{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:13Z","timestamp":1753893793135,"version":"3.41.2"},"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>Let $s(n)$ be the side of the smallest square into which it is possible pack $n$ unit squares.   We show that $s(10)=3+\\sqrt{1\\over 2}\\approx3.707$ and that $s(11)\\geq2+2\\sqrt{4\\over 5}\\approx3.789$. We also show that an optimal packing of $11$ unit squares with orientations limited to $0$ degrees  or $45$ degrees has side  $2+2\\sqrt{8\\over 9}\\approx3.886$.  These results prove Martin Gardner's  conjecture that $n=11$ is the first case in which an optimal result requires a non-$45$ degree packing.<\/jats:p>","DOI":"10.37236\/1701","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T21:29:49Z","timestamp":1578691789000},"source":"Crossref","is-referenced-by-count":3,"title":["Packing 10 or 11 Unit Squares in a Square"],"prefix":"10.37236","volume":"10","author":[{"given":"Walter","family":"Stromquist","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2003,3,18]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v10i1r8\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v10i1r8\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T00:09:56Z","timestamp":1579306196000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v10i1r8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,3,18]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2003,1,6]]}},"URL":"https:\/\/doi.org\/10.37236\/1701","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2003,3,18]]},"article-number":"R8"}}