Laertes and Roxane have before them a cake, freshly baked by their mother. It has been lightly scored so as to allow it, when cut, to be divided into $M \times N$ pieces. The two kids decide to play a game to cut the cake. Laertes will cut any rectangle into two pieces along the vertical lines, and Roxane along the horizontal lines. The game ends when the first who is unable to make a move loses.

Laertes always goes first.

Consider 256 games with cakes varying in size from $1 \times 1$ to $16 \times 16$. How many of these games can Laertes always win, if he plays with perfect strategy?