Have you ever played Tic-Tac-Toe? Did you win? Did you know that each Tic-Tac-Toe game will always end in a tie unless one player makes a mistake?

There are nine squares on a Tic-Tac-Toe game board, so the first player has nine choices for the first move. The second player has eight choices for the second move (one square is taken), which makes a total of 72 possible arrangements after the first two turns. After five turns, there are 15,120 possible arrangements.

Someone could possibly win on the sixth turn if the other player played really badly, so the nine possible winning positions must be subtracted from the total, leaving 60,471 arrangements after only six moves. No one can keep track of that many combinations, but that is not necessary. The successful Tic-Tac-Toe strategy is simply to block the other player's moves while hoping the other player makes a mistake.

A...