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?
Answer
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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