It’s the holiday shopping season, so I figured it couldn’t hurt to write a post or two on the puzzles I am selling.

Every mathematical puzzle designer worth his or her salt has an argument for their puzzle’s awesomeness using impressive sounding mathematical justifications. This, for L-topia, is mine.

There are 12 pieces in the set. Empirically, 12 is a good number of pieces for a mathematical puzzle. There are 12 pentominoes, and 12 hexiamonds.

The shape of the pieces, an l-tetromino, has some desirable properties. It is very highly tileable. Two factors that affect the tilability of a polyomino are its size and its symmetries. Smaller and less symmetrical polyominoes are the most tilable. The l-tetromino is the smallest asymmetrical polyomino, and the only asymmetrical tetromino, so it should be the most tilable of all.

A set of 12 l-tetrominoes tiles a 6×8 rectangle in 1114 ways. That’s probably the most for any set of 12 copies of a single polyomino tiling any rectangle, but it’s not that impressive compared to other sets containing multiple shapes. For example, the 12 pentominoes can tile a 6×10 rectangle 2339 ways.

But because the shapes *are* all the same, if you mark all of them in some way to distinguish them from each other, (as the holes on the L-topia pieces do) every permutation of placements of the 12 l-tetrominoes can create a distinct tiling. Now the total number of tilings is roughly 1114 · 12!. (Actually, it’s slightly less because some of the tilings of the rectangle are symmetrical: about 55 of the 1114 solutions are symmetrical by reflection or 180° rotation, so the total is about 1059 · 12! + 55 · 12! / 2, or about 520 billion.)

Well, that’s a pretty impressive number, but having an impressively large space of possibilities does not, by itself, make for a great puzzle. In this case, however, I do think it is helpful, and I’ll explain why presently.

Suppose I think of a proposition that can apply to any of the holes in the set. For example, that the hole appears in an odd numbered row. Because there are two different kinds of holes, it may be elegant to use either the opposite of that proposition, or some proposition that is complementary in some way, to apply to the second kind of hole; in the problem illustrated by the solution above, we have the round holes in odd rows, and the square holes in odd columns. Suppose the probability of the proposition being true is ½, and suppose that the probability for each hole is independent from the others. (One must take care that the placement of holes on the pieces doesn’t fatally interfere with independence; if, for example, we had asked for circles on odd rows and squares on even rows, there would have been pieces that could not have been placed legally anywhere.) Then the probability that the proposition is true for all of the holes is 1/2^{24}. Given this piece of information, we can get an expected number of tilings where the proposition is true by multiplying that probability by the total number of tilings.

The result is about 31,000. That number is tiny compared to the size of the total space of tilings, but I can say from experience that it makes for puzzles that are challenging but solvable. And it gives us wiggle room to use propositions with probabilities that are a little smaller than ½, or for which the probabilities are not entirely independent. The result is that we can come up with a wide variety of propositions to use in designing puzzles with the expectation that they will provide a good puzzle solving experience. L-topia isn’t just a puzzle, it’s a natural puzzle creation kit!

**Why L-Topia isn’t awesome, and Agincourt is**

Unfortunately, to be perfectly honest, being a “puzzle creation kit” interferes with L-topia’s accessability as a puzzle. Because the circular and square holes have no inherent meaning, but have to have their meanings imposed by a puzzle’s directions, you can’t simply take the pieces out of the box and start solving.

Agincourt, on the other hand, with its 64 squares with an arrow in each, practically begs to be turned into four 4×4 layers with the arrows aligned. Of course, there are other challenges to be found, but the one that literally comes out of the box is both elegant, and has a reasonable level of difficulty. (Some of the L-topia puzzles are better for hardcore puzzle solvers.)

Once again, I have both puzzles available for sale. Order soon for delivery by Christmas!