262:
207:
152:
971:
739:
566:
408:
2, 3, 5, 6, 7, 8, 10, 13, 14, 15, 24, 34, 35, 46, 47, and 48. For most of these numbers (with the exceptions only of 5 and 10), the packing is the natural one with axis-aligned squares, and
254:
199:
457:
783:
484:
291:
629:
490:(round up) function. The figure shows the optimal packings for 5 and 10 squares, the two smallest numbers of squares for which the optimal packing involves tilted squares.
835:
381:
887:
1036:
699:
1095:
1069:
117:
1122:
599:
517:
406:
1819:
991:
673:
426:
357:
337:
317:
141:
83:
63:
1765:
1815:
649:
261:
1978:
1321:
903:
1535:
1494:
1456:
1361:
704:
1672:
1587:
1952:
522:
1884:
1758:
568:. By contrast, the tightest known packing of 11 squares is inside a square of side length approximately 3.877084 found by
1879:
1869:
1823:
1811:
1268:
215:
160:
1973:
1859:
1751:
431:
1874:
793:
showed that for a different packing by tilted unit squares, the wasted space could be significantly reduced to
602:
744:
1533:
Gensane, Thierry; Ryckelynck, Philippe (2005), "Improved dense packings of congruent squares in a square",
1012:
Some numbers of unit squares are never the optimal number in a packing. In particular, if a square of size
1833:
1864:
462:
270:
608:
1926:
1788:
1451:
1396:
1283:
1273:
1002:
1408:
897:
796:
362:
1489:
848:
1356:
1015:
678:
646:
What is the asymptotic growth rate of wasted space for square packing in a half-integer square?
1931:
1828:
1327:
1317:
1278:
838:
1311:
1074:
1041:
575:
The smallest case where the best known packing involves squares at three different angles is
1774:
1681:
1646:
1596:
1544:
1503:
1461:
1418:
1370:
487:
92:
1695:
1610:
1558:
1515:
1475:
1430:
1382:
1339:
1100:
1691:
1606:
1579:
1554:
1511:
1471:
1426:
1378:
1335:
578:
519:. It is known that 11 unit squares cannot be packed in a square of side length less than
496:
386:
206:
23:
151:
1889:
1851:
1838:
1803:
1728:
976:
658:
411:
342:
322:
302:
126:
68:
48:
45:(squares of side length one) that can be packed inside a larger square of side length
1967:
1947:
1894:
1732:
1709:
1686:
1627:
1601:
1575:
1571:
1307:
790:
786:
1006:
994:
569:
741:
grid of axis-aligned unit squares, but this may leave a large area, approximately
1631:
1143:
with radius as small as possible. For this problem, good solutions are known for
1793:
1667:
998:
894:
42:
1650:
1663:
1549:
890:
120:
601:. It was discovered in 1998 by John Bidwell, an undergraduate student at the
1623:
842:
1531:
listed this side length as 3.8772; the tighter bound stated here is from
638:
86:
1140:
966:{\displaystyle \Omega {\bigl (}a^{1/2}(a-\lfloor a\rfloor ){\bigr )}}
27:
1005:
of the wasted space, even for half-integer side lengths, remains an
1743:
1413:
1507:
1374:
1331:
1422:
1466:
30:
can be packed into some larger shape, often a square or circle.
