Thoughts on a conjecture of Erdős. Issue 552 (16th October 2017)

• Record Type: Journal Article
• Authors: Dolan, Stan
• Is Part Of: Mathematical gazette. Volume 101:Issue 552(2017)
• Page Start: 449
• Page End: 457
• Publication Date: 2017-10-16
• Subjects: Mathematics -- Periodicals; Mathématique; Mathematik; Mathematics; Periodicals; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); Zeitschrift; Online-Publikation; 510
• DOI: 10.1017/mag.2017.126
• Languages: English
• ISSNs: 0025-5572

Abstract

Abstract : 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 k 2 + 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 .