1351:
156:
31:
494:
78:
543:
the square of the smallest size not yet used to get another, larger L-shaped region. The squares added during the puffing up procedure have sizes that have not yet appeared in the construction and the procedure is set up so that the resulting rectangular regions are expanding in all four directions, which leads to a tiling of the whole plane.
1358:
539:. For example, the plane can be tiled with different integral squares, but not for every integer, by recursively taking any perfect squared square and enlarging it so that the formerly smallest tile now has the size of the original squared square, then replacing this tile with a copy of the original squared square.
542:
In 2008 James Henle and
Frederick Henle proved that this, in fact, can be done. Their proof is constructive and proceeds by "puffing up" an L-shaped region formed by two side-by-side and horizontally flush squares of different sizes to a perfect tiling of a larger rectangular region, then adjoining
213:
When the constraint of all the squares being different sizes is relaxed, a squared square such that the side lengths of the smaller squares do not have a common divisor larger than 1 is called a "Mrs. Perkins's quilt". In other words, the
578:
in squares, the smallest square in this dissection does not lie on an edge of the rectangle. Indeed, each corner square has a smaller adjacent edge square, and the smallest edge square is adjacent to smaller squares not on the edge.
182:
discovered a simple perfect squared square of side 112 with the smallest number of squares using a computer search. His tiling uses 21 squares, and has been proved to be minimal. This squared square forms the logo of the
69:, which can be done even with the restriction that each natural number occurs exactly once as a size of a square in the tiling. The order of a squared square is its number of constituent squares.
168:
A "simple" squared square is one where no subset of more than one of the squares forms a rectangle or square. When a squared square has a square or rectangular subset, it is "compound".
379:
943:
906:
315:
201:
The perfect compound squared square with the fewest squares was discovered by T.H. Willcocks in 1946 and has 24 squares; however, it was not until 1982 that
Duijvestijn,
279:
146:
246:
654:
could be perfectly hypercubed then its 'faces' would be perfect cubed cubes; this is impossible. Similarly, there is no solution for all cubes of higher dimensions.
194:
Duijvestijn also found two simple perfect squared squares of sides 110 but each comprising 22 squares. Theophilus
Harding Willcocks, an amateur mathematician and
483:
463:
335:
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772:
668:
391:
1489:
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1245:
1034:
564:
Unlike the case of squaring the square, a hard yet solvable problem, there is no perfect cubed cube and, more generally, no dissection of a
2106:
1941:
1192:
34:
The first perfect squared square discovered, a compound one of side 4205 and order 55. Each number denotes the side length of its square.
2256:
2231:
2221:
2191:
2146:
2096:
2076:
1891:
1776:
733:
Gardner, Martin (November 1958). "How rectangles, including squares, can be divided into squares of unequal size". Mathematical Games.
198:
composer, found another. In 1999, I. Gambini proved that these three are the smallest perfect squared squares in terms of side length.
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2196:
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2101:
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be the smallest square in this dissection. By the claim above, this is surrounded on all 4 sides by squares which are larger than
159:
Lowest-order perfect squared square (1) and the three smallest perfect squared squares (2–4): all are simple squared squares
61:. Squaring the square is an easy task unless additional conditions are set. The most studied restriction is that the squaring be
2126:
2061:
2046:
1881:
1501:
2226:
2186:
2141:
2081:
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2031:
1392:
188:
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2011:
1866:
980:
756:
is not named as an author of the column, but it consists almost entirely of a long multi-paragraph quote credited to Tutte.
2021:
2006:
1966:
1896:
1846:
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514:
raised the question whether the whole plane can be tiled by squares, one of each integer edge-length, which he called the
535:
and G. C. Shephard stated that in all perfect integral tilings of the plane known at that time, the sizes of the squares
505:
3. Scaling the
Fibonacci tiling by 110 times and replacing one of the 110-squares with Duijvestijn's perfects the tiling.
1991:
1956:
1946:
1806:
1350:
815:
184:
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1961:
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1721:
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1106:
93:
121:
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1936:
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663:
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1906:
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1821:
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1187:
1123:
693:
647:, ... is infinite and the corresponding cubes are infinite in number. This contradicts our original supposition.
