Optimal Packings of 22 and 33 Unit Squares in a Square

TL;DR

This paper proves that the smallest square to pack 22 and 33 unit squares has side length equal to the square root of the number of squares, confirming the most efficient packings are trivial for these cases.

Contribution

It establishes that for m=5 and 6, the minimal square side length for packing m^2-3 unit squares equals m, using a novel modification of unavoidable point sets.

Findings

s(22)=5 and s(33)=6, confirming trivial packings are optimal

Introduces a modified method of unavoidable points with continuously varying families

Supports the conjecture s(m^2-3)=m for m≥3

Abstract

Let $s(n)$ be the side length of the smallest square into which $n$ non-overlapping unit squares can be packed. In 2010, the author showed that $s(13)=4$ and $s(46)=7$. Together with the result $s(6)=3$ by Keaney and Shiu, these results strongly suggest that $s(m^2-3)=m$ for $m\ge 3$, in particular for the values $m=5,6$, which correspond to cases that lie in between the previous results. In this article we show that indeed $s(m^2-3)=m$ for $m=5,6$, implying that the most efficient packings of 22 and 33 squares are the trivial ones. To achieve our results, we modify the well-known method of sets of unavoidable points by replacing them with continuously varying families of such sets.