Neurobics

Noughts and Crosses

By Russell Dear

Of all the board games ever devised, noughts and crosses and its near relatives have probably been the most popular -- at least in terms of the length of time they have been played. A British Museum exhibition of board games displayed one example dated to 850 BC.

Noughts and crosses, or ticktacktoe, remains popular. It is related to similar games such as Pong hau k'i from China, the Maori game Mu Torere, and Achi played by Ghanaian children. Games like Nine Men's Morris and Go-Moku are related to it. More complicated versions are played in three or more dimensions, or on boards of unlimited size.

In the standard game two players take turns to place a nought or a cross in one of nine unoccupied positions on a three-by-three grid. One player places only noughts, the other only crosses. The first player to achieve three in a line -- vertically, horizontally or diagonally -- wins the game.

If the game is played rationally by both players it will end in a draw (playing rationally here means having full knowledge of the game and playing as well as possible).

For many people, the excitement of the game centres on waiting for an opponent to make an error and then gleefully applying the coup de grace. In Game I, a game between two careful players has already begun. What is the outcome of the game?

There are many minor variations of the standard game. Looking for a winning strategy in some cases can be challenging. In the misère version, sometimes called Toetacktick, the first player to place three of his or her symbols in a straight line loses. Can you predict the outcome of Toetacktick if both players play rationally?

In Wild Ticktacktoe, players may use either symbol. The first to complete a line of three in either symbol wins. If you play the game you'll find that the first player can always win. What about the misre version, Wild Toetacktick, where the first player to make three in a line loses? What happens in this version of the game if both players play rationally?

A less interesting version of the game, perhaps, is better called noughts and noughts, or crosses and crosses. Here both the players use the same symbol. In this, and its misère version, the first player can always win. Which version of noughts and noughts is being played in Game II?

Two other versions of the game were devised in New Zealand. As far as I know both end in a draw if played rationally. In Smatick, both players in turn first play a nought, then both play a cross, then both a nought, and so on. The first to complete three in a line of either symbol wins the game. Smatoe is the misère version of Smatick.

Answers: Since we know both players in Game I are careful, it must be X's turn (why?). The game will end in a draw.

In Toetacktick the first player can force a draw.

In Wild Toetacktick either player can force a draw.

It is impossible to tell which version of noughts and noughts is being played in Game II. What can be said is that at least one of the players is inexperienced or not playing rationally, as this position would never be reached if both players were playing rationally.

Russell Dear is a Mathematician living in Invercargill