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202:
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215:
125:
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2026:
1996:
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172:
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The first perfect squared square to be published, a compound one of side 4205 and order 55, was found by
2136:
2041:
2001:
1986:
1981:
1976:
1971:
1726:
1516:
1231:
112:") at Cambridge University between 1936 and 1938. They transformed the square tiling into an equivalent
1193:
https://web.archive.org/web/20030419012114/http://www.math.niu.edu/~rusin/known-math/98/square_dissect
978:
Wynn, Ed (2014). "Exhaustive generation of 'Mrs. Perkins's quilt' square dissections for low orders".
2181:
1921:
1634:
1622:
1506:
1435:
1411:
1336:
527:
105:
499:
1. Tiling with squares with
Fibonacci-number sides is almost perfect except for 2 squares of side 1.
340:
155:
1926:
1746:
1592:
1546:
1426:
1110:
735:
519:
97:
85:
A "perfect" squared square is a square such that each of the smaller squares has a different size.
58:
1711:
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989:
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284:
177:
113:
582:
Now suppose that there is a perfect dissection of a rectangular cuboid in cubes. Make a face of
532:
251:
225:
2206:
1756:
1683:
1526:
1309:
1102:
1030:
109:
89:
50:
691:
Sprague, R. (1939). "Beispiel einer
Zerlegung des Quadrats in lauter verschiedene Quadrate".
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2016:
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techniques to that circuit. The first perfect squared squares they found were of order 69.
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65:, meaning the sizes of the smaller squares are all different. A related problem is
42:
1174:, Eindhoven University of Technology, Faculty of Mathematics and Computing Science
1136:
1118:
1051:
574:
To prove this, we start with the following claim: for any perfect dissection of a
493:
120:
that connected to their neighbors at their top and bottom edges, and then applied
17:
561:, the problem of dividing it into finitely many smaller cubes, no two congruent.
1573:
1114:
753:
522:
column and appeared in several books, but it defied solution for over 30 years.
195:
142:
101:
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1, 4, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, ... (sequence
1642:
1188:
http://www.math.uwaterloo.ca/navigation/ideas/articles/honsberger2/index.shtml
956:
919:
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is the analogue in three dimensions of squaring the square: that is, given a
1662:
1647:
1563:
1539:
1216:
651:
616:
is divided into a perfect squared square by the cubes which rest on it. Let
497:
Tiling the plane with different integral squares using the
Fibonacci series
77:
586:
its horizontal base. The base is divided into a perfect squared rectangle
502:
2. Duijvestijn found a 110-square tiled with 22 different integer squares.
1431:
117:
1080:
748:
879:
706:
54:
1357:
721:
An example of a dissection of the square into pairwise unequal squares
205:
and P. Leeuw mathematically proved it to be the lowest-order example.
317:. Computer searches have found exact solutions for small values of
222:
asks for a Mrs. Perkins's quilt with the fewest pieces for a given
116:– they called it a "Smith diagram" – by considering the squares as
994:
492:
154:
76:
29:
555:
465:
other than 2, 3, and 5, it is possible to dissect a square into
1619:
1469:
1369:
1265:
1227:
1223:
1183:
http://www.maa.org/editorial/mathgames/mathgames_12_01_03.html
518:. This problem was later publicized by Martin Gardner in his
720:
1177:
386:
1202:
45:
an integral square using only other integral squares. (An
401:
57:
length.) The name was coined in a humorous analogy with
403:
A square cut into 10 pieces (an HTML table)
145:
about the early history of squaring the square in his
1167:"Album of Simple Perfect Squared Squares of order 26"
471:
451:
343:
323:
287:
254:
228:
811:"A method for cutting squares into distinct squares"
1775:
1702:
1671:
1633:
1165:Bouwkamp, C. J.; Duijvestijn, A. J. W. (Dec 1994).
