1. Raj wants to paint a checkered pathway in a park (really really big) with red and green paints. He wants to paint all the squares on the pathway such that it has no rectangle with all the corners in the same color (Let us call them the mono-corner rectangles). The picture on the left shows a correct case and picture on the right shows a failure case i.e. a mono- corner rectangle (since all 4 corners of the rectangle have same color). (a) Prove that it is impossible for Raj to paint all the squares of the pathway without any mono-corner rectangles. (b) Prove that using 3 colors instead of 2 will not help avoid the mono- corner rectangles.

1. Raj wants to paint a checkered pathway in a park (really really big) with red and green paints. He wants to paint all the squares on the pathway such that it has no rectangle with all the corners in the same color (Let us call them the mono-corner rectangles). The picture on the left shows a correct case and picture on the right shows a failure case i.e. a mono- corner rectangle (since all 4 corners of the rectangle have same color). (a) Prove that it is impossible for Raj to paint all the squares of the pathway without any mono-corner rectangles. (b) Prove that using 3 colors instead of 2 will not help avoid the mono- corner rectangles.

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter17: Markov Chains

Section: Chapter Questions

Problem 12RP

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