Around the World in Eighty Games: From Tarot to Tic-Tac-Toe, Catan to Chutes and Ladders, a Mathematician Unlocks the Secrets of the World's Greatest Games

by Marcus du Sautoy

Pulling all of these threads together, why do we really play games? According to Caillois, it is “an occasion of pure waste: waste of time, energy, ingenuity, skill, and often of money.”

The game relies on a principle called nontransitivity to create a deck that is well balanced. As in Rock Paper Scissors, it is important that each card be strong against some cards even as they remain vulnerable against others.

Nontransitivity is often quite important in making an effective game. For example, in the role-playing game Dungeons & Dragons, each character is skillful at some things, perhaps fighting, but not so adept at others, like casting spells. The important quality is setting up a varied range of characters where each will be able to beat some monsters but ineffectual when encountering others. A good team has a balance of all traits so that, as with the dice, you can pick the right character to compete in any particular context.

If you are good enough, is it possible for this game to go on forever? Heidi Burgiel, a mathematician at the Geometry Center in Minneapolis, showed that a certain sequence of tetrominos would, despite a player’s best efforts, result in their losing the game. Her theorem proves that if the player encounters alternating Z pieces, then whatever they do to rotate and stack them, the game will be lost after placement of at most 69,600 pieces.

The first example of machine learning in game play also consisted of a machine that learned how to play tic-tac-toe. But rather than being made from electronics, this machine consisted of 304 matchboxes. Created by British computer scientist Donald Michie in 1961, it was called MENACE, which stood for Machine Educable Noughts and Crosses Engine. Each matchbox corresponded to all the possible configurations the game could be in from beginning to end. Again, you didn’t need to consider every state because you used symmetry to reduce things down to 304 different configurations. So a match box might have on its side