Greg Hunter

Logic Skills, General and Specific

In Puzzles on August 12, 2009 at 9:23 PM

Logic puzzles, perhaps more than any other type of game, are an inherently solitary activity. Word puzzles such as crosswords are fun in groups; video games, even single-player ones, are prime topics for discussion and commiseration; single-player board games (Solitaire, for instance), invite spectators. But nobody ever wants to talk about logic puzzles for particularly long.

So the circumstances under which I’m now starting to blog about these puzzles are somewhat bizarre. The talks I’ve had with people about logic puzzles have been mostly summary: “Oh, you play Kenken?” “I used to be really into Minesweeper.” “You should try Honeycomb Hotel. It’s all I played after getting my wisdom teeth out.” Discussions of the finer points are rare, and attempts to solve in groups often lead to conflict, with one party accusing the other’s markings of being too sparse or too messy. All of this means that I have no idea what the general population knows about puzzles, so I apologize if the material herein is too basic or too esoteric. Please correct my course in the comments, if necessary.

In some sense, solving logic puzzles is just like proving theorems. Almost everything you mark is the result of a “proof”: If there weren’t a 7 here, there couldn’t be a 7 anywhere in this column. Therefore, there’s a 7 here. Or, Suppose this space has no mine. Then this one does, so this one doesn’t, so this one does, so these two do, but they’re both adjacent to the same “1”. Therefore, there’s a mine here.

There’s not too much to say about this that can’t be said about math in general. How one solves logic puzzles, then, is more or less always just a matter of asking the right questions, making the wrong assumptions, and seeing what contradictions (and thence proofs) arise.

But of course there’s more to say about it than that, just as “you have to move in the right way” is a poor summary of sports. How does one “learn” a logic puzzle? What is it that we do when we get better at, say, Sudoku? Back to the math analogy, it’s mostly a matter of lemmas: little theorems you prove once, then reuse over and over to make your work go faster.

Normal 5x5 Slitherlink, #1,016,045.

Normal 5x5 Slitherlink, #1,016,045.

Let’s be concrete. Have you all played Slitherlink? (Wikipedia lists seven more names I’d never heard.) It looks like the thing on the left:

No? Go try a couple here. The instructions are on the left panel, so I won’t bother reprinting them.

Okay, how long did the 5x5s take you? Probably several minutes, if this is your first time. But if you’ve done this a lot, it probably took you less than 30 seconds. So what is the skill that lets us quarter (or better) our time? It’s the lemmas. The configurations we’ve seen before, not here but in other puzzles, that we fill in automatically without going through the logic in our heads. The skill comes in the conversion of our logical thinking to instinctual doing.

Figure B

Figure B

Figure A

Figure A

Let’s explore a particular lemma. See the 3 in the upper-left corner in the example? Three sides of that square have line segments, and one doesn’t. But the missing segment couldn’t be one of the two in the corner, as in Figure A, or else the path would have nowhere else to go. So you can immediately mark two segments as in Figure B.

After playing this for a while, you’ll pick up patterns for having a 2 in a corner, 3s that share an edge, 3s that share a corner, 0s next to 3s, long diagonal chains of 2s with 1s or 3s on the end, and so on. This is what it means to become “good” at a logic puzzle: familiarity with these patterns, and a large enough library of lemmas that the solving is close to automatic. And to be good at puzzles in general is to be good at inventing these lemmas, recognizing what patterns are useful, and incorporating them into your intuition.

So, yeah! That’s how logic puzzles work. I’ll be blogging later about specific puzzles, and the extent to which the “techniques” described above fall short or lead to surprising insights. Good times.

Until we figure out how to include post authorship in the headers,


  1. I’m working on the slither thing right now. It took me about five minutes to figure out the first one. We’ll see how long for the fifth.

  2. […] also apply to most logic puzzles, because by solving these one discovers the sorts of lemmas I talked about earlier. But this sort of misses the point, because those lemmas aren’t usually observed by the […]

  3. Excellent site, keep up the good work

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