The Kangaroo Hops for Two

The game Upright is played by two players on an \(m \times n\) board of squares and has the following rules:

  1. At the start of the game, a kangaroo game piece is placed on the bottom left square of the board.
  2. Players alternate turns moving the kangaroo, and the first player moves first.
  3. On the player's turn, they can either move the kangaroo some number of squares to the right, keeping it in the same row, or they can move it to the leftmost square on the row above.
  4. A player loses if they are unable to make a move.

If \(m\) and \(n\) can each be any number between 1 and 20 inclusive, for how many of the 400 possible game board sizes can the second player win if both players play optimally?


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