### I don't have the right math for this...

Jul. 26th, 2009 12:12 pm**mmanurere**

So, here's a description of a puzzle:

You have 16 square "tiles", each with a different combination of red and blue "sides". Since the same side is always up (no rotating the tiles), the 16 tiles represent all 16 possible combinations of "sides". Now, these 16 tiles are to be placed in a 4x4 square, such that touching sides must be of the same color and all exposed sides around the outside of the square must be blue.

From one set of relative placements of the all-blue and all-red tiles (all-blue in a corner, all-red diagonally adjacent; rotations and reflections cover all variations on this), I've come up with 56 solutions. It took awhile to completely exhaust the possibilities, though, with much note-taking and backtracking, so I don't

One weird pattern I noticed -- in making the "tiles" (er, pieces of paper), instead of making the whole side a color I just had basically a dot at the center of each side, with all sides of the same color in a given tile connected (so all-red has a red + taking up the whole square, the tile with a single blue side at the top has a red T-shape, the one with red on the sides and blue on the top and bottom a -, etc.). Each solution has a different "pattern" of the red "channels" this way (and you could actually put the tiles together based just on this pattern if you have it written down), but all 56 solutions I have so far have exactly two "loops" in the red channels. Going to poke around a bit and see if this might have something to do with solutions to this puzzle corresponding to 4x4 normal magic squares.

You have 16 square "tiles", each with a different combination of red and blue "sides". Since the same side is always up (no rotating the tiles), the 16 tiles represent all 16 possible combinations of "sides". Now, these 16 tiles are to be placed in a 4x4 square, such that touching sides must be of the same color and all exposed sides around the outside of the square must be blue.

From one set of relative placements of the all-blue and all-red tiles (all-blue in a corner, all-red diagonally adjacent; rotations and reflections cover all variations on this), I've come up with 56 solutions. It took awhile to completely exhaust the possibilities, though, with much note-taking and backtracking, so I don't

*especially*want to just brute-force the other six relative placements of all-red and all-blue. Unfortunately, I don't really have the right math to take a more analytic approach.One weird pattern I noticed -- in making the "tiles" (er, pieces of paper), instead of making the whole side a color I just had basically a dot at the center of each side, with all sides of the same color in a given tile connected (so all-red has a red + taking up the whole square, the tile with a single blue side at the top has a red T-shape, the one with red on the sides and blue on the top and bottom a -, etc.). Each solution has a different "pattern" of the red "channels" this way (and you could actually put the tiles together based just on this pattern if you have it written down), but all 56 solutions I have so far have exactly two "loops" in the red channels. Going to poke around a bit and see if this might have something to do with solutions to this puzzle corresponding to 4x4 normal magic squares.