Does anyone have some thoughts about generalizations of hex to various manifolds? On the torus, it seems to work out fairly nicely, although I haven't actually played any games with anyone. Take any tiling of the torus by hexagons (or for that matter, any tiling by polygons such that 3 polygons meet at any vertex. One reasonable method is to use a standard hex board, and "wrap it around" into a torus. Partition the possible "slopes" in first homology H_1(T^2) into two sets, for instance by using some coordinate system at an irrational slope and assigning the first and third quadrants to white, the second and fourth quadrants to black. If the torus is a wrapped hex board, a more direct rule is to say a slope is black if there is a line segment of that slope connecting the two black sides---you just need to be a little careful about the exact idetification when it's wrapped <--> the exact rule for acceptable slopes. Players alternate putting black and white stones on the hexagons, until one of them has a closed path in a homology class of their color. It's a nice little puzzle to extend the usual analysis of hex, to prove that if the board is totally filled with black and white stones, exactly one player has a curve in a homology class of the designated color. I think Dan Asimov brought up with me the question of doing it on higher-dimensional disks, long ago ... there were good definitions for even-dimensional disks, which I won't try to explain just here and now, since the boundary conditions make it a little more complicated than doing it on a closed manifold. What other manifolds have reasonable rules for hex generalizations? I think other surfaces should have good rules, but I have a hard trouble making simple definitions for these rules. For CP^2, a good rule seems to be to declare the winner to be anyone such that H^2(their colors) is non-zero. On the other hand, aI don't think there's any rule analogous to standard hex for T^3 ... anyway, I threw this out as a question in the topology class that I'm teaching, and I'd welcome any thoughts or insights. Bill Thurston