The game Slice is played using a \(m \times n\) rectangular piece of paper as a board. Players alternate turns, on each turn they choose a rectangle and cut it into two rectangles, each with integer side lengths. The last player who is able to cut a rectangle is the winner. If \(1 \leq m \leq 20\) and \(1 \leq n \leq 20\), for how many of the \(400\) different starting games does the first player have a winning strategy, no matter how the second player plays?
Details and assumptions
For a \(1 \times 1\) board, the second player is a winner.