## Counting edge-matching puzzles

I recently re-discovered a puzzle that I had mostly forgotten from when I was a kid. The problem is simple to state: rearrange and rotate the seven hexagonal pieces shown below so that each of the twelve pairs of facing edges have matching labels. (Currently, only the 0s at 2 o’clock and the 3s at 5 o’clock match as required.)

The original puzzle wasn’t exactly the one above; the edge labels were different, but the basic idea was the same. What snagged my interest here, decades later, was not solving this puzzle, but counting them. That is, if hexagonal pieces are distinguishable only by their pattern (up to rotation) of edge labels 0 through 5, then how many different possible puzzles– sets of seven such pieces packaged and sold as a product– are there?

I think this question is not “nice” mathematically– or at least, I was unable to make much progress toward a reasonably concise solution– but it was interesting computationally, because the numbers involved are small enough to be tractable, but large enough to require some thought in design and implementation of even a “brute force” approach.

(My Python solution is on GitHub. What I learned from this exercise: I had planned to implement a lazy k-way merge using the priority queue in the `heapq` module, but I found that it was already built-in.)

There are several variants of the question that we can ask. First and easiest, let’s ignore solvability. There are 5!=120 different individual hexagonal pieces, and so there are ${7+120-1 \choose 7}$, or 84,431,259,000 distinguishable sets of seven such pieces.

However, most of these puzzles do not have a solution. It turns out there are 4,967,864,520 different solvable puzzles… but there are at least a couple of ways that we might reasonably reduce this number further. For example, over a billion of these solvable puzzles have multiple solutions– 1800 of which have twenty different solutions each. If we constrain a “marketable” puzzle to have a unique solution, then there are… well, still 3,899,636,160 different possible puzzles.

Of course, many of these puzzles are only cosmetically different, so to speak. For example, the puzzle shown above has four identical pieces with the same 0-through-5 counterclockwise labeling. If we arbitrarily distinguish this “identity” piece, then although some puzzles have none of these pieces, they are not really “different” in a useful way, since we could simply relabel all of the edges appropriately so that they do contain at least one identity piece. There are only 281,528,111 different puzzles containing at least one identity piece, of which 221,013,350 have a unique solution.

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