Packing unit squares in a rectangle (Q2571280)

Let \(N\) be a positive integer. The edge length of the smallest square into which one can pack \(N\) unit squares (allowing individual squares to be rotated) is denoted by \(s(N)\). The cases \(s(n^2-1)=s(n^2-2)=n\) for \(n\geq 2\) are here settled (trivially, \(s(n^2)=n\)), and the general bound \(s(N) \geq \min\left\{\lceil\sqrt{N}\rceil,\sqrt{N-2\lfloor\sqrt{N}\rfloor+1}+1\right\}\) for \(N \geq 4\) is proved. To obtain these results, it is shown that for real numbers \(a,b\), at most \(ab-(a+1-\lceil a \rceil)-(b+1-\lceil b \rceil)\) unit squares can be packed into an \(a'\times b'\) rectangle with \(a'<a\) and \(b'<b\).