590:by the cubes which rest on it. The smallest square
944:Proceedings of the Cambridge Philosophical Society
907:Proceedings of the Cambridge Philosophical Society
477:
457:
373:
329:
309:
273:
248:square. The number of pieces required is at least
240:
218:of all the smaller side lengths should be 1. The
856:; Leeuw, P. (1982). "Compound perfect squares".
438:
435:
941:Trustrum, G. B. (1965). "Mrs Perkins's quilt".
768:"Simple perfect squared square of lowest order"
1239:
1050:Henle, Frederick V.; Henle, James M. (2008).
609:, cubes. Hence the upper face of the cube on
8:
1029:. Australian Mathematics Trust. p. 84.
337:(small enough to need up to 18 pieces). For
1119:"The dissection of rectangles into squares"
1630:
1616:
1466:
1366:
1262:
1246:
1232:
1224:
141:published an extensive article written by
1557:Dividing a square into similar rectangles
993:
828:
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773:Journal of Combinatorial Theory, Series B
686:
684:
669:Dividing a square into similar rectangles
470:
450:
342:
322:
295:
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259:
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227:
88:It is first recorded as being studied by
680:
571:into a finite number of unequal cubes.
485:squares of one or two different sizes.
187:. It also appears on the cover of the
719:English translation by David Moews, "
7:
1025:Henry, J. B.; Taylor, P. J. (2009).
421:
416:
413:
410:
407:
381:the number of pieces required is:
25:
1060:The American Mathematical Monthly
859:The American Mathematical Monthly
1356:
1349:
904:(1964). "Mrs. Perkins's quilt".
398:No more than two different sizes
516:heterogeneous tiling conjecture
189:Journal of Combinatorial Theory
1199:Nowhere-neat squared squares:
1073:10.1080/00029890.2008.11920491
872:10.1080/00029890.1982.11995375
766:Duijvestijn, A. J. W. (1978).
374:{\displaystyle n=1,2,3,\dots }
1:
1582:Regular Division of the Plane
1137:10.1215/S0012-7094-40-00718-9
1027:Challenge! 1999 - 2006 Book 2
830:10.1016/S0166-218X(99)00158-4
816:Discrete Applied Mathematics
787:10.1016/0095-8956(78)90041-2
220:Mrs. Perkins's quilt problem
185:Trinity Mathematical Society
81:Smith diagram of a rectangle
1490:Architectonic and catoptric
1388:Aperiodic set of prototiles
310:{\displaystyle 6\log _{2}n}
2357:
1004:10.1016/j.disc.2014.06.022
664:Square packing in a square
274:{\displaystyle \log _{2}n}
1629:
1615:
1476:
1465:
1378:
1365:
1347:
1274:
1261:
1161:Perfect squared squares:
1124:Duke Mathematical Journal
957:10.1017/s0305004100038573
920:10.1017/S0305004100037877
694:Mathematische Zeitschrift
241:{\displaystyle n\times n}
2341:Rectangular subdivisions
2336:Recreational mathematics
1178:http://www.squaring.net/
633:The sequence of squares
203:Pasquale Joseph Federico
122:Kirchhoff's circuit laws
1203:http://karlscherer.com/
1172:. EUT Report 94-WSK-02.
852:Duijvestijn, A. J. W.;
216:greatest common divisor
73:Perfect squared squares
1209:Mrs. Perkins's quilt:
630:and therefore higher.
507:
479:
459:
375:
331:
311:
275:
242:
164:Simple squared squares
160:
82:
35:
2331:Mathematical problems
809:Gambini, Ian (1999).
531:, published in 1987,
496:
480:
460:
376:
332:
312:
276:
243:
208:
158:
126:circuit decomposition
80:
33:
1213:Mrs. Perkins's Quilt
1052:"Squaring the plane"
981:Discrete Mathematics
528:Tilings and patterns
469:
449:
341:
321:
285:
252:
226:
209:Mrs. Perkins's quilt
173:A. J. W. Duijvestijn
106:collective pseudonym
27:Mathematical problem
736:Scientific American
650:If a 4-dimensional
520:Scientific American
404:
104:(writing under the
59:squaring the circle
39:Squaring the square
707:10.1007/BF01580305
566:rectangular cuboid
537:grew exponentially
508:
489:Squaring the plane
475:
455:
402:
371:
327:
307:
271:
238:
161:
152:of November 1958.
