A problem from the January 1990 issue of Quantum: Forty-one rooks are placed on a 10 × 10 chessboard. Prove that some five of them don’t attack one another. (Two rooks attack one another if they occupy the same row or column.)
Roll the board into a cylinder and paint each of its diagonals a different color. These 10 regions contain 41 = 4 × 10 + 1 rooks, so at least one region must contain 5 rooks. And rooks on the same diagonal don’t attack one another.
(Alexander Soifer and Edward Lozansky, “Pigeons in Every Pigeonhole,” Quantum 1:1 [January 1990], 25-28, 32.)