Simple Perfect Squared Squares (SPSSs);
Order 21 to 32 and higher orders

This collection of SPSSs from Order 21 to 32 is a complete collection. The number of squared squares grows exponentially with the order. For SPSSs of order 33 and above we have a partial (and incomplete) collection.

This collection of simple squared squares has been assembled from many sources, with material from published and unpublished collections, mathematics journals, books, and recent computer searches. The catalogue is presented, arranged by order, beginning with the single simple perfect squared square of order 21. SPSSs are viewable in javascript and pdf. Other selections of this collection are available for download. For definitions of squared squares and squared rectangles, including 'simple', compound', 'perfect' and 'imperfect' see here.

A chronology of the significant low(er) order simple perfect squared square (SPSS) discoveries and the people who made them;

• 1939 R.L. Brooks found a simple perfect squared square (SPSS), side 4920 of order 38. This was not published.
• 1940 R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T.Tutte published 'The Dissection of Rectangles into Squares" (1st page only), referring to an order 55 SPSS, side 5468 using theoretical methods involving the use of symmetry in electrical networks, attributed to all four authors.
• 1946-7 C.J. Bouwkamp, published 'On the dissection of rectangles into squares', 'Paper I', 'Papers II and Paper III' and 'On the construction of simple perfect squared squares' (Koninkl. Nederl. Akad. Wetensch. Proc. Ser. A)
• 1947 T.H.Willcocks, discovered an SPSS of order 37, side 1947. This was a variation on R.L. Brooks order 38 4920 SPSS.
• 1962 A.J.W. Duijvestijn, in his PhD thesis 'Electronic Computation Of Squared Rectangles', showed no SPSS exists with fewer than 20 squares.
• 1964 J.C. Wilson found an SPSS with side 503 of order 25.
• 1966 T.H. Willcocks constructed 2 SPSSs with sides 1415, 2606 of order 31.
• 1967 G.H. Morley's SPSS method published in Eureka. Eight examples, from 56:1118251A to 60:5629849A, can be found here. They include 60:616457A, wrongly stated in the article to have side 616,467.
• 1967 T.H. Willcocks constructs 2 SPSSs with sides 1360, 1372 of order 31.
• 1967 J.C. Wilson included in his PhD thesis 5 new SPSSs of order 25 (including the one he found in 1964) and 24 new SPSSs of order 26.
• 1969 T.H. Willcocks constructed an SPSS with side 900 of order 27.
• 1978 P.J. Federico found two SPSSs of order 25.
• 1978 Mar, A.J.W. Duijvestijn, SPSS order 21 enumerated; 1 SPSS (21 : 112 x 112) proved order minimal and unique.
• 1978 A.J. W. Duijvestijn found an SPSS side 110 of order 22. T.H. Willcocks was able to transform it into a different SPSS with side 110 of order 22. By Gambini's result these 2 and another 110 in order 23 are the smallest size perfect squared squares.
• 1990 J.D. Skinner constructed SPSSs of side 180 and side 188 of order 23.
• 1990 C.J. Bouwkamp constructed 2 SPSSs of order 24 of side 186 and side 288 of order 24.
• 1991 Jul, A.J.W. Duijvestijn enumerated the remaining SPSSs of orders 21 - 24.
• 1992 Jan, C.J. Bouwkamp constructed 5 SPSS of order 25 and 21 of order 26.
• 1992 A.J.W. Duijvestijn enumerated the remaining SPSSs of order 25 and published SPSSs of orders 21 to 25
• 1993 J.D. Skinner found 3 SPSSs of order 26.
• 1993 During the period of January 7, 1993 to March 15, 1993 A.J.W. Duijvestijn calculated the order-26 squared-square solutions by means of four HP workstations connected to the Eindhoven university network. Their speed was 75 Mflops. The machines were only available to him during the nights and the weekends. A.J.W. Duijvestijn published the results as Simple Perfect Squared Squares and 2 x 1 Squared Rectangles of Order 26 in 1996.
• 1999 I. Gambini published his doctoral thesis 'Quant aux carrés carrelés' on squared squares. He confirmed Duijvestijn's counts of SPSSs up to order 26. With a second algorithm a lot more efficient but incomplete he obtained more than 30,000 squared squares. He also published a paper
  'A method for cutting squares into distinct squares'
