Tic Tac Toe and the Levels Hypothesis |
The picture to the right depicts three different "physical environments" within which Tic Tac Toe can be played. The figure in the upper left depicts the familiar setting that involves a 3 x 3 matrix of squares where the moves of each player involve placing an X (Player X) or an O (Player O) on one of the empty squares of this matrix.
The stack of colors to the right depict a different setting within which the game could be played. X and O remain the moves, but the nine squares are arranged as a column rather than in a 3 x 3 matrix. The arrows show part of the mapping between these two settings. Note that color in the column setting of Tic Tac Toe corresponds to the row pattern in the 3 x 3 setting; and hue to the columns of the 3 x 3 setting.
Finally the red square filled with the integers 1 - 9 placed over two rectangles, one light and the other dark gray; depict yet another setting within which Tic Tac Toe could be played. The placement of the numbers in the red square shows the correspondence of the number to the squares in the 3 x 3 matrix setting of the game. In this setting, each player simply selects an integer in the range 1 - 9 that has not already been selected and places it in that player's rectangular area. The first player to have a subset of the numbers selected that sums to 15 is the winner of the game. Note that the mapping between this numeric setting of the game and the spatial versions of the game is not a simple mapping. Nonetheless, there is a clear sense in which Tic Tac Toe can be realized in each of these quite different physical settings. We can illustrate this by animating a game of TicTacToe that is depicted being played simultaneously in each of these settings. Click here to view this animation.