# Tic-Tac-Toe on a Torus: Move First or Second?

• maze
In summary, if you play tic-tac-toe on a torus, it is proven that the first player always has a non-losing strategy. This is because if the second player has a winning strategy, the first player can use a specific strategy to guarantee a win. However, variations to the game can potentially change this outcome.
maze
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 .. ..

maze said:
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

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.​

## 1. What is Tic-Tac-Toe on a Torus?

Tic-Tac-Toe on a Torus is a variation of the classic game of Tic-Tac-Toe, where the board is in the shape of a torus, or a donut. This means that the board is a 3x3 grid, but the top and bottom edges are connected, as well as the left and right edges.

## 2. How is Tic-Tac-Toe on a Torus different from regular Tic-Tac-Toe?

The main difference is the shape of the board. In regular Tic-Tac-Toe, the board is a flat 3x3 grid, while in Tic-Tac-Toe on a Torus, the board is a donut shape. This changes the way the game is played and adds new strategies.

## 3. Should I move first or second in Tic-Tac-Toe on a Torus?

There is no clear answer to this question. Some players argue that moving first gives an advantage, while others argue that moving second allows for more flexibility. The best strategy may vary depending on the player's style and the specific game situation.

## 4. Is Tic-Tac-Toe on a Torus a solved game?

No, Tic-Tac-Toe on a Torus is not a solved game. While regular Tic-Tac-Toe has been proven to always result in a draw with perfect play, the same cannot be said for Tic-Tac-Toe on a Torus. It is still an open problem for mathematicians to determine the optimal strategy for this game.

## 5. Can Tic-Tac-Toe on a Torus be played with more than 2 players?

Yes, it is possible to play Tic-Tac-Toe on a Torus with more than 2 players. However, the game becomes more complex and it may be more difficult to determine a winning strategy. It is more commonly played with 2 players, but can be adapted for more players if desired.

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