Editing Squaring the square (section)

== Simple squared squares ==

A "simple" squared square is one where no subset of more than one of the squares forms a rectangle or square. When a squared square has a square or rectangular subset, it is "compound".

In 1978, {{ill|A. J. W. Duijvestijn|de}} discovered a simple perfect squared square of side 112 with the smallest number of squares using a computer search. His tiling uses 21 squares, and has been proved to be minimal.{{r|duijvestijn}} This squared square forms the logo of the [[Trinity Mathematical Society]]. It also appears on the cover of the [[Journal_of_Combinatorial_Theory | Journal of Combinatorial Theory]].

Duijvestijn also found two simple perfect squared squares of sides 110 but each comprising 22 squares. Theophilus Harding Willcocks, an amateur mathematician and [[fairy chess]] composer, found another. In 1999, I. Gambini proved that these three are the smallest perfect squared squares in terms of side length.{{r|gambini}}

The perfect compound squared square with the fewest squares was discovered by T.H. Willcocks in 1946 and has 24 squares; however, it was not until 1982 that Duijvestijn, [[Pasquale Joseph Federico]] and P. Leeuw mathematically proved it to be the lowest-order example.{{r|dfl}}