• 2003 J.D. Skinner completed enumeration of SPSSs of order 27. By 2003, Skinner had also enumerated 94% of order 28 and 45% of order 29 and about 22% (estimated) of order 30 SPSSs.
• 2003 S.E. Anderson, in a collaboration with J.D. Skinner in 2002-3, found 60 SPSSs of order 28
• 2007 G.H. Morley published 130 SPSSs of order 31.
• 2010 S.E. Anderson and Ed Pegg Jr. enumerate all SPSSs up to order 28, and begin investigation of order 29. They confirm that the known SPSSs up to order 27 are complete, and find the remaining 30 of a total 3001. SPSSs of order 28..
• 2011 August S.E. Anderson, and Stephen Johnson finalise SPSSs and SISSs of order 29. They find a total of 7901 SPSSs and 326042 SISSs in order 29.
• 2011 October, Stephen Johnson discovers 500+ SPSSs from order 30 to order 36.
• 2012 April Lorenz Milla found SPSSs of sides 1375 and 2704 in order 31. Also in April 2012, Lorenz Milla completed searches of the 13 and 14 vertex, 31 edge polyhedral graph classes, using Stuart Anderson's noddy (node analysis on electrical nets) software and found 61 new SPSS of order 30. He also found an additional 8 SPSS in order 31.
• 2013 January and February; James Williams wrote a program which he ran for 6 weeks which produced over 15 million SPSSs from orders 21 to 44. The software was not exhastive, it did not find all SPSSs in a given order, the goal was to find as many perfect squares as possible in a practical amount of time. No particular effort was made to search for a particular kind of square ( low order, etc. ) but rather just to find the most. His algorithm looks for squares for which the associated electrical network can be cut roughly in half by cutting at exactly three nodes. The resulting two halves are connected together in a squared square. A number of tests or 'filters' are used to evaluate whether a pair of networks will form a perfect square and these tests speed up the processing considerably. This collection, the 'Williams Collection', is currently being added to the site, beginning with order 30 (no new discoveries were found below order 30).
• 2013 March and April; Lorenz Milla and Stuart Anderson enumerated simple squared squares of order 30. Lorenz used plantri (McKay/Brinkmann) to generate graphs, and Stuart Anderson's sqfind to find squared squares and his sqt to encode the dissections. Lorenz ran the programs on 17 dual core computers over the Easter school holidays. Some 6756 new SPSSs were found, combined with the known 13810 SPSSs of order 30 (including 9189 SPSSs found in Jan/Feb 2013 by James Williams) there are 20566 order 30 SPSSs in total. Simple Imperfect Squared Squares (SISSs) of order 30 were also enumerated. The total for order 30 SISSs is 667403.
• 2013 May ; Lorenz Milla and Stuart Anderson enumerated compound perfect squared squares of order 30.
• 2013 June, July and August ; Lorenz Milla and Stuart Anderson rewrote Stuart's software to incorporate a determinant factorisation technique recommended by William Tutte in his writings to speed up squared square searches, this was used by Duijvestijn in his 1960 thesis. Lorenz replaced the Boost library with C arrays and handwritten linear algebra routines; LU decomposition was replaced with LDL decomposition (planar maps are symmetrical, so LDL Cholesky decomposition is possible and twice as fast as LU decomposition). Lorenz also wrote a plantri plugin to filter graphs using the determinant factorisation technique as they were produced, he was also able to speed up the routines in Stuart's sqfind and sqt programs and combine the plantri plugin and sqfind into a single program mandrill (a combination of the names Milla and Anderson). The end result was squared square software 35 times faster than what was used several months ago. This made it possible to complete the enumeration of order 31 and 32 compound perfect (CPSSs), simple perfect (SPSSs) and simple imperfect squared squares (SISSs) in under 2 months. The computations were done by Lorenz on his computers. There are 54541 SPSSs in order 31 and 144161 SPSSs in order 32. There are 1627213 SISSs in order 31 and 3508496 SISSs in order 32, although these have not been completely verified and may be undercounts.

