Turning Turtles

At each move a player chooses an "O" and turns it into an "X". At the same time this player may, if he so wishes, change a letter in any other square to the left from the first one. In the left square, the player is allowed to turn "O" into "X" and also "X" into "O". To perform a move, the player should first click under the square he plans to change. After selecting 1 or 2 squares, click on the "Make Move" button. If you plan to change a single square (an "O" into an "X") you may click on that square directly. This is kind of a shortcut with which one should be cautious. You can't undo your moves. The player to make the last move wins.

In the original version, players turn turtles upside down and back to their feet. Not having any aptitude for painting, I settled on the TicTacToe symbols to present the two possible states of each square.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

The game is a Nim in disguise, Nim with eleven rows of lengths 1, 2, 3, ..., and 11. Each square corresponds to one row. "X" in a row indicates that a row is empty while "O" means that the row is full of check marks. Changing "O" into "X" is the same as removing a whole row of check marks. If you wish to remove m check marks from a row that has n > m of them, change the square number n and then change the square number n-m to the left. It does not matter whether at this point m contains "O" or "X".


  1. E. R. Berlekamp, J. H. Conway, R. K. Guy, Winning Ways for Your Mathematical Plays, v1, A K Peters, 2001.
  2. E. R. Berlekamp, J. H. Conway, R. K. Guy, Winning Ways for Your Mathematical Plays, v2, Academic Press, 1982.
  3. R. K. Guy, Fair Game, Comap's Explorations in Mathematics, 1989

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