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.
- 1939 Announcement by C. A. B. Smith that Tutte had found a perfect squared square with no perfect subrectangle. Minutes of the 204th Meeting of the Trinity Mathematical Society (Cambridge) (24 Apr 1939). Minute Books, vol. III, p. 248. (Singmaster)
- 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)
- 1950 R.L. Brooks discovery of a simple perfect squared square (SPSS), side 4920 of order 38 was published by W.T. Tutte in 1950, in a paper which demonstrated some advanced mathematical techniques. A simple perfect squared square of the 52nd order, due to Tutte and Smith is given, as is one of 70th order and side 384948 and the 69th order with a side of 7919535. The paper was submitted 18 March 1948.
- 1959 The addendum (pp. 162-4) to Tutte's chapter in Martin Gardner's "More Mathematical Puzzles and Diversions" says "The smallest published square that is both simple and perfect is a 38th-order square with a side of 4,920, discovered by R.L. Brooks. In 1959 this was bettered by T.H. Willcocks of Bristol, with a 37th-order square, 1,947 on the side."
- 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. Six examples, from 59:2568805 to 60:5629849, have been be found. They include 60:616457A, wrongly stated in the article to have side 616,467. By modifications of his method Morley has also found SPSSs of orders 48 and 49.
- 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 to September ; 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 1627218 SISSs in order 31 and 3508516 SISSs in order 32. The software was run for a second time in September by both Stuart and Lorenz due to a bug affecting a small number of determinant factorisation cases, CPSS and SPSS totals remained the same, but SISS totals were increased slightly.
- 2013 December ; Stephen Johnson collated the results of his squared square searches in early 2013 and made them available. A number of new SPSS discoveries in orders 33-37 were found, these have been included in the website.
- 2014 March ; Brian Trial uses a new 'Ell-Munch' method to find 114712 SPSSs in orders where there are large gaps in the current catalogues. By “ell,” we mean any six-sided figure whose sides are parallel to the coordinate axes (Henle 2008). Brian's lowest order find is of order 34 and the highest is order 236, with the majority of finds in the orders 30s to 70s. He wrote;There are 4 different types of Ell depending on what quadrant the cut out section is located, which could have any rectangular shape, some quite stretched in one direction as we can see in the solutions so far. Another "munch" can be done along any one of the 6 sides of the Ell depending on the sides' relative sizes.
- 2014 May June ; Jim Williams, with new software enumerated the SPSSs of order 33. In 2 weeks he found all 378197 SPSSs of this order.
- 2016 April ; Jim Williams, enumerated the SPSSs of order 34 and 35. He found 990981 SPSSs in order 34 and 2578081 SPSSs in order 35.
SPSS counts are listed at OEIS as the sequence;
A006983 Number of simple perfect squared squares of order n.
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.
- A&P - Stuart E. Anderson and Ed. Pegg Jnr
- 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
- B_T - Brian Trial (United States);
- CJB - Christoffel J. (Chris) Bouwkamp (Netherlands, 1916-2003);
- EPJ - Ed. Pegg Jnr (United States);
- F&W - Pasquale J. Federico and T.H. (Phil) Willcocks
- GHM - Geoffrey H. Morley (England);
- I_G - Ian Gambini (France);
- JBW - James B. Williams (United States)
- JCW - John C. Wilson (Canada)
- 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);
- SEA - Stuart E. Anderson (Australia)
- S_J - Stephen Johnson (United States);
- THW - T.H. (Phil) Willcocks (England);
- WTT - William T. (Bill) Tutte (Canada, 1917-2002);
The following publications have featured extensive collections of Simple Perfect Squared Squares;
Updated on ... May 1, 2016