Problem #89
The Horses of Diomedes
The game
Hercules and Abderus play a game with a cake (as described in Problem 84 of the preliminaries). Hercules eats squares horizontally and Abderus vertically. We define the value of a game as:
+1 if Hercules can win no matter who starts,
-1 if Abderus can win no matter who starts,
+i if the first player can always win, and
-i if the second player can always win.
Consider 30 games, played with cakes of sizes M \times N, with 1 \leq M \leq 3, and 1 \leq N \leq 10.
What is the sum of the values of these games? If your answer is a+ib, enter a,b.
The story
Oh, why is Abderus important? He was eaten alive by the crazed horses of Diomedes, and in revenge Hercules fed Diomedes to his own horses. This seemed to have calmed the crazed horses and they lived happily ever after in a meadow somewhere. Greek myths can be very weird sometimes.