![]() ![]() ![]() This is the number of different combinations of filling the grid with 5 Xs and 4 Os. This counts each X and O as distinct from other X and Os.ĩ choose 5 = 126. (First you have 9 choices of squares, then there are 8 choices of squares etc). This is the total number of ways that positions can be filled on the grid. This is the total number of possible game positions in a 3×3 grid – as every square will either be a O, X or blank.ĩ! = 362,880. Once this complete tree is drawn, any participant can work through this tree to see what is their optimal move at any one time from any position.Īn upper bound for the number of positions and number of different games is given by:ģ 9= 19,683. ![]() We can expand this game tree to cover every possible outcome for the game. This is the start of the game tree for Noughts and Crosses. The way to solve Noughts and Crosses is to use combinatorial Game Theory – which is a branch of mathematics that allows us to analyses all different outcomes of an event. In fact it’s so simple that it has been “solved” – before any move has been played we already know it should result in a draw (as long as the participants play optimal moves). It’s a very simple game – the first person to get 3 in a row wins. The game of Noughts and Crosses or Tic Tac Toe is well known throughout the world and variants are thought to have been played over 2000 years ago in Rome. ![]()
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