Packing 10 or 11 Unit Squares in a Square

Abstract

Let $s(n)$ be the side of the smallest square into which it is possible pack $n$ unit squares. We show that $s(10)=3+\sqrt{1\over 2}\approx3.707$ and that $s(11)\geq2+2\sqrt{4\over 5}\approx3.789$. We also show that an optimal packing of $11$ unit squares with orientations limited to $0$ degrees or $45$ degrees has side $2+2\sqrt{8\over 9}\approx3.886$. These results prove Martin Gardner's conjecture that $n=11$ is the first case in which an optimal result requires a non-$45$ degree packing.