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.
The torus twist on the pentomino coloring problem is such a neat idea – the tilted square fundamental domain in problem 47 feels like it should be impossible until you actually fold it in your head. I spent a happy evening on the two-coloring packing variant (problem 48) and the moat trick you mention is indeed the lazy but effective answer. It reminded me why polyomino puzzles are such a great gateway into spatial thinking; my kids got hooked through a simpler browser game called Pictomino that uses domino-style tiles, and now they are curious enough to look at real pentomino sets. Any chance of a post on hexiamond torus colorings next?
I’m glad you enjoyed looking at this! Did you see this post? That has the one hexiamond torus coloring I’ve done, and is the post I was referring to by the “Recently” remark at the top of the post. But it occurs to me that I neglected to make that a link, which I will fix presently. I have a big backlog of material I want to post about that doesn’t include colorings, so I don’t expect to get back to that for a while. But see the “coloring” tag for all of my posts that touch on the subject. Also note, (and this probably shows up better on a desktop browser than on mobile) that the header image and background image are both torus tilings. (The former is the same as the one in that post, with a different color scheme.)