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Can you tile a 6x6 chessboard with dominoes, without overlaps or gaps, so that the number of dominoes oriented horizontally is equal to the number of dominoes oriented vertically? Why or why not?

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The answer is

No

Proof/construction:

Paint the board with a horizontal stripe covering every other row, which colors 18 cells. In a tiling as described, each of the 9 vertical dominoes covers one colored cell, and each of the 9 horizontal ones covers an even number of colored cells (0 or 2). But that means an odd number of colored cells are covered, which can't be 18.

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  • $\begingroup$ Beautiful. BTW, I can across this problem while trying to form a proof to this question. $\endgroup$ – Tyler Seacrest Jun 6 '15 at 6:08

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