Squaring the square

Squaring the square is the problem of tiling an integral square using only other integral squares. (An integral square is a square whose sides have integer length.) The name was coined in a humorous analogy with squaring the circle. Squaring the square is an easy task unless additional conditions are set. The most studied restriction is that the squaring be perfect, meaning that the sizes of the smaller squares are all different. A related problem is squaring the plane, which can be done even with the restriction that each natural number occurs exactly once as a size of a square in the tiling. The order of a squared square is its number of constituent squares.

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

It is first recorded as being studied by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte at Cambridge University. They transformed the square tiling into an equivalent electrical circuit — they called it "Smith diagram" — by considering the squares as resistors 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 square, a compound one of side 4205 and order 55, was found by Roland Sprague in 1939.

Martin Gardner published an extensive article written by W. T. Tutte about the early history of squaring the square in his mathematical games column in November 1958.

A "simple" squared square is one where no subset of the squares forms a rectangle or square, otherwise it is "compound".

In 1978, A. J. W. Duijvestijn 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. This squared square forms the logo of the Trinity Mathematical Society.

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