A game is played on a $100\times 100$ board between three players, $A$, $B$, and $C$, in this order. Each turn, the player must paint a cell adjacent to (at least) one of the sides of the board. The player cannot paint a cell adjacent to a painted cell, or a cell symmetric to a painted cell with respect to the center of the board. The player who cannot paint loses. Can $B$ and $C$ combine to make $A$ lose?
My thought is that $B$ and $C$ could combine by, supposing there is an $x$- and $y$-axis with the center of the board as the origin, having $B$ paint the cell symmetric to what $A$ painted with respect to the $x$-axis, and similarly for $C$ and the $y$-axis. This would make $A$ lose because $A$ cannot paint the cell symmetric with respect to the origin. But the strategy doesn't work, because if $A$ paints a cell adjacent to an axis, the cell on the other side of the axis adjacent to it cannot be painted.
Hint: The problem was misleading you by presenting the game on a rectangular board, causing you to look for rectangular symmetries. Really, the game is just being played on a ring of 396 cells, where each cell is adjacent to its neighbors and its opposite. Since 396 is a multiple of 3, this ring has trilateral symmetry, which is more helpful since this is a three player game.
Please have another go using this hint, the solution is rather beautiful and satisfying to find.