Packing Unit Squares in a Rectangle - Nagamochi, Hiroshi (The Electronic Journal of Combinatorics, 2005)
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-a ceil) -(b+1-b ceil) 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)≥ {N ceil, N -2 N floor +1 +1}. In particular, for any integer n≥ 2, s(n^2)=s(n^2-1)=s(n^2-2)=n holds.