As a young boy in secondary school, I was supremely excited when I first encountered this most interesting subject - about the same time that Duijvestijn was doing his investigations. Thanks for helping me relive some of that excitement and for adding the rich bit of the perfect squared square discovery. 
Squaring the Square
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May 20, 2011
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#2
Why not design a chess board with squares of 64 different sizes, and a chess game to be played on it.
A simple perfect square of order 64 is shown near the end of the document at
www.squaring.net/sq/ss/spss/o31+/SPSSo31-75.pdf
The distribution of sizes is somewhat unbalanced, and it might be better to look for a different one. It would be necessary to use four colors for the squares, so that no two neighbor squares would have the same color.
Squared rectangles can be designed interactively and colored with four colors at
www.bjarne.altervista.org/rct/introduction.html
The rectangles cannot have much more than 30 squares. This is because the 16-digit precision offered on most computers is a limitation.
Somewhat embarrassing, given that a squared square of order 69 was once computed with pencil and paper.
Regards
Bjarne Pagh Byrnak
Squaring the Square
“Lady Isabel’s Casket”, one of the Canterbury Puzzles, appeared first in Strand Mag. 7 (Jan 1902) and is the first published reference dealing with the dissection of a square into smaller different sized squares.
Squaring the Square is a curious turn of phrase. It sounds as though it may have some relation to “squaring the circle” an activity which was proven to be impossible in 1882 by Ferdinand von Lindemann due to the transcendence of π (Pi). The phrase 'squaring the square' was coined as a humorous reference to squaring the circle by Tutte when he and Brooks, Smith, and Stone successfully attacked the problem, thought to be impossible, of dissecting a square into smaller squares, all of different sizes. Oddly enough, this problem, while having the appearance of classical geometry, did not appear anywhere in arts, science, mathematics or literature until the early 20th century. The earliest reference to the dissection of squares into squares is in 1902 and is due to Henry Dudeney, an English puzzler and writer of recreational mathematics.
Tiling by Squares
A square or rectangle is said to be 'squared' into n squares if it is tiled into n squares of sizes s1,s2,s3,..sn. A rectangle can be squared if its sides are commensurable (in rational proportion, both being integral multiples of the same quantity). The sizes of the squares si are shown as integers and the number of squares n is called the order. Squared squares and squared rectangles are called simple if they do not contain a smaller squared square or rectangle, and compound if they do. Squared squares and squared rectangles are called perfect if the squares in the tiling are all of different sizes and imperfect if they are not. Most of the square tilings we are familiar with in our everyday lives use repeating squares of the same size, such as fly screens, square floor tiles, square graph paper and the like. These repeating grid square tilings can be described using this terminology as 'imperfect' and 'compound'. Simple and perfect square tilings are quite different in that no square of the same size is repeated. This is not so familiar, nor is it immediately obvious how to show that they exist or to be able to make them as required. In fact simple perfect squared rectangles begin at order 9 and simple perfect squared squares at order 21.
Zbigniew Moroń(1904-1971), Wraclow, Poland
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. In the years 1925-28 he found further results in this domain; among others, certainly as difficult a conclusion as many other apparently simple questions in number theory, he proved that it is impossible to construct a rectangle with less than 9 different squares.
One way of attacking the problem of finding rectangles that can be squared is to make a sketch of a proposed partition into squares, labelling the edge-length of each square, writing down all the relations that the edge-lengths must satisfy in order to fit into the rectangle, and solving the system of equations thus obtained. Instead of carrying as many unknowns as there are squares in the subdivision, one can label neighbouring squares so that they fit together in the sketch; thus having fewer unknowns to eliminate later.
In Rectangle I the neighbouring squares x, y and z, are labelled as shown. Then the remaining squares are x + y, 2x + y, y - z, y - 2z, y - 3z, 2y - 5z. Thus one obtains relations between the unknowns; for example, one can equate the lengths of opposite sides of the containing rectangle.
The horizontal sides yield
2x + y + x + y = 2y - 5z + y - 2z + y - z, that is, (1) 3x - 2y + 8z = 0; and the vertical sides yield 2y - 5z + 2x + y = y - z + y + x + y, that is, (2) x - 4z = 0. So x = 4z and Y = 10Z.
If z = 1, the tiling shown is obtained; if z is set equal to any other positive quantity, the same configuration, blown up (or shrunk) by the factor z is obtained.
Adrianus Johannes Wilhelmus Duijvestijn (AJWD) (1927-1998)
He was born in the Hague, the Netherlands, December 10, 1927. In 1962 he earned the degree of Ph. D. at Technological University Eindhoven, under the supervision of Prof. Dr. C. J. Bouwkamp, his thesis advisor. Duijvestijn's 1962 thesis was titled the "Electronic computation of squared rectangles" but it was also a search for simple perfect squared squares. Duijvestijn and Bouwkamp were able to produce c-nets up to order 20, and investigated squared squares up to order 19. No simple perfect squared squares were found. So Duijvestijn and Bouwkamp established that the lowest order simple perfect squared square must be order 20 or higher.
It was not possible to continue the investigation any further at this time. Duijvestijn had used the processing power, memory and secondary storage of the computing equipment to its limits. As c-nets and squared rectangles increase by order, the number of objects increases exponentially so that the computer hits a wall where it cannot process the volume of data required. Throughout the sixties, many of the discoveries made were based on the tables produced by Duijvestijn and Bouwkamp but using human construction techniques.
AJWD's order 21 SPSS, 112 x 112
Computer storage and processing power had started to grow according to Moore's Law, pushing the computational limits back, so in 1978 Duijvestijn decided to run his software on current hardware, to continue the search for the lowest order simple perfect squared square. During the night of March 22, 1978 a line of computer output from Duijvestijn's program showed he had found an order 21: 112 x 112 square (Rectangle II), the lowest order simple perfect squared square.