Tic-Tac-Toe on a Finite Plane
Maureen T. Carroll, Steven T. Dougherty · Mathematics Magazine · 2004
Everyone knows how to play tic-tac-toe. On an n × n board, if a player is able to place their marks "n in a row" either horizontally, vertically, or diagonally then they have won the game. What if we keep the rules of the game the same but increase the number of possible lines to include some that would not fit our standard description of "in a row"? We have done this in a systematic way by including only those lines prescribed by a finite affine plane. Voila! The game which grew tiresome for us as children is transformed into an interesting, geometrically motivated game. For example, the game shown on the 3 × 3 board below is a win for X while the 4 × 4 board shows a win for O on the affine planes of order 3 and 4, respectively... As shown by these examples, it is difficult to identify wins on an affine plane because lines in a finite plane do not have to be "straight".