I continue last week’s discussion of tiling hexagons with the four trillows below.
Instead of curved triangles I am using the markings with curved arrows that have to match in direction when the tiles are put together. This creates directed graphs like so.
Last time we saw that one can systematically solve such puzzles by first drawing the undirected graphs on the region to be tiled, and then directing the edges. Today we will use the concept of deficits to look at this more closely. The goal is to solve puzzles that tile the hexagon of edge length two with an equal number of green and gray triangles. Above we see two solutions that use 12 and 6 green and gray triangles, respectively. Let’s try it with 10 each.
For this, we need to leave two purple and brown triangles. A solution is up above to the right. To the left is the undirected graph. Each edge comes with a deficit, which counts how many more left turns one takes than right turns when following that edge. This number is determined up to sign only as long as the edges aren’t oriented. The orientation needs to be chosen so that the deficits cancel, as left turns give not unexpectedly green triangles, and right turns grey ones. Below is another example with the brown purple triangles located differently.
Can we do it with 11 green and gray triangles each? Below is the complete graph, from which purple diamonds have to be removed until only two purple triangles are left. There are, up to symmetry, only two possibilities, and we see that the deficits cannot be combined to give 0, Therefore this version of the puzzle has no solution. Thanks for trying.
This can be continued, of course. Try it with 9 green and gray triangles each. Below are two solutions with six triangles of each kind.
The Inner Frame 