There are (2n - 1) rooks on a (2n - 1)×(2n - 1) board placed so that none of them threatens another. Prove that any n×n square contains at least one rook.

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Copyright © 1996-2017 Alexander BogomolnyThere are (2n - 1) rooks on a (2n - 1)×(2n - 1) board placed so that none of them threatens another. Prove that any n×n square contains at least one rook.

### Solution

Assuming an n×n square is void of the rooks, there remain

(This is Problem 11.5 from the 1993-1994 St Petersburg Regional Mathematical Olympiad, grade 11.)

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