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 $ab -(a+1-\lceil a\rceil) -(b+1-\lceil b\rceil)$ 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 $N≥ 4$, $s(N)≥ \min\{\lceil \sqrt{N} \rceil, \sqrt{N -2 \lfloor \sqrt{N}\rfloor +1 }+1\}$. In particular, for any integer $n≥ 2$, $s(n^2)=s(n^2-1)=s(n^2-2)=n$ holds.