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.
Mutilated board and dominoes
Riddle statement
On an \(8\times 8\) chessboard, two opposite corner squares are removed.
Can the remaining 62 squares be covered exactly with \(1\times 2\) dominoes, without overlaps or gaps?
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Solution
Answer: You can't.
A domino, placed in any orientation, always covers one white and one black square: the adjacent squares on a chess board are always different colors.
On a \(8 \times 8\) board there are 32 white squares and 32 black squares. The two opposite corners are the same color - both white or both black depending on the choice -, so that when they are removed there are 30 squares of one color and 32 of the other.
To cover the remaining 62 squares with 31 dominoes, the colors would need to be in equal numbers. Since they are not, coating is impossible.