# Wrap around Tic-Tac-Toe

If you play tic-tac-toe on a torus (the board wraps around), would you prefer to move first or second, or does it matter?

EG,
x wins:
X O ..
.. O X
.. X ..

o wins:
X .. X
O O O
X .. ..

## Answers and Replies

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If you play tic-tac-toe on a torus (the board wraps around), would you prefer to move first or second, or does it matter?
It seems that if you go first, you can always force a win. You could lose if you tried to (while going 1st), but you can always win if you go first. And actually, I think you're destined never to have a tie game, either.

DaveE

Hurkyl
Staff Emeritus
Gold Member
It's a theorem that for all games of this type, the first player has a non-losing strategy.

The proof goes as follows: suppose player 2 has a winning strategy. Then player one has a winning strategy as follows:

1. Place his first piece randomly (this will now be called the 'extra' piece)

2. Pretend the extra piece doesn't exist
Note that, when pretending this, he becomes player 2 in his pretend game​

3. Use player 2's winning strategy to win
Note that if the winning strategy ever asks him to play a piece where he's already put his extra piece, then he just stops pretending it's extra, and makes a random play, now considering that piece the extra piece

Since both players cannot win, we have a contradiction. Therefore, there exists a player 1 strategy that guarantees player 2 cannot win.

Of course, there are variations you can make to defeat this technique... but you didn't make one.​