Math in… Solved Games

Many games people play are mathematical in some way. While you might not have to do arithmetic to play a game, just finding a winning strategy is a mathematical problem in its own right!

If you’ve played enough games of tic-tac-toe, you probably know it’s good to play in a corner as the first player. If the second player doesn’t play in the middle, the first player can win within a few moves. If the second player plays in the middle, however, they can force a tie.

If it is possible to predict how a game will end when both players play perfectly, the game is said to be solved.

Tic-tac-toe is solved: perfect play from both players ends in a draw. Connect Four and checkers are also solved games. Assuming perfect play, the first player wins Connect Four while checkers ends in a tie.

Chess and Go are not yet solved.

Solving a game occasionally yields a nice strategy that a human player can follow, but strategies often come from analyzing all configurations the game board could have. This can often be simplified by making use of symmetry.

Without making use of symmetry, tic-tac-toe has only 549,946 states that can occur during play. Compare that to the 500,995,484,682,338,672,639 states in checkers and you’ll see why kids can master tic-tac-toe while it took hundreds of computers 18 years to map out a strategy for checkers!

What games do you like to play?