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Simple Perfect Squared Squares (SPSSs);
Order 21 to 35 and higher orders

This collection of SPSSs from Order 21 to 35 is a complete collection. The number of squared squares grows exponentially with the order. For SPSSs of order 36 and above we have a partial, 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This collection of simple squared squares has been assembled from many sources, with material from published and unpublished collections, mathematics journals, books, and recent computer searches.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.

The origin of the problem of squaring the square

The first appearance of a reference to this problem occurred in the early twentieth century. Ernest Dudeney was a writer who published mathematical puzzles in magazines. In 1907 he published The Canterbury Puzzles which contained a puzzle called Lady Isabel's Casket which concerns the dissection of a square into different sized squares and a rectangle. According to David Singmaster 'Lady Isabel's Casket' appeared first in The London Magazine 7 (Jan 1902) 584 and is the first published reference dealing with the dissection of a square into smaller different sized squares.

Max Dehn was the first mathematician to do work on the dissection of rectangles into squares. Dehn studied at Göttingen under Hilbert's supervision obtaining his doctorate in 1900. Max Dehn had studied the squaring problem in 1903 and proved;

A rectangle can be squared if and only if its sides are commensurable.

If a rectangle can be squared then there are infinitely many perfect squarings.

From 1921 until 1935 he held the chair of Pure and Applied Mathematics at the University of Frankfurt but he was forced to leave his post by the Nazi regime in 1938.

Samuel Loyd was the creator of many famous mathematical puzzles and recreations. Loyd produced over 10,000 puzzles in his lifetime many involving sophisticated mathematical ideas. In 1914 The Patch Quilt Puzzle appeared in Sam Loyd's “Cyclopedia of Puzzles”, in it, 'a square quilt made of 169 square patches of the same size is to be divided into the smallest number of square pieces by cutting along lattice lines'. Gardner states that this problem first appeared in 1907 in a puzzle magazine edited by Sam Loyd. David Singmaster lists it as first appearing in 1914 in Cyclopedia by Loyd but credits Loyd with publishing Our Puzzle Magazine in 1907 - 08. This puzzle also appeared in a publication by Henry Dudeney as Mrs Perkins Quilt. Problem 173 in Amusements in Mathematics 1917.

In 1925 Zbigniew Moroń published a paper, 'O Rozkladach Prostokatow Na Kwadraty' (On the Dissection of a Rectangle into Squares). Moroń gave the first examples of rectangles divided into unequal squares in his paper. He doesn't indicate how they were obtained. Rectangle I is 33 x 32 in size and is divided into 9 unequal squares. Rectangle II is 65 x 47 and has ten squares. According to correspondence quoted by P J Federico, Prof Wladyslaw Orlicz wrote of Moroń in a letter to Dr. Stanislaw Dobrzycki of Lubin, Poland;

“Zbigniew Moroń was my younger schoolmate when studying mathematics at the University of Lwów; about 1923-24 we were both junior assistants in the Institute of Mathematics. Professor Stanislaw Ruziewicz (who was then professor of mathematics at the University) communicated to us the problem of the dissection of a rectangle into squares. He had heard of it from the mathematicians of the University of Krakow who took interest in it. As young men we enthusiastically engaged ourselves in investigating this problem, but after some time we all came to the conclusion that it was certainly as difficult as many other apparently simple questions in number theory. The examples found by Moroń were to us a great surprise. Before the World War II Moroń was a teacher in secondary schools; after it he was too, and dwelt in Wraclow, where he died some 5 years ago.”

Further, in a translation of a later paper by Zbigniew Moroń, by Dr Dobrzycki, Zbigniew Moroń states,

"In the years 1925-28 I found further results in this domain; among others I proved that it is impossible to construct a rectangle with less than 9 different squares; I also knew of the dissection of a square which was later given by Sprague. Nevertheless I did not publish them, but only exposed them at meetings of the mathematical seminar of Professor Ruziewicz."

It is possible that Zbigniew Moroń discovered Sprague's square more than 10 years before Sprague himself, as the Sprague square is partly composed of the Moroń Rectangles I and II, but no other evidence has emerged to support the claim.

In the 1930s the problem "can one decompose a square into a finite number of squares all different?" was added to the Scottish Book as problem 59 by Professor Ruziewicz. The Scottish Book was named after the Scottish Cafe - which was in Lwów Poland (now Lviv, Ukraine),- where Lwów mathematicians would converse and work on math problems, recording them in the Scottish Book . No solution was found to problem 59. Professor Stanislaw Ruziewicz, along with other Lwów mathematicians and academics were murdered by the Nazis and their collaborators in 1942.