1747:
1670:(1978), "Inefficiency in packing squares with unit squares",
734:{\displaystyle \lfloor a\rfloor \!\times \!\lfloor a\rfloor }
1452:"Packing unit squares in squares: a survey and new results"
1397:"Optimal packings of 13 and 46 unit squares in a square"
701:
square remains unknown. It is always possible to pack a
561:{\displaystyle \textstyle 2+2{\sqrt {4/5}}\approx 3.789}
123:– amount of unfilled space for an arbitrary non-integer
26:
where the objective is to determine how many congruent
1632:"Efficient packings of unit squares in a large square"
526:
1103:
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979:
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163:
129:
95:
71:
51:
675:, the exact number of unit squares that can pack an
41:
is the problem of determining the maximum number of
1940:
1919:
1903:
1850:
1802:
1781:
997:, the wasted space is at least proportional to its
1116:
1089:
1063:
1030:
985:
965:
881:
829:
777:
733:
693:
667:
623:
593:
560:
511:
478:
451:
420:
400:
375:
351:
331:
311:
285:
248:
193:
135:
111:
77:
57:
721:
717:
1357:"Efficient packing of unit squares in a square"
1759:
958:
912:
8:
1071:unit squares, then it must be the case that
950:
944:
769:
763:
728:
722:
714:
708:
473:
466:
446:
435:
249:{\displaystyle 3+1/{\sqrt {2}}\approx 3.707}
194:{\displaystyle 2+1/{\sqrt {2}}\approx 2.707}
1490:"Packing 10 or 11 unit squares in a square"
1301:
1299:
452:{\displaystyle \lceil {\sqrt {n}}\,\rceil }
267:11 unit squares in a square of side length
212:10 unit squares in a square of side length
1766:
1752:
1744:
900:, all solutions must waste space at least
157:5 unit squares in a square of side length
1685:
1600:
1548:
1465:
1412:
1355:Kearney, Michael J.; Shiu, Peter (2002),
1147:up to 35. Here are minimum solutions for
1108:
1102:
1076:
1049:
1043:
1017:
978:
957:
956:
925:
921:
911:
910:
905:
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814:
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798:
746:
706:
680:
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610:
580:
541:
536:
524:
498:
469:
464:
445:
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433:
413:
396:
388:
366:
364:
359:is a perfect square (in which case it is
344:
324:
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272:
233:
228:
217:
178:
173:
162:
128:
100:
94:
70:
50:
1528:
1445:
1443:
1441:
1439:
1153:
1580:"On packing squares with equal squares"
1401:The Electronic Journal of Combinatorics
1295:
778:{\displaystyle 2a(a-\lfloor a\rfloor )}
650:(more unsolved problems in mathematics)
1350:
1348:
1313:Research Problems in Discrete Geometry
1639:Discrete & Computational Geometry
1536:Discrete & Computational Geometry
655:For larger values of the side length
7:
845:further reduced the wasted space to
1495:Electronic Journal of Combinatorics
1457:Electronic Journal of Combinatorics
1362:Electronic Journal of Combinatorics
1316:, New York: Springer, p. 45,
907:
479:{\displaystyle \lceil \,\ \rceil }
14:
785:, uncovered and wasted. Instead,
286:{\displaystyle a\approx 3.877084}
1369:(1), Research Paper 14, 14 pp.,
1135:is a related problem of packing
493:The smallest unresolved case is
260:
205:
150:
1673:Journal of Combinatorial Theory
1588:Journal of Combinatorial Theory
1124:unit squares is also possible.
641:Unsolved problem in mathematics
624:{\displaystyle a\approx 4.6756}
16:Two-dimensional packing problem
1895:Sphere-packing (Hamming) bound
1306:Brass, Peter; Moser, William;
953:
935:
876:
855:
824:
803:
772:
754:
1:
1979:Unsolved problems in geometry
1687:10.1016/0097-3165(78)90005-5
1602:10.1016/0097-3165(75)90099-0
1488:Stromquist, Walter (2003),
830:{\displaystyle o(a^{7/11})}
376:{\displaystyle {\sqrt {n}}}
339:unit squares is known when
319:that allows the packing of
1995:
1651:10.1007/s00454-019-00088-9
1269:Circle packing in a square
1133:Square packing in a circle
1128:Square packing in a circle
882:{\displaystyle O(a^{3/5})}
119:but the precise – or even
39:Square packing in a square
34:Square packing in a square
1550:10.1007/s00454-004-1129-z
1031:{\displaystyle a\times a}
694:{\displaystyle a\times a}
1820:isosceles right triangle
1739:, Erich's Packing Center
1450:Friedman, Erich (2009),
1395:Bentz, Wolfram (2010),
1090:{\displaystyle a\geq n}
1064:{\displaystyle n^{2}-2}
1834:Circle packing theorem
1118:
1097:and that a packing of
1091:
1065:
1038:allows the packing of
1032:
1003:asymptotic growth rate
987:
973:. In particular, when
967:
883:
831:
779:
735:
695:
669:
625:
605:, and has side length
595:
562:
513:
480:
453:
422:
402:
377:
353:
333:
313:
299:The smallest value of
287:
250:
195:
137:
113:
112:{\displaystyle a^{2},}
79:
59:
1119:
1117:{\displaystyle n^{2}}
1092:
1066:
1033:
988:
968:
884:
841:). Later, Graham and
832:
780:
736:
696:
670:
626:
603:University of Hawaiʻi
596:
563:
514:
481:
454:
423:
403:
378:
354:
334:
314:
288:
251:
196:
143:is an open question.
