The order 27 term 1152 was found by Jasper Skinner and announced in 2003, after exhaustive search of squared rectangle dissections, derived from electrical networks made from c-nets of order 28 using a program supplied by A. J. Duijvestijn and adapted by Jasper Skinner. - Stuart E Anderson, Apr 16 2013 The order 28 term 3001 was found in 2010 by Stuart Anderson and Ed Pegg Jr after exhaustive search of 29 edge 3-connected planar graphs (produced by McKay/Brinkmann's plantri software), translated into electrical networks and then into dissections of squares with Anderson's software. Anderson and Pegg only found 31 new SPSSs, the majority already having been found by Skinner. During 2002 and 2003, Anderson produced 60 SPSSs of order 28, while he corresponded with Jasper Skinner, who by then had already contributed 2775 SPSSs to order 28 (92% of the total). Other contributers to order 28 SPSSs include Duijvestijn(101), Bouwkamp(32), Willcocks(1), and Ian Gambini (1). - Stuart E Anderson, Apr 16 2013 The order 29 term 7901 was found by exhaustive enumeration of 30 edge 3-connected planar graphs, converted into electrical networks, then into dissections of squares. The enumeration was commenced by Stuart Anderson and Ed Pegg Jr in 2010 and completed in August 2011 by Stuart Anderson and Stephen Johnson, using plantri and software written by Anderson. Jasper Skinner also contributed 4389 SPSSs to the order 29 total, and Anderson contributed 6 SPSSs from enumeration they had both separately completed almost a decade earlier. Other contributers to the order 29 total are Duijvestin (138), Bouwkamp (16) and Gambini (1). - Stuart E Anderson, Apr 16 2013 The order 30 term 20566 was found by Lorenz Milla on Easter 2013, using plantri and Anderson's (rewritten) software to process 31 edge 3-connected planar graphs. Milla completed 13 and 14 vertex graph classes in April 2012, finding 61 new SPSSs, and the 15 and 16 vertex graph classes over the Easter holidays in 2013, finishing in April 2013, finding 6756 new SPSSs. James Williams contributed 9189 new discoveries to this total in February 2013 using software he wrote that constructed the electrical nets of squared squares from 2 smaller nets, joined at 3 nodes. Jasper Skinner also contributed 4311 SPSSs to the order 30 total from enumeration he had completed a decade earlier. Other contributers to the order 30 total are Duijvestin (168), Bouwkamp (10), Johnson (70) and Gambini (1). - Stuart E Anderson, Apr 16 2013 From June to September 2013, Lorenz Milla and Stuart Anderson incorporated into the software a technique recommended by William Tutte in his writings, which involved factoring the determinant of the Kirchhoff/discrete Laplacian matrix of a graph into a product 2fS, where f is a squarefree number and S is a square number. This was followed with further software optimization by Lorenz Milla which resulted in a 35 times speed up of the search for perfect squared squares. Lorenz ran the software and completed the computation required to enumerate all SPSSs (simple perfect squared squares), SISSs (simple imperfect squared squares) and CPSSs (compound perfect squared squares) of order 31 and 32. A second run with partly rewritten software was undertaken by Milla and Anderson as there was a possibility some squared squares could have been missed on the first run due to a software bug. The second run found nothing new or different in SPSS totals and confirmed the existing SPSS results from order 21 to order 32. The February 2013 computer search by James Williams had already contributed nearly 90,000 SPSSs to orders 31 and 32. Several other researchers have also made prior discoveries in these orders over the last five decades. - Stuart E Anderson, Sep 29 2013 In June 2014 Jim Williams wrote his own software and completed enumeration of order 33. In April 2016 Jim Williams completed enumeration of order 34 and 35. The totals by discoverer for orders 31 - 35 are as follows: 31 32 33 34 35 SPSSs enumerated by; ----------------------------------------------------------- 27154 81209 13715 11324 4010 Milla & Anderson 26961 62835 364445 979575 2573957 James B Williams 276 0 0 0 0 Geoffrey H Morley 142 115 36 62 41 Stephen Johnson 4 0 0 0 0 PJ Federico 2 0 0 0 0 Lorenz Milla 2 2 1 2 2 Ian Gambini 0 0 0 18 71 Brian Trial ---------------------------------------------------------- 54541 144161 378197 990981 2578081 TOTAL ---------------------------------------------------------- Dissections from these and other orders can be viewed from Anderson's Squaring.net link. _Stuart E Anderson_, May 02 2016