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 column of colored cells on the right of the picture depicts 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. Thus, the location of the X in the top left cell of the 3 x 3 corresponds to the top cell in the column and color setting. And, the location of the O in the top right cell corresponds to the cell that is third from the top in the column and color setting.

     In this case, the mapping between the two settings is quite regular...row 1 of the 3 x 3 matrix corresponds to the top 3 cells of the column and color setting; row 2 of the 3 x 3 matrix corresponds to the next 3 cells of the column and color setting; and so on. This mapping could have been arbitrary in the sense that any one of the cells in the 3 x 3 matrix could have been mapped to any one of the cells in the column and color setting. From a purely mathematical point of view, all that is required to maintain the functional isomorphism in this case is that the mapping be one-to-one; and, we would need to remember the various triples of cells that constitute a winning situation. Color and hue were used for this purpose in the column and color setting. That is, three cells of the same color or three cells of the same hue or three cells of differing color but adjacent hue constitute a winning situation.

     Finally, the red square filled with the integers 1 - 9 placed over two rectangles, one light and the other dark gray; depicts yet another setting within which Tic-Tac-Toe could be played. We will refer to this setting as the Numbers Setting. (I have seen it referred to as Number Scrabble.) 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 numbers 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 one. Nonetheless, there is a clear sense in which Tic-Tac-Toe can be realized in each of these quite different physical settings.
     Clicking on any of these depictions will create a pop-up window within which an animated version of a tic-tac-toe game is played in that setting . Click here to view this animation an animation of the game played simultaneously in each of these three settings.
     Now that we have illustrated these 3 settings you may want to read over p. 68 of your text again and study the definition of an isomorphism that is provided there. Additionally, you may want to click here to view yet another setting which is homomorphically related to our standard Tic-Tac-Toe game.

Levels Hypothesis | Introduction - Table of Contents
 © Charles F. Schmidt