# Puzzle solving skills

Back when I was looking for a job, I did a lot of interview prep with other people. Among other things, that means practicing brainteasers in front of white boards. But as it turns out, I am extremely good at this already and don’t need the practice. Recreational math was a hobby in my youth, I participated in math competitions in college, and I’m mildly competitive in the US Puzzle Championship. I don’t want to brag, but friends have told me I ought to brag more often, so I am bragging. I am ridiculously good at puzzles.

So if I’m “good” at puzzles, what skills does that mean I have?  It’s hard to say.  (Does this imply that I’m also good at the jobs I interview for? Eh.)

There are, of course, many very different kinds of puzzles, and perhaps each category of puzzles requires unique skills. To name a few categories: programming puzzles where you seek to write an algorithm; logic problems where you deduce a solution from structured clues (e.g. Sudoku); puzzles where you make multiple moves in sequence (e.g. Sokoban); math problems; jigsaw puzzles; and what I call “linchpin” puzzles, where the goal is to have a particular realization. Some puzzle solving “skills” are more properly understood as puzzle-solving knowledge–knowing the solution to a bunch of common puzzles gives you a set of tools to solve new puzzles. But I also think there are a few general puzzle-solving skills, which I’ll try to describe.

Scaffolding

We learn many skills by observation, but it’s important to understand the limitations of observing others solve puzzles. Some puzzles have very clever solutions, and you think, that’s so smart, but how did anyone ever come up with that? Often, the answer is scaffolding. As you build a solution, you don’t make one giant clever leap, you often work your way there incrementally. Then, once you find your solution, you can remove all the intermediate steps, leaving an answer that seems impossibly clever. With a lot of experience, you can do this really fast, or by intuition, and make it look like one giant leap towards the solution. But observing these giant leaps isn’t always a great way to learn puzzle-solving, because all the intermediate steps are hidden.

Bottlenecks

Finding a solution to a problem is a matter of eliminating incorrect solutions. So a general strategy in puzzling is to identify the part of the puzzle that eliminates the most incorrect solutions. Another way to think about this, is that a correct solution needs to jump through a series of hoops. If you identify the smallest hoop, then that imposes the most constraints on the solution. In other words, you want to identify the puzzle’s bottleneck (or bottlenecks).

In a Sokoban-style game, this is easy to do by just trying stuff, and observing where your attempted solutions fail. But sometimes it also helps to look ahead for bottlenecks that you haven’t yet reached, because that might tell you something about the intended solution to the earlier parts of the puzzle.

Solve a simpler problem

As mentioned above, a lot of puzzle solving “skills” are really more like puzzle solving knowledge, gained by solving similar puzzles.  So it stands to reason, if you encounter a difficult puzzle that you don’t have prior experience with, then you can make up for it by posing similar and simpler puzzles to yourself.

This is easiest to illustrate with math problems. For instance, if a question asks you to calculate how many numbers less than 2021 have a factor in common with it, you should first try the same problem but replace 2021 with a smaller number. Try 2, then 3, then 4, and so on, until you recognize a pattern.

Brute force is a legitimate strategy

If you’re accustomed to linchpin puzzles, you might think that every puzzle is about hitting upon some clever idea. You might think that doing an exhaustive search of all possible solutions (known as a “brute force” solution) is inelegant and unintended.

First of all, that’s not even always how linchpin puzzles work (see scaffolding). And second of all, most problems in the world require some form of solution search.  Linchpin puzzles are created by carefully selecting questions that fit a certain aesthetic, but not all puzzles are created with that same aesthetic in mind.  There is no sense in trying to solve, for example, a jigsaw puzzle, as if it were a linchpin puzzle.

There are multiple methods to do a solution search. “Brute force” usually refers to an exhaustive search, systematically trying each possible solution. But you can also do a random search—basically trying out things to see what works. Or a heuristic search, which first tries possibilities that seem most likely to work. Generally, you would start out with a random search, just to get a sense of what works and what doesn’t. Once you’ve identified the major bottlenecks, hopefully that only leaves a few possibilities, which you can then search exhaustively.

Part of the skill in solving puzzles is knowing when to switch from a bottleneck search to an exhaustive search. If there are 1000 possible solutions, and it takes 10 seconds to check each one, that’s almost 3 hours to solve. If you can spend 5 minutes to identify a bottleneck that blocks even half of the solutions, that saves you a lot of time.

How do you perform an exhaustive search? You can think of solutions as being in a tree structure. For example in a Sudoku puzzle, maybe there’s a square that could be a 1 or an 8. So you create a tree, with one branch assuming that it’s a 1, and the other branch assuming it’s an 8. Each branch may further split into more branches.

