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{\bf Hiroshi Nagamochi}
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{\bf Packing Unit Squares in a Rectangle}
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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 $a,b\geq 2$, no more than $ab
-(a+1-\lceil a\rceil) -(b+1-\lceil b\rceil)$ unit squares can be
packed in any $a'\times b'$ rectangle $R$ with $a'<a$ and $b'<b$.
From this, we can deduce that, for any integer $N\geq 4$, $s(N)\geq
\min\{\lceil \sqrt{N} \rceil, \sqrt{N -2 \lfloor \sqrt{N}\rfloor +1
}+1\}$.  In particular, for any integer $n\geq 2$,
$s(n^2)=s(n^2-1)=s(n^2-2)=n$ holds.

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