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Wrap around Tic-Tac-Toe

  1. Feb 9, 2009 #1
    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 .. ..
     
  2. jcsd
  3. Feb 10, 2009 #2
    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
     
  4. Feb 10, 2009 #3

    Hurkyl

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    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.​
     
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