In 1930 Michio Abe, published a paper "Covering the square by squares without overlapping," in the Journal of Japan Mathematical Physics. Three things have been said about Abe's work; Working in apparent isolation, he produced over 600 simple perfect rectangles. It appears he was aware of the literature of his day, in particular Z Moron´. He was far ahead of the Germans in researching the topic. Abe published a second paper in 1931. An English translation was published in 1932. This was his last known paper, in it he showed that an infinite series of compound perfect rectangles can be built up (from a simple perfect rectangle [191 x 195]) whose size-ratios approach the limit of one. Abe noted that the problem of squaring the square has not yet been solved.

Tutte wrote in the 1950s, "In 1936 there were a few references in the literature to the problem of cutting up a rectangle into unequal squares. Thus it was known that a rectangle of sides 32 and 33 can be dissected into nine squares with sides of 1,4,7,8,9,10,14,15, and 18 units. Stone was intrigued by a statement in Dudeney's Canterbury Puzzles which seemed to imply that is is impossible to cut up a square into unequal smaller squares. He tried to prove the impossibility for himself, but without success." R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T.Tutte then teamed up to work on the problem. In 1990 Cedric A. B. Smith wrote in personal communication to J.D. Skinner, "W. R. Dean (later Professor of Applied Mathematics at University College, London) is largely responsible for the development of the idea of squaring the square. Dean visited Arthur Stone's shool before Arthur came to Cambridge and said that an unsolved problem was to show that a square cannot be dissected into a finite number of unequal squares". This was at or prior to 1935. In 1997 Cedric A. B. Smith wrote a paper; 'Did Erdös save western civilization?'. In an anecdotal manner the author describes how he became acquainted in the 1930s with a geometric conjecture by Paul Erdös, on the dissection of a square into a finite number of smaller squares. Erdös conjectured such a dissection must contain a least two squares the same size. There was a related conjecture known as Lusin's conjecture. In 1930 Kraitchik published personal communication from Russian mathematician N. N. Lusin that is was impossible to dissect a square into a finite number of dfferent elements. Perhaps Lusin was the originator of the conjecture, perhaps Erdös heard of it from the Polish mathematicians, perhaps Professor Dean heard it from Erdös, in any case Erdös publicised the problem as he often transmitted mathematical ideas from place to place in his travels. As a student in Cambridge England, Smith was introduced to the problem by Arthur Stone. They jointly discussed with R. L. Brooks and then chemist William T. Tutte the relation of that conjecture to Kirchhoff's theory of electrical networks. They finally (1940) showed Erdös's conjecture to be incorrect. Tutte, later founding President of the Institute of Combinatorics and Its Applications, became involved into the work at Bletchley Park during World War II, where the German codes were broken, hastening the end of WWII and saving millions of lives. Since he had been drawn into mathematics by Erdös's conjecture the author is led to the provoking title of this paper. It should be noted the Official Secrets Act prevented any detailed discussion of Tutte's role or the contributions of others to breaking German codes.

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

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.

These at least 3-connected, minimum degree 3, simple planar map classes (v,f,e) where f ≥ v were processed to produce Simple Perfect Squared Squares (SPSSs) up to Order 32.
Face Count in row and Vertex Count in column (or vice-versa)
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
4 1
5 1 1
6 1 2 2 2
7 2 8 11 8 5
8 2 11 42 74 76 38 14
9 8 74 296 633 768 558 219 50
10 5 76 633 2 635 6 134 8 822 7 916 4 442 1 404 233
11 38 768 6 134 25 626 64 439 104 213 112 082 79 773 36 528 9 714 1 249
12 14 558 8 822 64 439 268 394 709 302 1 263 032 1 556 952 1 338 853 789 749 306 470
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
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
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
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
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
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
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
Total (column) graphs 1 2 7 34 257 2 606 32 300 440 564 6 384 634 96 262 938 1 496 225 352 23 833 988 129 387 591 510 244 6 415 851 530 241 107 854 282 197 058
The column node graph total is OEIS sequence A000944.
The edge graph total is OEIS sequence A002840.
The number of edges is the sum of the row and columns indices minus 2 (Euler's Polyhedral formula; v + f = e + 2).        Gérard P. Michon  &  Stuart E. Anderson 2000-2001, 2011  
edgesTotal polyhedral
graphs by edge:
A002840
Total graphs to process
to enumerate all SPSSs
of order = edges - 1:
order
6115
7006
8117
9218
10229
114210
12121011
13221112
14585013
151587914
1644837215
171 34267116
184 1993 41717
1913 3846 69218
2043 70834 66719
21144 81072 40520
22485 704377 04921
231 645 576822 78822
245 623 5714 280 53323
2519 358 4109 679 20524
2667 078 82833 539 41425
27233 800 162116 900 08126
28819 267 086409 633 54327
292 884 908 4301 442 454 21528
3010 204 782 9567 421 166 51429
3136 249 143 67618 124 571 83830
32129 267 865 14492 880 679 56731
33462 669 746 182231 334 873 09132
341 661 652 306 5391 180 993 869 91733
355 986 979 643 5422 993 489 821 77134
36        35

Credit for Discovery

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

Updated on ... May 1, 2016