{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T11:59:15Z","timestamp":1790683155523,"version":"4.1.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 $a' &lt; a$ and $b' &lt; b$. 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.<\/jats:p>","DOI":"10.37236\/1934","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:11:06Z","timestamp":1578701466000},"source":"Crossref","is-referenced-by-count":5,"title":["Packing Unit Squares in a Rectangle"],"prefix":"10.37236","volume":"12","author":[{"given":"Hiroshi","family":"Nagamochi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"23455","published-online":{"date-parts":[[2005,7,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r37\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r37\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:48:04Z","timestamp":1579304884000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v12i1r37"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,7,30]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2005,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1934","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,7,30]]},"article-number":"R37"}}