Two friends, $A$ and $B$, play a game with one single game piece on a rectangular board with $m$ rows and $n$ columns. $A$ begins the game by moving the game piece from its starting point $(1, 1)$ to either $(1, 2)$ or $(2,1)$ i.e one can only move in horisontal or vertical direction, one step every move. No point is allowed to be entered twice. The player that no longer can make any move loses.
Is there any winning strategy?
Intuitively it feels like we need to consider rows and columns with equal and non-equal parities, and I tried to reduce the problem to smaller cases, but it gave me nothing. Any suggestions?