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A 10×10 chessboard contains 41 rooks. Prove that there are five rooks that don’t attack one another.

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Roll the board into a cylinder. The cylinder has 10 diagonals; because there are 41 rooks, some five of them must occupy one diagonal, and thus do not attack one another. From Alexander Soifer and Edward Lozansky, “Pigeons in Every Pigeonhole,” Quantum, January 1990.