cp's OEIS Frontend

A squared rectangle (which may be a square) is a rectangle dissected into a finite number of two or more squares. If no two squares have the same size, the squared rectangle is perfect. A squared rectangle is simple if it does not contain a smaller squared rectangle. The order of a squared rectangle is the number of constituent squares. - Geoffrey H. Morley, Oct 17 2012

a(n) is the number of solutions to the classic problem of 'squaring the square' by n unequal squares. A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. The order of a squared rectangle is the number of constituent squares. A squared rectangle is simple if it does not contain a smaller squared rectangle, and compound if it does.

A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. A squared rectangle is simple if it does not contain a smaller squared rectangle. The order of a squared rectangle is the number of constituent squares. - Geoffrey H. Morley, Oct 17 2012

Orders 15 to 19 were enumerated by C. J. Bowkamp and A. J. W. Duijvestijn (1962). Orders 20 to 29 were enumerated by Stuart Anderson (2010-2012). Orders 30 to 32 were enumerated by Lorenz Milla and Stuart Anderson (2013). - Stuart E Anderson, Sep 30 2013

A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size, the squared rectangle is perfect. A squared rectangle is compound if it contains a smaller squared rectangle. The order of a squared rectangle is the number of constituent squares. - Geoffrey H. Morley, Oct 17 2012

The smallest perfect compound squared square was published by T. H. Willcocks in 1948, has 24 squares and has one rectangle as a sub-dissection; however, it was not until 1982 that A. J. W. Duijvestijn, P. J. Federico and P. Leeuw proved it to be the lowest-order example.

In 2010 Stuart Anderson and Ed Pegg Jr generated all 2-connected minimum degree 3 planar graphs up and including 29 edges, using B. D. McKay and G. Brinkmann's plantri software, then applied electrical node analysis to the graphs to obtain complete counts of compound perfect squares in orders 24, 25, 26, 27 and 28, along with all the members of each equivalence class of each compound square.

In 2011 S. E. Anderson and Stephen Johnson commenced order 29 CPSSs, and processed all plantri generated 2-connected minimum degree 3 planar graph embeddings with up to 15 vertices. This left the largest graph class, the 16 vertex class. In 2012 S. E. Anderson processed the remaining graphs, using the Amazon Elastic Cloud supercomputer and new software which he wrote to find a(29). - Stuart E Anderson, Nov 30 2012

In May 2013 Lorenz Milla and Stuart Anderson enumerated a(30) (CPSSs of order 30), using the same process and software as used on order 29 CPSSs, with the addition of a technique recommended by William Tutte in his writings which resulted in a 3x speed-up of the search for perfect squared squares by 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. - Stuart E Anderson, May 26 2013

From June to September 2013, Lorenz Milla further optimized the process and software and completed the computation required to enumerate all CPSSs of order 31 and 32. A second run with enhanced software was undertaken by Milla and Anderson as there was a possibility some CPSSs could have been missed on the first run. The second run found nothing new or different and confirmed the result. - Stuart E Anderson, Sep 29 2013

In April 2014, Jim Williams wrote software and used it to complete the enumeration of CPSS orders 33, 34, 35 and 36. - Stuart E Anderson, May 02 2016

In August 2018, Jim Williams completed the enumeration of CPSS orders 37, 38 and 39. - Stuart E Anderson, Sep 17 2018.

A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. The order of a squared rectangle is the number of constituent squares.

A squared rectangle is simple if it does not contain a smaller squared rectangle, compound if it does, and trivially compound if a constituent square has the same side length as a side of the squared rectangle under consideration.

A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. The order of a squared rectangle is the number of constituent squares.

A squared rectangle is simple if it does not contain a smaller squared rectangle, compound if it does, and trivially compound if a constituent square has the same side length as a side of the squared rectangle under consideration.

The isomer count of a compound perfect squared square (CPSS) is the number of ways its squared subrectangle and constituent squares can be arranged, up to symmetry of the CPSS. A squared square is perfect if none of its constituent squares are the same size. A squared square is compound if it contains a smaller squared subrectangle. Note that the squared subrectangle can be a squared square. Specific concrete examples of CPSSs with isomer counts under 100 of 4, 7, 8, 11, 12, 16, 19, 20, 23, 24, 28, 31, 32, 35, 36, 39, 40, 47, 48, 56, 60, 63, 64, 68, 72, 76, 80, 88 and 96 exist. Geometric constructions based on a suitable pair of perfect squared rectangles each with up to 4 isomers suggests additional isomer counts up to 100 of 14, 21, 22, 33, 42, 44, 66 and 99, but no actual examples are known. As the number of squares in a squared square - the order - increases new arrangements appear. It is conjectured that expected CPSS subrectangle isomer arrangements will eventually appear if the order is high enough.

The term a(6)=14 is based on a theoretical construction, not on known or existing CPSSs. These terms have been included to distinguish the sequence from others. Considering all the ways two or more subrectangles can be arranged within a CPSS it does not appear possible for a CPSS with 5, 6, 9, 10 or 13 isomers to exist but even this much has not been proved.