138:
114:
80:
60:
1816:equilateral triangle
1733:"Squares in Squares"
1527:The 2000 version of
1502:, Research Paper 8,
1460:, Dynamic Survey 7,
1139:unit squares into a
1101:
1075:
1042:
1016:
977:
904:
849:
797:
745:
705:
679:
659:
609:
594:{\displaystyle n=17}
579:
523:
512:{\displaystyle n=11}
497:
463:
432:
412:
401:{\displaystyle n={}}
387:
363:
343:
323:
303:
271:
216:
161:
127:
93:
69:
49:
1953:Slothouber–Graatsma
1284:Moving sofa problem
1274:Squaring the square
1711:Squares in Circles
1157:Number of squares
1114:
1087:
1061:
1028:
983:
963:
879:
827:
775:
731:
691:
665:
635:Asymptotic results
621:
591:
558:
557:
509:
476:
449:
418:
398:
383:), as well as for
373:
349:
329:
309:
283:
246:
191:
133:
109:
75:
55:
1961:
1960:
1920:Other 3-D packing
1904:Other 2-D packing
1829:Apollonian gasket
1708:Friedman, Erich,
1279:Rectangle packing
1260:
1259:
986:{\displaystyle a}
839:little o notation
837:(here written in
668:{\displaystyle a}
549:
472:
443:
421:{\displaystyle a}
371:
352:{\displaystyle n}
332:{\displaystyle n}
312:{\displaystyle a}
238:
183:
136:{\displaystyle a}
78:{\displaystyle a}
58:{\displaystyle a}
1986:
1974:Packing problems
1842:
1782:Abstract packing
1775:Packing problems
1768:
1761:
1754:
1745:
1740:
1715:
1714:
1705:
1699:
1698:
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1654:
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1620:
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1613:
1604:
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1323:978-0387-23815-9
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1120:
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1037:
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1034:
1029:
992:
990:
989:
984:
972:
970:
969:
964:
962:
961:
934:
933:
929:
916:
915:
888:
886:
885:
880:
875:
874:
870:
836:
834:
833:
828:
823:
822:
818:
784:
782:
781:
776:
740:
738:
737:
732:
700:
698:
697:
692:
674:
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671:
666:
642:
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622:
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184:
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154:
142:
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118:
116:
115:
110:
105:
104:
89:, the answer is
84:
82:
81:
76:
64:
62:
61:
56:
1994:
1993:
1989:
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1987:
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1984:
1983:
1964:
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1962:
1957:
1936:
1915:
1899:
1846:
1840:
1839:Tammes problem
1798:
1777:
1772:
1729:Friedman, Erich
1727:
1724:
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1718:
1707:
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1702:
1662:
1661:
1657:
1634:
1622:
1621:
1617:
1582:
1570:
1569:
1565:
1532:
1529:Friedman (2009)
1526:
1522:
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1486:
1482:
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1437:
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1393:
1389:
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1353:
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1324:
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1304:
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1292:
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1104:
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1072:
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1039:
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1013:
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974:
917:
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794:
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742:
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702:
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606:
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520:
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409:
385:
384:
361:
360:
341:
340:
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293:
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268:
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257:
256:
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201:
159:
158:
155:
125:
124:
96:
91:
90:
67:
66:
47:
46:
36:
24:packing problem
17:
12:
11:
5:
1992:
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1982:
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1937:
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1929:
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1916:
1914:
1913:
1911:Square packing
1907:
1905:
1901:
1900:
1898:
1897:
1892:
1890:Kissing number
1887:
1882:
1877:
1872:
1867:
1862:
1856:
1854:
1852:Sphere packing
1848:
1847:
1845:
1844:
1836:
1831:
1826:
1808:
1806:
1804:Circle packing
1800:
1799:
1797:
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1791:
1785:
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1779:
1778:
1773:
1771:
1770:
1763:
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1723:
1722:External links
1720:
1717:
1716:
1700:
1680:(2): 170–186,
1668:Vaughan, R. C.
1655:
1645:(3): 690–699,
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1160:Circle radius
1158:
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1126:
1111:
1107:
1086:
1083:
1080:
1060:
1057:
1052:
1048:
1027:
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1021:
1001:. The precise
982:
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952:
949:
946:
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937:
932:
928:
924:
920:
914:
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889:. However, as
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20:Square packing
15:
13:
10:
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6:
4:
3:
2:
1991:
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1885:Close-packing
1883:
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1880:In a cylinder
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1577:
1576:Graham, R. L.
