Gareth asks: << [I wrote]:
Game of Squares: Players take turns placing a marker of their own color (say white or black) on the vertices of an 8x8 square grid until there is some square (tilted or not) with all four vertices having the same-colored markers on them. The player whose color it is loses. ... Game of Toral Squares: Same game but on the 64 vertices of GxG, where G is a regular octagon.
How are you defining squares formed from the vertices of GxG?
A square is defined here as four *distinct* points of GxG -- P,Q,R,S -- such that the segments PQ, QR, RS, ST are of equal length, and the angles PQR, QRS, RSP, SQP are right. A square is determined by its set of vertices. Oh -- and the metric used on GxG is just the cartesian product of that on G with itself, where G is a regular octagonal curve. --Dan