I've also seen several different variants for chess on a torus (imagined -- still played on a flat board, but you play as if each edge is connected to the oppos

I've also seen several different variants for chess on a torus (imagined -- still played on a flat board, but you play as if each edge is connected to the opposite one). Pieces end up having basically the same allowed moves (IIRC the only required change is that a piece making a move where it loops around the board and back to its original position is disallowed), but a different initial setup is required, because given the regular one, both sides start the game already check-mated. For further brain-bending, you can do a Klein bottle instead of a torus.