Take an ordinary chessboard and cut off two diagonally opposite corners. Now: Is it possible to tile the remaining 62 squares with 31 dominoes?
This calls for inspiration rather than trial and error. Most people see the solution immediately or not at all.
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No, it can’t be done.
Each domino covers two squares, a light one and a dark one. Therefore a board can be tiled perfectly only if it has an equal number of light and dark squares.
The original board met this requirement, but we’ve removed two squares of the same color (diagonal corners). So the task is impossible.
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