SPSS (Simple Perfect Squared Square), Order 21: 112 x 112 (AJWD)	SISS (Simple Imperfect Squared Square), Order 13: 23 x 23	CPSS (Compound Perfect Squared Square), Order 24: 175 x 175(THW)
	Simple Perfect Squared Squares (SPSSs) are defined as Simple by having no smaller squared squares or squared rectangles in the dissection. and Perfect if the squares are all different sizes. They are considerably rare compared to perfect simple squared rectangles of the same order. According to David Gale[1], "Bouwkamp says that there are about 5,000,000 perfect simple squared rectangles to every such [squared] square (for order greater than 20)!". The lowest order SPSS appears alone in order 21.	Simple Imperfect Squared Squares (SISSs) are Imperfect as not all the squares are different sizes (at least two squares are the same size). These are more numerous than Simple Perfect Squared Squares. The first one appears in order 13. Having squares of the same size can result in SISSs having symmetrical arrangements. SISSs can be used to 'derive' SPSSs two orders down.	Compound Perfect Squared Squares (CPSSs) are defined as Compound if there is a rectangle or a smaller dissected square in the dissection. These squared squares are rarer at any given order (>= 24) than SPSSs.

A chronology of the significant square dissection discoveries and the people who made them;

• 1902 H.E. Dudeney published a puzzle called Lady Isabel's Casket that concerns the dissection of a square into different sized squares and a rectangle. According to David Singmaster 'Lady Isabel's Casket' appeared first in Strand Mag. 7 (Jan 1902) 584 and is the first published reference dealing with the dissection of a square into smaller different sized squares. It was also published in The Canterbury Puzzles in 1907.
• 1903 M. Dehn studied the squaring problem in 1903 and proved;
  1. A rectangle can be squared if and only if its sides are commensurable. [It follows a squared rectangle can always be given integer sides]
  2. If a rectangle can be squared then there are infinitely many perfect squarings.
• 1907-1914 S. Loyd published The Patch Quilt Puzzle. 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. The answer, which is unique, is composed of 11 squares with sides 1,1,2,2,2,3,3,4,6,6,7 within a square of 13. It is neither perfect nor simple. 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
• 1923-24 Z. Moroń started work on the problem of squaring rectangles and squares. "S. Ruziewicz communicated to us (Z. Moroń and W Orlicz) the problem of the dissection of a rectangle into squares. He had heard it from the mathematicians of the University of Krakow who took an interest in it." Moroń later claimed in a paper, translated by Dr Dobrzycki "I also knew of the dissection of a square which was later given by R Sprague. Neverless I did not publish ..." (Skinner, Who's Who, & What's What p 9).
• 1925 Z. Moroń published a paper, 'On the Dissection of a Rectangle into Squares' (translated from Polish). 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. Moroń raised the question "For what squares is it possible to dissect them into squares?" He then observes “if there exists a rectangle (of different sides) for which there are two dissections R1 and R2 such that;
  1. in neither of these dissections does there appear a square equal to the smaller side of the rectangle and,
  2. each square of dissection R1 is different from each square in dissection R2, then the square is dissected into squares, all different
• 1930 M. Kraitchik published a personal communication from N. Lusin which stated it was impossible to dissect a square into a finite number of different elements.
• 1931 M. Abe, working in isolation in Japan published his first paper on squared rectangles, he produced over 600 squared rectangles apparently in search of perfect squared squares. He published a second paper in 1932.
• 1933-35 A.H. Stone hears about the problem, and later communicates it to R.L. Brooks, C.A.B. Smith, and W.T.Tutte. According to C.A.B. Smith (1990 correspondence to Skinner). "Prof. W.R. Dean is largely responsible for the development of the idea of squaring the square - he visited Arthur Stone's school before Arthur came to Cambridge and said that an unsolved problem was to show that a square can not be dissected into a finite number of unequal squares."
• 1938 R.P. Sprague found a solution to the problem of squaring the square. Sprague constructed his solution using several copies of various sizes of Z. Moroń's Rectangle I (33x32), Rectangle II (65x47) and a third 12 order simple perfect rectangle and five other elemental squares to create an order 55, compound perfect squared square (CPSS) with side 4205.
• 1939 R.L. Brooks found a simple perfect squared square (SPSS), side 4920 of order 38.
• 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 and an empirically constructed order 26 CPSS, side 608, attributed to Tutte. In the same year A.H. Stone uses 2 order 13 SPSRs as R1, R2 to construct and publish a Moroń dissection CPSS of order 28, side 1015.
• 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.
• 1948 T.H.Willcocks, discovered a CPSS side 175, of order 24. It is one of 4 isomers (squarings with the same elements), since the subrectangle has 4 possible orientations.
