Jacob and Pavel are playing a game on an $m \times n$ chessboard: there is a pawn at the bottom left corner of the chessboard, and starting with Jacob, they move it in a straight line as many squares as they wish either upwards or to the right. The first to reach the top right corner wins. For which $(m, n)$ does Jacob have a winning strategy?
Jacob has a winning strategy for all $(m, n)$ such that $m \neq n$.

It consists of coloring the squares with coordinates $(m, n),(m-1, n-1),(m-2, n-2), \ldots$ At each turn, Jacob can place the pawn on one of these squares (unless it is already there at the start, hence the condition $m \neq n$), and Pavel cannot reach a colored square from another colored square.

When $m=n$, Pavel can apply this strategy and therefore he has a winning strategy.