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Double Tic-Tac-Toe Strategy

  1. Feb 15, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider the game of `double move tic-tac-toe' played by the usual rules of tic-
    tac-toe, except that each player makes two marks in succession before relinquishing
    his turn to the other player (you may know tic-tac-toe by the name `noughts and
    crosses'). Prove that there exists a strategy by which the first player always wins.


    2. Relevant equations
    None that I can think of.


    3. The attempt at a solution
    I have no clue how to prove this. The obvious strategy is that player one places an X at one of the corners of the board and then one in the center. Player two can't block all the winning strategies with their two moves.

    The question is, how do I show this in "math speak"?

    Thanks! Sorry for all the posting lately, I'm just terrible at Discrete Math.
     
  2. jcsd
  3. Feb 15, 2012 #2
    After the first turn the board looks like


    +-+-+-+
    |X| | |
    +-+-+-+
    | |X| |
    +-+-+-+
    | | | |
    +-+-+-+


    Where do player 2 need to place O's for you not to be able to win in your turn? Can this be done with just 2 O's? If you can find three disjoint sets of spots that must each be blocked, then you are finished.

    In other words, can you partition the last 7 empty spaces into 3 disjoint subsets such that if any of the 3 disjoint subsets are left untouched by player 2, then you can win in your turn?
     
  4. Feb 15, 2012 #3

    Deveno

    User Avatar
    Science Advisor

    obviously, player 2 cannot win in 1 move (he only can place two marks).

    since player 2 cannot win on their first move, their best strategy is to prevent player 1 from winning on player 1's second move.

    player 2 must place a mark at (3,3), or else player 1 will on the next move, and then win.

    there are 4 possible ways player 1 might place two marks and win on their next move, given that (3,3) is taken: complete the center row, complete the center column, or complete the left column, or the top row. player 2 must block all 4 of these possibilities with a single move.

    show that player 2 can at most only block 2 of these.

    a slightly more challenging question is: suppose player 1 allows player 2 to make his first move for him (still using X's for player 1, and O's for player 2. player 2 does NOT get to place 4 O's). does player 1 still always have a winning strategy?
     
  5. Feb 16, 2012 #4
    +-+-+-+
    |x| | |
    +-+-+-+
    | | | |
    +-+-+-+
    | | |x|
    +-+-+-+

    In this case player two can never win because you have three rows to win, right?
     
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