SPSS counts are listed at OEIS as the sequence;

A006983    Number of simple perfect squared squares of order n.

Counting Polyhedra and Simple Square Tilings.

Using Brendan McKay and Gunnar Brinkmann's published data and plantri software, Stuart E Anderson updated Gérard P. Michon's original table of polyhedral graphs (also known as c-nets and 3-connected planar graphs). These are processed to produce simple, perfect and simple imperfect squared rectangles and squared squares. The number of graphs to process is less than the total for an edge class as edge classes have a dual class which produces the same tilings (the only difference being the tiles are rotated 90 degrees). Only graphs classes where v ≤ f need to be processed. The classes outlined in black are self-dual and are also processed.

Vertex Count in row and Face Count in column, (or vice-versa) Total (row) 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 4 1 2 5 1 1 7 6 1 2 2 2 34 7 2 8 11 8 5 257 8 2 11 42 74 76 38 14 2 606 9 8 74 296 633 768 558 219 50 32 300 10 5 76 633 2 635 6 134 8 822 7 916 4 442 1 404 233 440 564 11 38 768 6 134 25 626 64 439 104 213 112 082 79 773 36 528 9 714 1 249 6 384 634 12 14 558 8 822 64 439 268 394 709 302 1 263 032 1 556 952 1 338 853 789 749 306 470 70 454 7 595 96 262 938 13 219 7 916 104 213 709 302 2 937 495 8 085 725 15 535 572 21 395 274 21 317 178 15 287 112 7 706 577 2 599 554 1 496 225 352 14 50 4 442 112 082 1 263 032 8 085 725 33 310 550 94 713 809 193 794 051 292 182 191 328 192 346 274 542 869 168 992 630 23 833 988 129 15 1 404 79 773 1 556 952 15 535 572 94 713 809 388 431 688 1 134 914 458 2 447 709 924 3 981 512 855 4 939 809 506 4 686 995 652 387 591 510 244 16 233 36 528 1 338 853 21 395 274 193 794 051 1 134 914 458 4 637 550 072 13 865 916 560 31 277 856 206 54 271 705 726 73 247 405 678 6 415 851 530 241 17 9 714 789 749 21 317 178 292 182 191 2 447 709 924 13 865 916 560 56 493 493 990 172 301 697 581 404 008 232 288 741 171 341 224 107 854 282 197 058 18 1 249 306 470 15 287 112 328 192 346 3 981 512 855 31 277 856 206 172 301 697 581 700 335 433 295 2 173 270 387 051 5 270 785 332 349 ~ 19 70 454 7 706 577 274 542 869 4 939 809 506 54 271 705 726 404 008 232 288 2 173 270 387 051 ~ ~ ~ 20 7 595 2 599 554 168 992 630 4 686 995 652 73 247 405 678 741 171 341 224 5 270 785 332 349 ~ ~ ~ 21 527 235 74 424 566 3 380 569 040 77 220 397 213 1 075 323 264 149 10 150 757 285 258 ~ ~ ~ 22 49 566 22 229 616 1 823 658 612 63 443 012 728 1 240 159 791 730 15 683 069 986 564 ~ ~ ~ 23 4 037 671 713 331 098 40 232 230 880 1 136 847 700 529 19 547 663 107 721 ~ ~ ~ 24 339 722 191 283 058 19 322 611 431 8 237 88 552 428 19 682 306 885 581 ~ ~ ~ 25 31 477 887 6 799 902 944 466 224 664 031 15 962 912 975 720 ~ ~ ~ 26 2 406 841 1 654 924 768 201 829 738 768 10 348 108 651 919 ~ ~ ~ 27 249 026 400 64 563 924 319 5 288 847 843 415 ~ ~ ~ 28 17 490 241 14 386 939 428 2 084 335 836 704 ~ ~ ~ 29 1 994 599 707 611 239 308 239 ~ ~ ~ 30 129 664 753 125 619 037 674 ~ ~ ~ 31 1 6147 744 792 ~ ~ ~ 32 977 526 957 ~ ~ The number of edges is the sum of the row and columns indices minus 2 (Euler's Polyhedral formula; v + f = e + 2). see OEIS sequences of the graph/map counts A000944, A000109, A049337        Gérard P. Michon  &  Stuart E. Anderson 2000-2001, 2011