148:Mathematical Games
114:electrical circuit
83:
67:squaring the plane
41:is the problem of
36:
18:Squaring the plane
2326:Discrete geometry
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2312:
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1611:
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1502:Computer graphics
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1036:978-1-876420-23-9
601:is surrounded by
478:{\displaystyle n}
458:{\displaystyle n}
443:
442:
330:{\displaystyle n}
110:Blanche Descartes
53:whose sides have
16:(Redirected from
2348:
1631:
1617:
1569:Conway criterion
1496:Circle Limit III
1467:
1400:Einstein problem
1367:
1360:
1353:
1289:Schwarz triangle
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1234:
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1156:External links
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823:(1–2): 65–80.
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743:(5): 136–144.
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139:Martin Gardner
133:Roland Sprague
94:C. A. B. Smith
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1103:Brooks, R. L.
1098:
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934:
929:
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903:
902:Conway, J. H.
897:
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619:
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382:
368:
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56:
52:
48:
44:
40:
32:
19:
1593:Substitution
1588:Regular grid
1580:
1551:
1494:
1427:Quaquaversal
1325:Kisrhombille
1255:Tessellation
1128:
1122:
1115:Tutte, W. T.
1111:Stone, A. H.
1097:
1064:
1058:
1045:
1026:
1020:
985:
979:
973:
948:
942:
936:
911:
905:
896:
866:(1): 15–32.
863:
857:
847:
820:
814:
804:
777:
771:
761:
740:
734:
728:
698:
692:
649:
641:
634:
632:
624:
617:
610:
606:
602:
598:
591:
587:
583:
581:
575:
573:
568:
563:
558:
551:
550:
541:
526:
524:
515:
509:
444:
219:
212:
200:
193:
170:
167:
147:
137:
130:
90:R. L. Brooks
87:
84:
66:
62:
46:
38:
37:
1623:vertex type
1481:Anisohedral
1436:Self-tiling
1279:Pythagorean
1131:: 312–340.
1067:(1): 3–12.
914:: 363–368.
754:W. T. Tutte
701:: 607–608.
196:fairy chess
176: [
143:W. T. Tutte
102:W. T. Tutte
98:A. H. Stone
2320:Categories
1527:Pentagonal
675:References
1635:Spherical
1603:Voderberg
1564:Prototile
1531:Problems
1507:Honeycomb
1485:Isohedral
1372:Aperiodic
1310:honeycomb
1294:Rectangle
1284:Rhombille
1217:MathWorld
995:1308.5420
988:: 38–47.
652:hypercube
576:rectangle
510:In 1975,
369:…
302:
266:
233:×
171:In 1978,
135:in 1939.
118:resistors
1717:V3.4.3.4
1552:Squaring
1547:Heesch's
1512:Isotoxal
1432:Rep-tile
1422:Pinwheel
1315:Coloring
1268:Periodic
1117:(1940).
1089:26663945
1081:27642387
951:: 7–11.
