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Applications of the pigeon-hole principle

Published online by Cambridge University Press:  01 August 2016

Kiril Bankov
Kiril Bankov*
Affiliation:
Department for Education of Mathematics and Informatics, University of Sofia, 1126 Sofia, Bulgaria School of Education, University of Southampton, United Kingdom
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Extract

Consider the statement (whose truth is obvious):

If n objects are placed in k boxes, where n = qk + r, q and r are positive integers, and 0 < r < k, then at least one box contains more than q objects.

This is known as a pigeon-hole principle (or Dirichlet’s). It is one of the strategies which is often used for problem-solving. Although it is one of the most powerful tools of combinatorics, the principle can be used successfully in other branches of mathematics. Below you can find a collection of problems which can be solved easily by applying this principle to geometrical objects.

Type
Articles
Information
The Mathematical Gazette , Volume 79 , Issue 485 , July 1995 , pp. 286 - 292
Copyright
Copyright © The Mathematical Association 1995

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References

1. Rakovska, D., Tonov, I., Bankov, K. and Vitanov, T., Mathematics competitions 4–7 grade, Regalia 6, Sofia (1993) (in Bulgarian).Google Scholar
2. Tabov, J. and Bankov, K., Mathematical competitions around the world, Nauka i izkustvo, Sofia (1988) (in Bulgarian).Google Scholar
3. Taylor, P., International Mathematics Tournament of the Towns: questions, strategies and solutions, Book 1, Australian Mathematics Foundation Ltd (1989).Google Scholar