarism and deception, but because a computer is unable to come up with creative output -- the same reasoning I gave earlier in this section. The output is not really any work of art; a lot of the time, the output strongly resembles an image from the dataset! But if a human comes in and looks at the various images spat out by the AI, they can come up with a spark of *inspiration* -- and this, in turn, is what drives human creativity. It is also possible a human uses an AI's output as a base, improving on it in ways only a human's creative mind can do.

In a way, puzzle-making is similar. A computer cannot generate puzzles on its own, but a human can help sift through the output and improve on it. I'm willing to call these computer-assisted puzzles puzzles.

A puzzle wants to convey an idea.

Another core tenet of my definition is that a puzzle is telling something. If the "puzzle" does not have anything to say, it is not a puzzle. Now, what qualifies as "something" is actually a very low bar, but sometimes there are things that still go under.

Jigsaws are commonplace, and some people find pleasure in doing jigsaws, whether as a relaxing, meditative activity or as a competitive thing where speed is all that matters. Are they puzzles? My answer is no. What are they trying to say? The final picture? It's already on the box! A jigsaw does not say anything whatsoever, it has no idea. It is not a puzzle.

But wait! Does that mean that if you do not have the picture, then a jigsaw is a puzzle? Yes! In that case, the idea is the picture itself, and the author conveys the picture after you've done the work of assembling it. That said, doing this by itself is kind of weak; it usually doesn't leave much impression. More often, jigsaws are used as a component of a larger whole. One recent example is The Greatest Jigsaw (MITMH 2021), which started by assembling a jigsaw (with the final picture not given), although there were more steps to the puzzle after the jigsaw. My favorite example is perhaps P.I.HUNT 8, an entire puzzle hunt served as a jigsaw puzzle.

In the same vein, Tetris is not a puzzle. Surprisingly, I find a lot of comments that say Tetris is a puzzle game; I think it's very clearly not, and my definition makes it clear. There is no idea being conveyed. In fact, I can say more: in any single game of Tetris, there is no author. The creator of the game, Alexey Pajitnov, created the game, not any single instance of it; each instance is mostly just the random number generator spitting out pieces all the time. But let's return to ideas.

Tetris is fun; I play it competitively along with Puyo Puyo. I know openers, T-spin setups, and more. I know I can learn more strategies to improve as a player. Are these all ideas intended by Pajitnov? I'd say no, these are not ideas. These are incidental things that arise unplanned. Pajitnov didn't intend to say that we should learn the TKI; it's something found by chance later. Tetris isn't telling us any idea, and so it is not a puzzle game.

In addition, I also talked about mathematical competitions earlier. Problems in such competitions have authors, and they have intended solutions and thus ideas they are trying to convey. However, earlier I said that I don't think they are puzzles, and I'll elaborate why now.

You probably have seen puzzles that are not unique. Maybe a Sudoku puzzle ends up having multiple solutions (or no solution). Maybe you have solved a level in a puzzle game in an unintended manner, sometimes called a "cheese" solution if you bypass most things in the level.

Are they puzzles? I would say they aren't, because they are not conveying the idea properly: the communication is lost along the way. The solver does not get what the author is trying to say, and so the puzzle has failed its purpose. This is why, in many puzzle games, the developer occasionally says "patched out an unintended solution in level X". They want to convey the idea, so they want to remove all other unintended ways to solve the puzzle; this way, the player can only solve the level by seeing the intended idea.

What about mathematical competitions? Here lies the problem. Mathematics is much, much wider, and there are usually other ways to solve a problem — not necessarily easier, but such other solutions exist. They have to be marked correct, since any valid proof in mathematics is valid. But they likely miss the intended solution, including any key idea in it. Does this problem manage to convey its idea to the solver? Not really. Mathematical competitions are not puzzles (although they are definitely close).

A puzzle obfuscates its intent.

This part is pretty clear. If there is nothing hidden, it's called talking; you're stating what you want to say in a clear and plain manner. A puzzle should make the solver work for it.

A puzzle should be solvable by any solver.

Someone once said, in a puzzle, the author and the solver are playing a game in which the author wants to see the solver struggle, but ultimately the author wants to lose; the solver should prevail. I don't remember the exact wording or who said it, but it left an impression on me. Puzzles are meant to be solved. If a puzzle isn't designed to be solved, what is it for? However well the solution is hidden, it should eventually be found.

If a message is obfuscated in a way that is not meant to be solved, it's called an encrypted message. The intended recipient is supposed to be able to read the message without putting in any work, and any other person is not supposed to be able to read the message regardless of how much work they put in.

It's worth noting that "any" solver isn't quite a literal "any"; you may target your puzzle to a more specific intended audience. But you shouldn't actively make your puzzle inaccessible to other solvers.

Why so strict?

You may ask, why is my definition so strict, why am I so easily cutting away things that you would call puzzles?

My main motivation is to make it worth it. A puzzle satisfying my definitions has a high chance of being a pleasure to solve. I know there is a human that left a specific trail in this puzzle; I just have to find it. And when I persevere and figure it out, when everything clicks together, I will get this euphoric "a-ha!" feeling. (Even if I failed to solve it and read up on the solution or walkthrough, I would still be able to see what the puzzle was supposed to say and could still appreciate it that way.) Without any such satisfaction, I would be left thinking, "that's it?" I might feel incomplete, as if I missed something from the puzzle.

I mentioned in the lead that some people think this is a definition of a good puzzle instead of just a puzzle. I may agree with that; in that case, my view is that bad puzzles don't deserve to be called puzzles in my eyes.

That said, once again, this is my opinion, this is my definition of puzzles. If you enjoy jigsaws or computer-generated Sudokus, that's great! I also enjoy Tetris, which I explicitly called out as not a puzzle above. I'm just saying, the reason I might enjoy such non-puzzles will *not* be because they are puzzles; there will be another reason why I enjoy them, and if I crave my puzzles, I'm going to get my fill elsewhere.

I'll conclude with a few puzzles I really like:

• A Nurikabe puzzle I wrote. It's small and very straight to the point: I found a new theory on Nurikabe puzzles, and I wrote this puzzle to cleanly show off this theory. It's very much conveying a specific idea.
• Count Mover, a Sokoban game with a move counter. The motivation is that move counters in puzzle games are generally not interesting unless they actually form part of the puzzle, like in this one.
• Make10, a game where you have to make 10. This is a kind of game that exhibits the "convey an idea" aspect so well: as you figure things out, you'll just immediately know everything is completely intended and the author wants to tell you so.