On a square packing conjecture of Erd\H{o}s

TL;DR

This paper investigates the maximum total side length of non-overlapping squares and triangles packed inside a unit square or equilateral triangle, exploring properties, conjectures, and related shapes like parallelograms.

Contribution

It proves the equivalence between Erd ext{"o}s' conjecture on square packing and the convergence of a related series, and examines properties across different shapes.

Findings

Erd ext{"o}s' conjecture is equivalent to a series convergence.

Properties of packing functions are similar for squares and triangles.

Parallelogram packing shares analogous characteristics.

Abstract

Let $f(n)$ be the maximum sum of the sides of non-overlapping squares (or equilateral triangles) packed inside a unit square or (unit equilateral triangle). In this paper, we explore some properties of $f$ and examine how the square and triangle cases are similar. We prove that a conjecture of Erd\H{o}s, which says that $f(k^2+1) = k$ for all $k$, is equivalent to the convergence of the series $\sum_{k\geqslant 1}(f(k^2+1)-k)$. We also explore the case of parallelograms and discuss how that is similar to the case of unit square and triangle.