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Math Help - Complete the chessboard

  1. #1
    Moo
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    Complete the chessboard

    Hi !

    Try this :
    Remove the top-left corner and the bottom-right corner of a chessboard (thus remain 62 cases).
    You have a set of double 9 dominoes. Pick up 31 of these dominoes. How many ways are there to fill in the chessboard with these 31 dominoes ? This only includes the position of the dominoes, not the way they're placed (that means if you have 1-9, it matters that it's in A-6 and A-7, but it doesn't matter whether the 1 is on A-6 or A-7).

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  2. #2
    Super Member wingless's Avatar
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    Zero. You can't fill such board with 31 dominos.

    Even if we could, we still couldn't find how many ways there are because a set of dominos up to 9 makes 36 dominos and we take 31 of them. But the possibilities depend on the dominos we choose.


    The proof of this is easy. Assume we have the chessboard below:



    Now, how many tiles left? 30 white and 32 black.
    But every domino will cover 1 white and 1 black tile. They can cover 31 white and 31 black, but not 30 white and 32 black. Thus it's impossible to fill such board.
    Last edited by wingless; August 23rd 2008 at 12:25 PM.
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  3. #3
    Moo
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    Correct answer !

    Even if we could, we still couldn't find how many ways there are because a set of dominos up to 9 makes 36 dominos and we take 31 of them. But the possibilities depend on the dominos we choose.
    You just have to multiply the answer by the number of possible pickings. But I thought it was *clear* in my text that we didn't bother with the tiles we were taking :x
    Last edited by Moo; August 24th 2008 at 02:09 AM.
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  4. #4
    Super Member wingless's Avatar
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    I have a similar question too.

    You have a normal 8x8 checkered board. You should start from the top left tile and go to the bottom right tile by going on every tile once. In how many ways can you do this?
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  5. #5
    Moo
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    Quote Originally Posted by wingless View Post
    I have a similar question too.

    You have a normal 8x8 checkered board. You should start from the top left tile and go to the bottom right tile by going on every tile once. In how many ways can you do this?
    Yup.
    Zero too.
    Assuming that you cannot go diagonally (so you must go from a tile to another with a different color), since you start from a white tile and finish on a white tile and that there is an equal number of white and black tiles, there must be a part where you jumped from a black tile to a black tile, which is impossible, considering the assumption.


    (hmm sorry if it's not clearly stated :x)
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  6. #6
    Super Member wingless's Avatar
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    Ah yes, you can't go diagonally and your answer is correct =) We have to make 63 moves and an odd number of moves must finish on a different color.
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