Optimal Packings of 22 and 33 Unit Squares in a Square
Abstract
Let $s(n)$ be the side length of the smallest square into which $n$ nonoverlapping 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 $s(m^23)=m$ 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 $s(m^23)=m$ 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 wellknown method of sets of unavoidable points by replacing them with continuously varying families of such sets.
 Publication:

arXiv eprints
 Pub Date:
 June 2016
 arXiv:
 arXiv:1606.03746
 Bibcode:
 2016arXiv160603746B
 Keywords:

 Mathematics  Combinatorics;
 05B40;
 52C15