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Given two disks, one smaller than the other. Each disk is divided into 200 congruent sectors. In the larger disk 100 sectors are chosen arbitrarily and painted red; the other 100 sectors are painted blue. In the smaller disk each sector is painted either red or blue with no stipulation on the number of red and blue sectors. The smaller disk is placed on the larger disk so that the centers and sectors coincide. Show that it is possible to align the two disks so that the number of sectors of the smaller disk whose color matches the corresponding sector of the larger disk is at least 100.

Solution

Copyright © 1996-2008 Alexander Bogomolny

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution

We fix the larger disk first, then place the smaller disk on the top of the larger disk so that the centers and sectors coincide. There are 200 ways to place the smaller disk in such a manner. For each such alignment, some sectors of the two disks may have the same color. Since each sector of the smaller disk will match the same color sector of the larger disk 100 times among all the 200 ways and there are 200 sectors in the smaller disk, the total number of matched color sectors among the 200 ways is 100 × 200 = 20,000. Note that there are only 200 ways. Then there is at least one way that the number of matched color sectors is 20,000 / 200 = 100 or more.

Reference

  1. Beifang Chen, The Pigeonhole Principle

    Copyright © 1996-2008 Alexander Bogomolny

29628753Page copy protected against web site content infringement by Copyscape


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