Packing Unit Squares in a Rectangle

For a positive integer N , let s(N) be the side length of the minimum square into which N unit squares can be packed. This paper shows that, for given real numbers a, b ≥ 2, no more than ab− (a + 1− dae)− (b + 1− dbe) unit squares can be packed in any a′ × b′ rectangle R with a′ < a and b′ < b. From this, we can deduce that, for any integer N ≥ 4, s(N) ≥ min{d√Ne, √ N − 2b√Nc + 1 + 1}. In particular, for any integer n ≥ 2, s(n2) = s(n2 − 1) = s(n2 − 2) = n holds.