Here's a fun problem: take a hexagonal grid on a torus and put 0 or 1 into every hexagon, so that every permutation of 7-hexagon neighborhood occurs exactly once.
To clarify, rotations by multiples of 60⁰ are considered different. For example, both of these configurations should be present (individual hexagons are "vertical", like ⬢ ):
1 0 0 0
0 0 0 1 0 0
0 0 0 0
This is a known problem for the orthogonal grid: De Bruijn Torus, but a quick search yields no relevant results for hexagonal grids.
Is this even possible with hexagons? Turns out, it is!
I found a 32×4 torus relatively quickly, using backtracking:
Finding a 16×8 proved a bit harder, but then I shared my woes with a colleague. Next day, he found this 8×16:
And, here's a 16×8 torus:
For the orthogonal grid, De Bruijn toruses can be assembled directly (see [1] and [2]). For the hexagonal grid, I don't know if there's a construction method better than backtracking.
Update (2024-05-08): Twisted Hexagonal De Bruijn Torus.
[1] Cock, J.C., 1988. Toroidal tilings from de Bruijn-Good cyclic sequences. Discrete mathematics, 70(2), pp.209-210. (link)
[2] Iványi, A., 1988. Construction of infinite de Bruijn arrays. Discrete applied mathematics, 22(3), pp.289-293. (link)