thought of as being composed of 64 squares, with rules that tell you how the squares are connected up. The game is quite different in practice; for starters, one wouldn't want to commence one of these chess games with the pieces in their traditional positions, since the kings would begin on adjacent squares! Pritchard [59] gives various possible starting positions on the torus. Also, on the Klein bottle and the projective plane, one doesn't have the usual black-white pattern; the black square al is immediately above the black square h8. We could fix this if we were willing to use a 9 x 8 board, for example.