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Thoughts on a conjecture of Erdős

Published online by Cambridge University Press:  16 October 2017

Stan Dolan
Stan Dolan*
Affiliation:
126A Harpenden Road, St Albans AL3 6BZ
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Extract

If two squares with no interior point in common are drawn inside a unit square then prove that the sum of their side-lengths is at most 1.

This problem was posed in the 1930s by Paul Erdős [1]. It is the simplest case of a still unsolved conjecture.

If k2 + 1 squares with no interior point in common are drawn inside a unit square then the maximum possible sum of their side-lengths is k [2].

We shall use the notation S(n) to denote the maximum possible sum of the side-lengths for n squares drawn with no interior point in common inside a unit square. The main aim of this article will be to develop an approach to the study of the function S which will give surprisingly simple proofs of a number of known results. This approach will then be used to prove a new result about the asymptotic behaviour of S.

Type
Articles
Information
The Mathematical Gazette , Volume 101 , Issue 552 , November 2017 , pp. 449 - 457
Copyright
Copyright © Mathematical Association 2017 

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References

1. Erdős, P. and Graham, R. L., On packing squares with equal squares, Journal of Combinatorial Theory, Series A 19 (1975) pp. 119–123.Google Scholar
2. Erdős, P. and Soifer, A., Squares in a square, Geombinatorics 4 (1995) pp. 110–114.Google Scholar