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### 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?

- Doctors Nick and Sarah, The Math Forum
http://mathforum.org/dr.math/
```
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High School Geometry
High School Higher-Dimensional Geometry
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