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Perfect Stategy -- Placing picked numbers on two rows of a game

  1. Sep 30, 2017 #1
    1. The problem statement, all variables and given/known data
    My teacher gave our class this problem to do Two players take turns placing an unused number from {1; 2; 3; 4; 5; 6; 7; 8} into one of the empty squares in a 2 by 4 array. The game ends once all the squares are tiled. The 1st player wins if the product of the numbers in the top row is greater. The second player wins if the product of the numbers in the bottom row is greater. What is the perfect strategy for each player? Please help me, if you cannot provide the answer please give me a hint. This problem has been bugging me for so long.

    Reference https://www.physicsforums.com/threads/help-with-some-problems.927068/
    2. Relevant equations


    3. The attempt at a solution
    I am stuck, because the perfect strategy depends on the other person's play. And there are 8! ways this game can be played out.
    I know the first player should put 1 on the second row.
     
  2. jcsd
  3. Sep 30, 2017 #2

    berkeman

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    Staff: Mentor

    Welcome to the PF. :smile:

    Of course we cannot give you the answer -- that is against the PF rules. If I understand the problem statement, the strategy seems straight-forward. The players each have one of the two empty rows assigned to them at the start of the game, right?

    Tell us your thinking so far, so we can guide you a bit...
     
  4. Sep 30, 2017 #3

    Ray Vickson

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    How do you know that player 1 should put "1" in row 2? Might it not be better for player 1 to put "8" in row 1?
     
    Last edited: Sep 30, 2017
  5. Sep 30, 2017 #4
    I have found the Solution.
    I realized something very important while solving the problem
    Thanks anyways
     
  6. Oct 5, 2017 #5
    Sorry, never mind I still need help
     
  7. Oct 10, 2017 #6
    The game tree you need to analyse is actually smaller than that of tic-tac-toe. easily done by hand with the following:
    - each player has only two possible moves to consider on each move.
    - You can see really early if the game is already won. Work out what the product is you need to get to win and see if each player can still get it by using the maximum of the numbers that are left.
     
  8. Oct 11, 2017 #7

    kuruman

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    It's not about winning, it's about forcing your opponent to lose. There is a simple strategy that guarantees that the opponent of the player who goes first loses. Can you find it?
     
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