perplexus.info :: Games : Rook Placement

Rook Placement (Posted on 2015-09-21)

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.

Submitted by K Sengupta

Rating: 5.0000 (1 votes)

Solution: (Hide)

For an explanation, refer to the solution submitted by Ady TZIDON in this location.
Steve Herman writes in response to the counterexample of 40 rooks as follows:
"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."

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(2): the “official” solution - spoiler	Ady TZIDON	2015-09-27 06:41:47
re: the “official” solution - spoiler	Steve Herman	2015-09-27 06:39:20
the “official” solution - spoiler	Ady TZIDON	2015-09-26 16:45:00
re: Detailed solution (SPOILER)	Charlie	2015-09-26 07:51:35
Detailed solution (SPOILER)	Ady TZIDON	2015-09-26 06:10:45

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