• 1960 C.J. Bouwkamp, A.J.W. Duijvestijn, P. Medema [7] Enumerated all the polyhedral graphs up to 15 edges to produce simple perfect and simple imperfect squared rectangles (SPSRs & SISRs) to order 14.
• 1961 M. Gardner published an article in Mathematical Games, Scientific American magazine on squared squares written by William Tutte.
• 1962 A.J.W. Duijvestijn, in his PhD thesis 'Electronic Computation Of Squared Rectangles', showed no SPSS exists with fewer than 20 squares.
• 1963 R. Ellis produced a 2x1 non-trivial compound squared rectangle 282 x 564 of order 25
• 1964 C.J. Bouwkamp, A.J.W. Duijvestijn, J. Haubrich, publish a catalog of SPSRs orders 9 to 18.
• 1964 P.J. Federico constructed a CPSS with side 235 of order 25.
• 1964 J.C. Wilson found an SPSS with side 503 of order 25.
• 1965-66 E. Lainez constructed 2 CPSSs with sides 360, 460 of orders 26 and 27 respectively.
• 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.
• 1968 I.M Yaglom published the first book How to Dissect a Square on squared rectangles and squared squares (in Russian).
  links to book; djvu (1.7M) or pdf (9.5M).
• 1969 T.H. Willcocks constructed an SPSS with side 900 of order 27.
• 1972 N.D. Kazarinoff and R. Weitzenkamp proved the nonexistence of a CPSS of order less than 22.
• 1978 P.J. Federico found the other CPSS (side 344) of order 25, and 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 are the smallest size perfect squared squares.
• 1982 A. J. W. Duijvestijn, P. J. Federico, P. Leeuw established the lower limit of the order of CPSSs and searched to find all that can exist up to and including order 24. The main results were; There are no CPSSs below order 24. and there is one and only one (THW 24:175) CPSS of order 24.
• 1983 A. Augusteijn and A. J. W. Duijvestijn publish 'Simple Perfect Square-Cylinders of Low Order' and find 2 simple perfect square-cylinders with sides 79 and 81 of order 20
• 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 and 2x1 SPSRs of orders 21 - 24.
• 1992 Jan, C.J. Bouwkamp constructed 5 SPSS of order 25 and 21 SPSS of order 26.
• 1992 A.J.W. Duijvestijn enumerated the remaining SPSSs of order 25 and 2x1 SPSRs and published SPSSs of orders 21 to 25
• 1993 J.D. Skinner found 3 SPSSs of order 26.
• 1993 S. J. Chapman published 'The dissection of rectangles, cylinders, tori, and Möbius bands into squares.'
• 1993 J.D. Skinner published "Squared Squares, Who's Who & What's What"
• 1996 A.J.W. Duijvestijn enumerated the remaining SPSSs and 2x1 SPSRs of order 26.
• 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. He also found the known CPSSs up to order 26. With a second algorithm a lot more efficient but incomplete he obtained more than 30,000 squared squares. He also found solutions to two closely related problems; the decomposition using distinct sized squares of the cylinder and of the torus. He also published a paper 'A method for cutting squares into distinct squares' (Discrete Applied Mathematics 98 (1999) 65-80)
• 2003 J.D. Skinner completed enumeration of SPSSs of order 27. Skinner also enumerated 97% of order 28 SPSSs and significant amounts of order 29 and 30, and a variety of CPSSs in low and higher orders.
• 2003 S.E. Anderson, using plantri and software he wrote, 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 and CPSSs up to order 28. 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.
  In counting isomer classes of CPSS they find;
  1. 1 CPSS of order 24, with 4 isomers
  2. 2 CPSSs of order 25, with 12 isomers
  3. 16 CPSS of order 26, with 100 isomers, including 1 CPSS with side 512 not previously known ( 8 isomers), which completes this order.
  4. 46 CPSSs of order 27, with 220 isomers, including 7 CPSSs not previously known, which completes the order.
  5. 143 CPSSs of order 28, with 948 isomers, including 50 CPSSs not previously known, which completes the order.
• 2011 August S.E. Anderson, and Stephen Johnson finalise SPSSs and SISSs of order 29. They found a total of 7901 SPSSs and 326037 SISSs in order 29.
• In January 2012 a duplicate was detected in order 29 SPSSs, all tilings were regenerated from the graphs and a new count was done. The total for order 29 SPSS was adjusted down to 7901. This was incorrectly reported by Stuart Anderson as 7902 SPSSs until 2012. Order 28 SPSSs were also recounted in January 2012 and the total of 3001 was reconfirmed.
• In January 2012 Geoffrey Morley finds 31 new CPSSs in orders 30-34.