Tic-Tac-Toe on a TorusDate: 03/29/2001 at 13:31:15 From: JD Subject: Tic Tac Toe Mu Fest is coming up soon, on April 4th. I want to make the game tic tac toe not end up in a tie. Do you know how to do that? I want to make this project creative. Do you have an idea that will be useful? Date: 03/30/2001 at 12:33:21 From: Doctor Nbrooke Subject: Re: Tic Tac Toe Good afternoon JD, and thanks for writing to Dr. Math. You can take a cue from topology to make some interesting math tic-tac-toe games. Don't worry; it's not as hard as it probably sounds. Take a look at your ordinary tic-tac-toe board: | O | ---+---+--- O | X | X ---+---+--- | X | O Now pretend that you've glued the opposite edges to each other. This forms a shape that mathematicians call a torus. To visualize this easily you can take a copy of the board and paste it to each side of the board, like this: | | O | | |---+---+---| | O | X | X | |---+---+---| | | X | O | -----------+-----------+----------- | O | | | O | | | O | ---+---+---|---+---+---|---+---+--- O | X | X | O | X | X | O | X | X ---+---+---|---+---+---|---+---+--- | X | O | | X | O | | X | O -----------+-----------+----------- | | O | | |---+---+---| | O | X | X | |---+---+---| | | X | O | Can you find where O has tic-tac-toe, and has won the game? After a little practice, you and your peers can learn to play this game without copying the board. You'll learn to imagine the copies instead. You can also play tic-tac-toe on a torus on the Web at Jeff Weeks' site. For an introduction start here: http://www.geometrygames.org/TorusGames/ You can make this game more interesting by using bigger grids. Try getting 6 in a row on a 6x6 board. X | O | O | X | X | X ---+---+---+---+---+--- O | X | X | X | O | O ---+---+---+---+---+--- O | X | O | O | O | X ---+---+---+---+---+--- X | O | X | O | O | X ---+---+---+---+---+--- O | X | X | O | O | X ---+---+---+---+---+--- O | X | O | X | X | O Exercise: How many 6-in-a-rows can you find on the above board? Good luck with your project. - Doctors Nick and Sarah, The Math Forum http://mathforum.org/dr.math/ |
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