Total polyhedral graphs by edge;A002840 Total graphs to process to enumerate all SPSSs of order = edges - 1: 6 1 1 7 0 0 8 1 1 9 2 1 10 2 2 11 4 2 12 12 10 13 22 11 14 58 50 15 158 79 16 448 372 17 1 342 671 18 4 199 3 417 19 13 384 6 692 20 43 708 34 667 21 144 810 72 405 22 485 704 377 049 23 1 645 576 822 788 24 5 623 571 4 280 533 25 19 358 410 9 679 205 26 67 078 828 33 539 414 27 233 800 162 116 900 081 28 819 267 086 409 633 543 29 2 884 908 430 1 442 454 215 30 10 204 782 956 7 421 166 514 31 36 249 143 676 18 124 571 838 32 129 267 865 144 ~ 33 462 669 746 182 ~ 34 1 661 652 306 539 ~ 35 5 986 979 643 542 ~

Credit for Discovery

• AHS - Arthur H. Stone (United States, 1916-2000);
• AJD - A.J.W.D. (Arie) Duijvestijn (Netherlands, 1927-1998);
• AJP - Stuart Anderson, Stephen Johnson, Ed Pegg Jnr
• A&M - Stuart Anderson and Lorenz Milla
• CJB - Christoffel J. (Chris) Bouwkamp (Netherlands, 1916-2003);
• EPJ - Ed. Pegg Jnr (United States);
• SEA - Stuart E. Anderson (Australia)
• A&P - Stuart E. Anderson and Ed. Pegg Jnr
• JCW - John C. Wilson (Canada)
• GHM - Geoffrey H. Morley (England);
• I_G - Ian Gambini (France);
• JDS - Jasper D. Skinner II (United States);
• L_M - Lorenz Milla (Germany);
• M&A - Lorenz Milla (Germany) & Stuart E. Anderson (Australia);
• PJF - Pasquale J. Federico (United States, 1902-1982);
• RLB - R. Leonard Brooks (England, 1916-1993);
• F&W - Pasquale J. Federico and T.H. (Phil) Willcocks
• S_J - Stephen Johnson (United States);
• THW - T.H. (Phil) Willcocks (England);
• WTT - William T. (Bill) Tutte (Canada, 1917-2002);
• JBW - James B. Williams (United States)

The following publications have featured extensive collections of Simple Perfect Squared Squares;

• 'Squared Squares, Who's Who & What's What', Jasper Dale Skinner, II, Ph.D., 1993
• C.J. Bouwkamp and A.J.W. Duijvestijn, Catalogue of simple perfect squared squares of orders 21 through 25, EUT Report 92-WSK-03, Eindhoven, The Netherlands, 207 pp., 1992
• A. J. W. Duijvestijn: Simple perfect squared squares and 2 x 1 squared rectangles of order 26. Math. Comput. 65(215): 1359-1364, (1996)
• C.J. Bouwkamp, Simple Perfect Squared Squares of order 27, EUT Report 98-WSK-01 Eindhoven, July 1998

Updated on ... April 15, 2013