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Math Help - Tiling a chessboard

  1. #1
    Senior Member Pinkk's Avatar
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    Tiling a chessboard

    Suppose we have an 8 by 8 chessboard where the squares always alternate color from black to white. Suppose two squares that are directly next to each other are removed from each of the two opposite corners (so in total, two black squares and two white squares are removed). Can the remaining 60 squares, 30 black and 30 white, can be covered by 15 T-shaped figures (of three whites and one blue one or three blue squares and one white one)?

    So here is my attempt of proof:

    Suppose we can. Without loss of generality, let us consider the black tiles. So there exists some combination of the 3 black, 1 white T-shapes and 1 black, 3 white T-shapes that cover 30 black squares. Obviously, this combination must add up to 15 (perhaps 15 3-black t-shapes and 0 3-white t-shapes, etc). So the following system of equations must have integer solutions:

    3X + Y = 30, X + Y = 15, where X is the number of 3-black t-shapes and Y is the number of 1-black t-shapes). However, this is a contradiction. So there is no combination of the 3-black t-shapes and 1-black t-shapes that cover exactly 30 black squares and cover the total 60 squares, since we must use 15 t-shapes to cover the 60 squares. So therefore it cannot be done. Q.E.D.

    Is this correct? Thanks.
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  2. #2
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Pinkk View Post
    Suppose we have an 8 by 8 chessboard where the squares always alternate color from black to white. Suppose two squares that are directly next to each other are removed from each of the two opposite corners (so in total, two black squares and two white squares are removed). Can the remaining 60 squares, 30 black and 30 white, can be covered by 15 T-shaped figures (of three whites and one blue one or three blue squares and one white one)?

    So here is my attempt of proof:

    Suppose we can. Without loss of generality, let us consider the black tiles. So there exists some combination of the 3 black, 1 white T-shapes and 1 black, 3 white T-shapes that cover 30 black squares. Obviously, this combination must add up to 15 (perhaps 15 3-black t-shapes and 0 3-white t-shapes, etc). So the following system of equations must have integer solutions:

    3X + Y = 30, X + Y = 15, where X is the number of 3-black t-shapes and Y is the number of 1-black t-shapes). However, this is a contradiction. So there is no combination of the 3-black t-shapes and 1-black t-shapes that cover exactly 30 black squares and cover the total 60 squares, since we must use 15 t-shapes to cover the 60 squares. So therefore it cannot be done. Q.E.D.

    Is this correct? Thanks.
    Looks correct to me.

    By the way, you wrote blue in a couple places you meant black.
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  3. #3
    Senior Member Pinkk's Avatar
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    My mistake, I was trying to find some solutions at other places and other sites used blue and white instead of black and white.
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