Editing Squaring the square (section)

== Perfect squared squares ==
[[File:Smith diagram.svg|right|thumb|upright=1.1|Smith diagram of a rectangle]]

A "perfect" squared square is a square such that each of the smaller squares has a different size.

{{anchor|Smith diagram}}Perfect squared squares were studied by [[R. Leonard Brooks|R. L. Brooks]], [[Cedric Smith (statistician)|C. A. B. Smith]], [[Arthur Harold Stone|A. H. Stone]] and [[W. T. Tutte]] (writing under the [[collective pseudonym]] "[[Blanche Descartes]]") at Cambridge University between 1936 and 1938. They transformed the square tiling into an equivalent [[electrical circuit]] – they called it a "Smith diagram" – by considering the squares as [[resistor]]s that connected to their neighbors at their top and bottom edges, and then applied [[Kirchhoff's circuit laws]] and [[circuit decomposition]] techniques to that circuit. The first perfect squared squares they found were of order 69.

The first perfect squared square to be published, a compound one of side 4205 and order 55, was found by [[Roland Sprague]] in 1939.{{r|sprague}}

[[Martin Gardner]] published an extensive article written by [[W. T. Tutte]] about the early history of squaring the square in his [[List of Martin Gardner Mathematical Games columns|''Mathematical Games'' column]] of November 1958.{{r|gardner-tutte}}

[[File:smallest_perfect_squared_squares.svg|thumb|upright=1.35|Lowest-order perfect squared square (1) and the three smallest perfect squared squares (2&ndash;4): all are simple squared squares]]