Improved Packings of $n(n − 1)$ Unit Squares in a Square

Abstract

Let $s(n)$ be the side of the smallest square into which we can pack $n$ unit squares. The purpose of this paper is to prove that $s(n^2-n)<n$ for all $n\geq 12$. Besides, we show that $s(18^2-17) < 18, s(17^2-16) < 17,$ and $s(16^2-15) < 16.$