Know when to fold ‘em

You can’t solve every puzzle. At some point, it is correct to give up, and come to terms with not having solved everything. If you’re only incidentally into puzzles, looking up answers might feel like a missed opportunity, a puzzle that you’ll never again be able to solve for yourself. But there are lots of other puzzles in the world, an infinite number, in fact. You need to find the ones that match your skill level, and personal goals.

Most people’s personal goal is to have fun. In which case, you should give up when it’s no longer fun. But, even in a competitive context, or an interview context, there still comes a time to give up and save time for the next question.

It’s also worth mentioning that sometimes puzzles are bad. A good puzzle requires some degree of trust in the designer, that they picked a reasonable question and reasonable answer. Or sometimes a designer is just going for a different aesthetic than you were expecting, or the puzzle is removed from the context that made it originally work.  Just like with other hobbies, being a puzzle-lover doesn’t mean loving and solving every puzzle out there. It means having discernment, an understanding of what puzzles you appreciate, and what puzzles you don’t.

1. says

@Rob,
I don’t do crosswords, and it seems to me that trivia is a distinct skillset. But, some of the same skills apply! I think intuitively, crossword solvers go for bottlenecks first (i.e. words that already have some letters filled), for example.

2. Jazzlet says

Hmmm Siggy I think that this is an instance where American English and English English are leaving you in ignorance of what Rob is talking about. A well clued cryptic doesn’t require you to know any trivia, the clue should give you everything you need to solve it. A classic, although from a different paper, is “gsge, 9, 4” which has all you need to reach the answer.

3. says

@jazzlet,
I have never heard the term “clued cryptic”, I only looked at the website, and saw clues that looked like trivia. Now that I look at it further, I see that the aesthetic of a cryptic crossword is to contain all information within the clue, and not contain trivia. However, I don’t think I’m entirely wrong to call it trivia either. For example, one clue might be to give a straight definition of a word, along with an anagram. However, vocabulary absolutely is trivia.

Actually now that I understand the format, I had a realization that these kinds of puzzles were used extensively in 11th Hour–an old Myst era game. I never did get very far in that one–but mostly because the game crashed.

4. avalus says

Zachtronic Games are my puzzle source. Points of breaking vary.

5. Rob Grigjanis says

Siggy @4:

vocabulary absolutely is trivia

What an odd thing to say. Aren’t numbers trivia as well?

6. says

Two more strategies:

Cross referencing: On their own, two separate techniques or methods might not produce a data point or solution, but will when combined. Sudoku, Towers, Dominosa and others can be solved by cross referencing.

Uniqueness: In some puzzles, a clue or a piece of information is solitary, it cannot be interpreted or placed another way. In easier puzzles (e.g. Logic Puzzles), these are things that will be part of the solution and given at the start. In others, they’re hidden in boolean logic or combinations of two statements, or unspoken facts are universally known (e.g. gender and names, time sequences, numbers).

For example in a figure logic, if it says “A cube” and it’s a five digit number, you know the root is somewhere between 22 and 46, and if you have even one of the five digits, there will be at most six solutions, down from 90,000 possibilities.

I like Simon Tatham’s puzzle collection. They’re freeware, unlimited randomly generated puzzles, and I can play them on windows, mac, linux or android.

I have a pile of free mathematical puzzles if you want them. And there’s also Sokoban.

7. says

@Rob Grigjanis,
Absolutely, you could say that there’s trivia in numerical problems as well. For example, the problem I gave in the OP (how many natural numbers less than 2021 that share a factor)–that’s related to Euler’s Totient function, and that’s something you might just know, or you might not know. It’s like I said, a lot of puzzle solving skill is in fact puzzle solving knowledge.

But what I mean, is that if you’re given a definition and have to identify a word, that’s kind of prototypical trivia? It wouldn’t be out of place in a crossword or quiz show. Granted, I don’t have much familiarity with the cryptic clue format–it looks like the “straight” definition often underdetermines the answer, and that you have to lean on the “cryptic”. Anyway, I’m not saying there’s anything wrong with this kind of puzzle, only that I don’t have experience with it, and it doesn’t appeal to me personally.

That kind of puzzle would not appear in the US Puzzle Championship, because it’s a qualifying competition for the World Puzzle Championship, and therefore tries to be culturally independent (although usually Eurocentric). It contains word puzzles of the sort where you’re given the exact words (usually in the language of the country hosting the WPC), and told to fit them into a grid.

8. says

@Intransitive #7,
I don’t know what you mean by “uniqueness”. You might be thinking in the context of a particular kind of puzzle. Not all puzzles have unique solutions–programming puzzles almost never do.

