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Math Help - Maximum number of 5 cell figures on an 8x8 board

  1. #1
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    Question Maximum number of 5 cell figures on an 8x8 board

    I wasn't entirely sure where to post this so I guessed here would be the best place.

    See the figure attached for the problem statement.

    How do you go about solving a problem like this? There must some sort of proof to go along with it explaining how this undoubtedly the maximum number of 5 cell figures, right?

    I'm not doing a math degree or anything so maybe this is a common type of problem in a certain area of mathematics.

    Can somebody help clear things up? A start at the problem would be nice!
    Attached Thumbnails Attached Thumbnails Maximum number of 5 cell figures on an 8x8 board-5cellfigures.jpg  
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  2. #2
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    This a probability problem. you have 16 cells out of which u need to take out 5 cells that are somehow connected. So find the total number of cases then eliminate those you don't need such as the corner cells. or you can simply count the cases...
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  3. #3
    MHF Contributor Unknown008's Avatar
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    Maybe these will help you

    Polyomino - Wikipedia, the free encyclopedia

    Pentomino - Wikipedia, the free encyclopedia

    Those are called polyominoes. You are certainly familiar with dominoes (d - two, omino), well, the ones in your problem involves pentominoes (pent - five, omino)
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  4. #4
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    If you want to try something else (kinda similar), go here:
    Problem #19
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  5. #5
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    You can fit eight of these X pentominoes onto an 8x8 board, as in the attachment. I don't see any way to improve on that, but it would be nice to have a proof that this is the maximum number.
    Attached Thumbnails Attached Thumbnails Maximum number of 5 cell figures on an 8x8 board-chessboard.gif  
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  6. #6
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    Quote Originally Posted by jegues View Post
    I wasn't entirely sure where to post this so I guessed here would be the best place.

    See the figure attached for the problem statement.

    How do you go about solving a problem like this? There must some sort of proof to go along with it explaining how this undoubtedly the maximum number of 5 cell figures, right?

    I'm not doing a math degree or anything so maybe this is a common type of problem in a certain area of mathematics.

    Can somebody help clear things up? A start at the problem would be nice!
    Mathematically speaking,
    on an 8x8 chessboard, there are 5(12)+4 small square cells.

    Therefore, at least 4 cells will necessarily be unoccupied.
    Hence the maximum number of arbitrarily-shaped non-overlapping 5-cell figures is 12,
    which Opalg has shown.
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  7. #7
    MHF Contributor undefined's Avatar
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    Alternate way to place 8 figures, of course the figures can be shifted to get some more arrangements.

    Edit: sorry the image is so big, i don't have photoshop on this computer and it's a bit of a hassle to resize in a way that looks nice.
    Attached Thumbnails Attached Thumbnails Maximum number of 5 cell figures on an 8x8 board-5cellfituresgrid.gif  
    Last edited by undefined; September 26th 2010 at 01:45 PM.
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