1573:
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1546:
1543:(1): 97–109,
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1508:10.37236/1701
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1375:10.37236/1631
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1211:
1207:
1204:
1203:
1199:
1196:
1195:
1191:
1188:
1187:
1183:
1180:
1179:
1175:
1172:
1171:
1167:
1164:
1163:
1159:
1156:
1155:
1152:
1150:
1146:
1142:
1138:
1134:
1127:
1125:
1109:
1105:
1084:
1081:
1078:
1058:
1055:
1050:
1046:
1025:
1022:
1019:
1010:
1008:
1004:
1000:
996:
980:
947:
941:
938:
930:
926:
922:
918:
899:
896:
892:
871:
867:
863:
859:
852:
844:
840:
819:
815:
811:
807:
800:
792:
791:Ronald Graham
788:
766:
760:
757:
751:
748:
725:
718:
711:
688:
685:
682:
662:
651:
634:
632:
618:
615:
612:
604:
588:
585:
582:
573:
571:
554:
551:
546:
542:
538:
533:
530:
527:
506:
503:
500:
491:
489:
440:
415:
393:
390:
368:
346:
326:
306:
280:
277:
274:
263:
243:
240:
235:
229:
225:
222:
219:
208:
188:
185:
180:
174:
170:
167:
164:
153:
144:
130:
122:
106:
101:
97:
88:
72:
52:
44:
40:
33:
31:
29:
25:
21:
1910:
1822: /
1818: /
1814: /
1736:
1710:
1703:
1677:
1676:, Series A,
1671:
1658:
1642:
1638:
1618:
1592:
1591:, Series A,
1586:
1566:
1540:
1534:
1523:
1499:
1493:
1483:
1455:
1423:10.37236/398
1404:
1400:
1390:
1366:
1360:
1312:
1148:
1144:
1136:
1132:
1131:
1011:
1007:open problem
995:half-integer
654:
574:
570:Walter Trump
492:
298:
43:unit squares
38:
37:
19:
18:
1927:Tetrahedron
1870:In a sphere
1841:(on sphere)
1812:In a circle
1664:Roth, K. F.
1628:Graham, Ron
1595:: 119–123,
1467:10.37236/28
1308:Pach, János
999:square root
895:Bob Vaughan
1968:Categories
1860:Apollonian
1624:Chung, Fan
1414:1606.03746
1332:2005924022
1290:References
1151:up to 12:
891:Klaus Roth
787:Paul Erdős
121:asymptotic
1932:Ellipsoid
1875:In a cube
1572:Erdős, P.
1256:2.236...
1248:2.214...
1240:2.121...
1232:2.077...
1224:1.978...
1216:1.802...
1208:1.688...
1200:1.581...
1192:1.414...
1184:1.288...
1176:1.118...
1168:0.707...
1082:≥
1056:−
1023:×
951:⌋
945:⌊
942:−
908:Ω
843:Fan Chung
770:⌋
764:⌊
761:−
729:⌋
723:⌊
719:×
715:⌋
709:⌊
686:×
616:≈
552:≈
474:⌉
467:⌈
447:⌉
436:⌈
278:≈
241:≈
186:≈
1630:(2020),
1578:(1975),
1407:(R126),
1310:(2005),
1263:See also
459:, where
281:3.877084
1941:Puzzles
1696:0487806
1611:0370368
1559:2140885
1516:2386538
1476:1668055
1431:2729375
1383:1912796
1340:2163782
488:ceiling
486:is the
87:integer
28:squares
1948:Conway
1865:Finite
1824:square
1737:Github
1694:
1609:
1557:
1514:
1474:
1429:
1381:
1338:
1330:
1320:
1141:circle
898:proved
619:4.6756
471:
85:is an
1635:(PDF)
1583:(PDF)
1409:arXiv
993:is a
555:3.789
244:3.707
189:2.707
65:. If
22:is a
1328:LCCN
1318:ISBN
893:and
789:and
1794:Set
1789:Bin
1682:doi
1647:doi
1597:doi
1545:doi
1504:doi
1462:doi
1419:doi
1371:doi
1253:12
1245:11
1237:10
428:is
1970::
1735:,
1731:,
1692:MR
1690:,
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1666:;
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1607:MR
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1593:19
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1510:,
1500:10
1498:,
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1472:MR
1470:,
1454:,
1438:^
1427:MR
1425:,
1417:,
1405:17
1403:,
1399:,
1379:MR
1377:,
1365:,
1359:,
1347:^
1336:MR
1334:,
1326:,
1298:^
1229:9
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1684::
1649::
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1464::
1421::
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1373::
1367:9
1149:n
1145:n
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1106:n
1085:n
1079:a
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1026:a
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981:a
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220:3
181:2
175:/
171:1
168:+
165:2
131:a
107:,
102:2
98:a
73:a
53:a
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