Deductive puzzles generally obey the aesthetic of having unique solutions, and a deductive puzzle with multiple distinct solutions would be considered defective. By convention, you aren’t supposed to use uniqueness itself as a clue, and yet uniqueness informs the solving strategy–you treat the solution as a single needle in a haystack, that you need to systematically search for. However, it’s important to remember that this is merely a convention of the format–if the solution were not unique, then the correct strategy would be different, relying more on guessing and heuristics. And actually some deductive puzzles do depart from these conventions, Numberlink being a notable example.

9. jenorafeuer says

For ‘uniqueness’, there’s always things like the ages puzzle. Someone is asked for the ages of their three children, and tells the census worker “I won’t tell you that, but all of them are under ten, and I’ll tell you the sun of their ages and the product of their ages.” After receiving that information, the census worker says, “Sorry, I still don’t know the answer.” “The youngest one likes ice cream.” “Ahh, that’s enough, thank you.”

The key point here is that there is the only real combination that works here is a sum of 13 and a product of 36, which can be either 1,6,6 or 2,2,9 and so doesn’t have a unique solution at first… but add in the constraint that there is a youngest child and 1,6,6 becomes the only solution.

Raymond Smullyan had a variant on the Knight/Knave puzzle that relied on this taken a couple of steps further. A judge in the case asks a single question, gets an answer, and makes a decision; someone else gets told this and can’t figure it out, asks a different question, gets an answer, and then knows the solution. (I don’t remember all the details at the moment.) But the series of situations in which you know that somebody did not have enough information at one point and then one answer was able to grant enough information is sometimes enough to solve the problem even without knowing what the answer was, because only one possible answer will result in a unique solution.

The bit about scaffolding reminds me of a time when my uncle sent me a puzzle involving light bulb dropping. (100 storey building, light bulb is rated to survive being dropped from n storeys and is guaranteed to survive that but break if dropped from n+a storeys, you have two light bulbs, what is the minimum number of drops required to guarantee that you can determine n.) It was just plain obvious to me that the answer would involve triangle numbers (there is a maximum number of drops, and each drop of the first bulb means one fewer drops available for the second) so I started from there, which led my mother the retired teacher to comment on how a lot of my teachers had complained in the past about me jumping ahead to some ‘obvious’ conclusion without actually describing how I got there.

10. says

If you saw the Dominosa puzzle game, it’s exactly what the name infers, dominoes from 0-0 to 6-6 (or whatever upper limit you choose) in a rectangle. Using logic, you have to identify all the two part pieces.

Because of how pieces are placed, there could be multiple 1-2s in the puzzle. But if there’s only one 2-4, it eliminates any 1-2 involving that digit. The hard part is finding unique instances in a large playing area.

https://www.puzzle-dominosa.com/

11. Jazzlet says

@ Siggy
Sorry that was shorthand for a “well clued cryptic crossword puzzle”.

The answer to “gsge, 9, 4” is “scrambled eggs”.

A cryptic CWP may have a theme from the plays of Shakespear to reality stars, but tht is simply an additional overall clue, each individual clue should be solvable without reference to anything but the English language and the conventions that paper it is in uses for shorthand, though many of those are shared.

12. says

@Jazzlet #12
“without reference to anything but the English language”

“But the English language” is a hell of a qualifier, and one that would include many crossword clues as well. I don’t get it, are you just trying to explain a difference because you think I can’t already see it, or is there a thing where cryptic crossword solvers don’t like their puzzles to be compared to crosswords? The comparison is really obvious, but it’s okay, crosswords are a respectable puzzle genre.

@Intransitive #11
Uniqueness, as in Dominosa’s rule that each domino is unique, well it’s an element of many logic puzzles. Sudoku says that each row/column/box has a unique copy of each digit 1-9. Unruly says that each row and column has a unique pattern of black and white. Kakuro has no more than one copy of each digit in every row/column grouping. But that’s obviously dependent on the rules of the puzzle, and I don’t see uniqueness rules as being different from any other kinds of rules in these puzzles.

There’s a whole lot of intricacies to these logic puzzles, and that’s well beyond the scope of what I wanted to discuss in the OP. The way I organize my thoughts about logic puzzles might be different from yours, because “cross-referencing” and “uniqueness” aren’t meaningful categories to me. For me, the two main categories are deductive rules–which are distinct for each type of puzzle–and brute-force guessing, which is a solving strategy that applies everywhere.

13. says

If readers are interested, I could write more puzzle tips that are particular to the logic puzzle genre (Sudoku, Kakuro, Slitherlink, Nurikabe, etc.)