Tic-Tac-Toe on a Torus

Date: 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/