In a certain two-player game, you start with a rectangular $m \times n$ grid of squares. On each turn, a player either makes a horizontal cut and takes away the portion of the rectangle above the cut, or makes a vertical cut and takes away the portion to the right. Whichever player takes the last square (in the bottom left corner) loses. If both players play perfectly, determine for which values of $m$ and $n$ the first player will win.
The first player wins whenever the grid is not a square (i.e. $m \neq n$ ). The strategy is to always leave the other player with a square grid (where $m=n$ ), since then they will eventually be left with a $1 \times 1$ square and will lose. This works since starting from a square, it is impossible to get to a smaller square, but starting from a non-square, it is always possible to get to a square. Thus, the first player wins whenever the starting grid is not a square, and the second player wins whenever the starting grid is a square.