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Wednesday, July 16, 2008

the chessboard

You have a chessboard (8×8) plus a big box of dominoes (each 2×1). I use a marker pen to put an “X” in the squares at coordinates (1, 1) and (8, 8) - a pair of diagonally opposing corners. Is it possible to cover the remaining 62 squares using the dominoes without any of them sticking out over the edge of the board and without any of them overlapping? You cannot let the dominoes stand on their ends

Answer: Firstly the answer to this question is NO you can't cover the board. There is a trick here, look at the board below. Do you notice anything about the squares at 1,1 and 8,8? Well they are both the same colour. If you think about a Domino placed any where on the board it will necessarily cover a black square and a white square. We have in our example a 32 black squares and 30 white squares to cover. So it just can't be done.

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