749:24944827
658:See also
2177:6.4.8.4
2132:5.4.6.4
2092:4.12.16
2082:4.10.12
2052:V4.8.10
2027:V4.6.16
2017:V4.6.14
1917:3.6.4.6
1912:3.4.∞.4
1907:3.4.8.4
1902:3.4.7.4
1897:3.4.6.4
1847:3.∞.3.∞
1842:3.4.3.4
1837:3.8.3.8
1832:3.7.3.7
1827:3.6.3.8
1822:3.6.3.6
1817:3.5.3.6
1812:3.5.3.5
1807:3.4.3.∞
1802:3.4.3.8
1797:3.4.3.7
1792:3.4.3.6
1787:3.4.3.5
1742:3.4.6.4
1712:3.4.3.4
1705:regular
1672:Regular
1598:Voronoi
1522:Packing
1453:Truchet
1448:Socolar
1417:Penrose
1412:Gilbert
1337:Wythoff
1145:0003040
1012:3240464
965:0170831
928:0167425
888:0639770
880:2320990
839:1723687
796:0511994
715:0000470
439:
436:
431:
417:
414:
411:
408:
390:in the
387:A005670
63:perfect
55:integer
2067:4.8.16
2062:4.8.14
2057:4.8.12
2047:4.8.10
2022:4.6.16
2012:4.6.14
2007:4.6.12
1777:Hyper-
1762:4.6.12
1535:Domino
1441:Sphinx
1320:Convex
1299:Domino
1143:
1087:
1079:
1033:
1010:
963:
926:
886:
878:
837:
794:
747:
713:
607:higher
603:larger
428:
425:
422:
150:column
51:square
43:tiling
2182:(6.8)
2137:(5.6)
2072:4.8.∞
2042:(4.8)
2037:(4.7)
2032:4.6.∞
2002:(4.6)
1997:(4.5)
1967:4.∞.4
1962:4.8.4
1957:4.7.4
1952:4.6.4
1947:4.5.4
1927:(3.8)
1922:(3.7)
1892:(3.4)
1887:(3.4)
1779:bolic
1747:(3.6)
1703:Semi-
1574:Girih
1471:Other
1170:(PDF)
1085:S2CID
1077:JSTOR
1055:(PDF)
990:arXiv
876:JSTOR
745:JSTOR
180:]
49:is a
2267:8.16
2262:8.12
2232:7.14
2202:6.16
2197:6.12
2192:6.10
2152:5.12
2147:5.10
2102:4.16
2097:4.14
2087:4.12
2077:4.10
1937:3.16
1932:3.14
1752:3.12
1737:V3.6
1663:V4.n
1653:V3.n
1540:Wang
1517:List
1483:and
1434:and
1393:List
1308:and
1031:ISBN
556:cube
392:OEIS
124:and
100:and
2297:∞.8
2292:∞.6
2257:8.6
2227:7.8
2222:7.6
2187:6.8
2142:5.8
2107:4.∞
1942:3.∞
1867:3.4
1862:3.∞
1857:3.8
1852:3.7
1767:4.8
1757:4.∞
1732:3.6
1727:3.∞
1722:3.4
1658:4.n
1648:3.n
1621:By
1215:on
1133:doi
1069:doi
1065:115
1000:doi
986:334
953:doi
916:doi
868:doi
825:doi
782:doi
741:199
703:doi
597:in
525:In
293:log
257:log
2322::
1141:MR
1139:.
1127:.
1121:.
1113:;
1109:;
1105:;
1083:.
1075:.
1063:.
1057:.
1008:MR
1006:.
998:.
984:.
961:MR
959:.
949:61
947:.
924:MR
922:.
912:60
910:.
884:MR
882:.
874:.
864:89
862:.
835:MR
833:.
821:98
819:.
813:.
792:MR
790:.
778:25
776:.
770:.
739:.
723:".
711:MR
709:.
699:45
697:.
683:^
640:,
191:.
178:de
96:,
92:,
2287:∞
2282:∞
2277:∞
2272:∞
2252:8
2247:8
2242:8
2237:8
2217:7
2212:7
2207:7
2172:6
2167:6
2162:6
2157:6
2127:5
2122:5
2117:5
2112:5
1992:4
1987:4
1982:4
1977:4
1972:4
1882:3
1877:3
1872:3
1694:6
1689:4
1684:3
1679:2
1643:2
1247:e
1240:t
1233:v
1147:.
1135::
1129:7
1091:.
1071::
1039:.
1014:.
1002::
992::
967:.
955::
930:.
918::
890:.
870::
841:.
827::
798:.
784::
751:.
717:.
705::
645:2
642:s
638:1
635:s
628:2
625:s
621:2
618:s
614:1
611:s
599:R
595:1
592:s
588:R
584:C
569:C
559:C
473:n
453:n
394:)
366:,
363:3
360:,
357:2
354:,
351:1
348:=
345:n
325:n
305:n
297:2
289:6
269:n
261:2
236:n
230:n
108:"
20:)
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