Reducing Moser's Square Packing Problem to a Bounded Number of Squares

TL;DR

This paper reduces Moser's Square Packing Problem to a finite case by computing a number N, simplifying the problem to a finite set of squares, which could facilitate solving the longstanding open problem.

Contribution

The paper introduces a method to limit the problem to a finite set of squares by computing a natural number N, making the problem more tractable.

Findings

Reduction of the infinite problem to a finite case

Explicit computation of a bounding number N

Potential pathway to solving Moser's problem

Abstract

The problem widely known as Moser's Square Packing Problem asks for the smallest area $A$ such that for any set $S$ of squares of total area $1$, there exists a rectangle $R$ of area $A$ into which the squares in $S$ permit an internally-disjoint, axis-parallel packing. It was formulated by Moser in 1966 and remains unsolved so far. The best known lower bound of $\frac{2+\sqrt{3}}{3}\leq A$ is due to Novotn\'y and has been shown to be sufficient for up to $11$ squares by Platz, while Hougardy and Ilhan have established that $A < 1.37$. In this paper, we reduce Moser's Square Packing Problem to a problem on a finite set of squares in the following sense: We show how to compute a natural number $N$ such that it is enough to determine the value of $A$ for sets containing at most $N$ squares with total area $1$.