Problem:

A standard 8x8 chessboard has two opposite corners removed. Can you cover the remaining board with 31 dominoes, each covering exactly two squares?

Scroll to the bottom of the page for the answer

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Solution:

No, you cannot. Removing two opposite corners leaves an imbalance between the number of black and white squares (30 of one and 32 of the other), but each domino would cover one black and one white square.