Recently I revisited one of my old pentomino coloring problems, modified to apply to a tiling of a torus rather than a rectangle. That worked out well, so I might as well shamelessly continue to mine this vein.
There are 18 one-sided pentominoes. Six of them have reflection symmetry, and the other 12 are 6 sets of mirror pairs. A while back, I asked if there was a tiling with a three-coloring where the 6 with reflection symmetry share a color, and each mirror pair has one pentomino of each of the remaining two colors. Patrick Hamlyn found that there was no rectangle tiling that could be colored in this way, but there was such a tiling of the shape below:
The one-sided pentominoes have area 90, which is the area of a tilted square on a grid:
Problem #47: Find a tiling of a torus with this tilted square as its fundamental domain by the one-sided pentominoes with a three-coloring as described above. If possible, find a tiling with no crossroads.
Another older problem that could be adapted to a torus is the minimal 2-colored packing problem. Here’s my conjectured minimal 2-colored pentomino packing of a rectangle:
(I had forgotten that this was problem #1 on this very blog!)
Problem #48: Find a two-colored packing by the pentominoes of a torus with minimal area.
Obviously, you could just take the rectangular packing above and add a one unit “moat” around it to get a torus with a 14×6 rectangle as its fundamental domain, but surely we can do better.