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The “Perkinson Guest Bathroom Tile” Puzzle

A week before Thanksgiving, 2006, Kathleen and I drove down to Portland for the weekend, to see some dear friends who'd driven up that far from the Bay Area, and to attend a games party being thrown in their honor (or at least using them as an excuse) by Dave and Diane Perkinson. Dave and Diane are consummate hosts, and they'd invited us to stay in the guest room of their wonderful Craftsman-style house. The bathroom adjoining the guest room still has the great original tile, consisting of squares that have had one of their edges tilted, half one way and half the other. The tiles are laid in straight rows with the tilt of the edges alternating back and forth across the floor, making a kind of angular sine-wave pattern.

At some point as I was sitting in the bathroom, contemplating the world around me, I started to wonder how many ways there were to select three adjacent tiles from the floor in an L shape. (Now, their floor had the sine-wave lines from one row out of phase with those from the next row, but I allowed for both in-phase and out-of-phase patterns in my wondering.) Assuming that, like the real tiles, you couldn't flip a piece over to get its mirror image, I was pleased to discover that there are exactly twelve possibilities. Twelve pieces, times 3 tiles per piece, makes 36, a perfect square. This therefore led immediately to the question of whether or not a complete set of the pieces could be assembled to fill a six-by-six square. I resolved to find out as soon as I returned home, using my packing-puzzle design program.

Once I'd enhanced the software to support pieces that can't be flipped over, I was saddened to discover that the perfect six-by-six square is not, in fact, solvable. Happily, though, a complete set of pieces will nicely fill a variant of the square that has in-phase sine-wave lines at top and bottom. There are two distinct solutions, plus their mirror images. The "Perkinson Guest Bathroom Tile" puzzle is non-trivial to solve, but not wildly difficult either.

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