On a \(8\times 8\) chessboard, two opposite corners are eliminated.
Is it possible to cover exactly the remaining 62 squares with \(1\times 2\) dominoes, without overlaps or gaps?
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Mutilated board and dominoes
It is an elegant impossibility: it resists not because of excess combinations, but because there is something on the board that no coating can correct.
Hints
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- Don't try to try placements one by one.
- Each domino always covers one square of each color.
- Look at what color the two opposite corners that are removed are.
Solution
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Answer: It is not possible. A domino always covers 1 white and 1 black square. On a normal 8x8 board there are 32 white and 32 black. If you remove two opposite corners (same color), there are 30 of one color and 32 of the other. Since each domino covers one of each color, a complete coverage would require equality of colors, which is not met here.
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