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Rook Placement (Posted on 2015-09-21) Difficulty: 3 of 5
We place 41 rooks on a 10 x 10 chessboard. Prove that one can choose five of them that do not attack each other.

*** Two rooks "attack" each other if they are in the same row or column of the chessboard.

Source: A problem appearing in Colorado Mathematical Olympiad.

No Solution Yet Submitted by K Sengupta    
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re: the “official” solution - spoiler | Comment 4 of 5 |
(In reply to the “official” solution - spoiler by Ady TZIDON)

Thanks, Ady, for the clear and witty editing of the "official" solution.


The counter-example that you requested is having all of the rooks in 4 rows (or having them all in 4 columns).  With 40 rooks so placed, no set of 5 rooks can be found that do not attack each other.  By the pigeonhole principle, if they are all in 4 rows, then at least two of them must be in the same row.

  Posted by Steve Herman on 2015-09-27 06:39:20
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