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 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.

Perfect squared squares

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

Perfect squared squares were studied by R. L. Brooks, C. A. B. Smith, 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 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 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.

Martin Gardner published an extensive article written by W. T. Tutte about the early history of squaring the square in his ''Mathematical Games'' column of November 1958.

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, 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. It also appears on the cover of the 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.

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.

Mrs. Perkins's quilt

When the constraint of all the squares being different sizes is relaxed, a squared square such that the side lengths of the smaller squares do not have a common divisor larger than 1 is called a "Mrs. Perkins's quilt". In other words, the greatest common divisor of all the smaller side lengths should be 1. The Mrs. Perkins's quilt problem asks for a Mrs. Perkins's quilt with the fewest pieces for a given $n\times n$ square. The number of pieces required is at least $\log_2 n$, and at most $6\log_2 n$. Computer searches have found exact solutions for small values of $n$ (small enough to need up to 18 pieces). For $n=1,2,3,\dots$ the number of pieces required is:

No more than two different sizes

For any integer $n$ other than 2, 3, and 5, it is possible to dissect a square into $n$ squares of one or two different sizes.

Squaring the plane

In 1975, Solomon Golomb raised the question whether the whole plane can be tiled by squares, one of each integer edge-length, which he called the heterogeneous tiling conjecture. This problem was later publicized by Martin Gardner in his Scientific American column and appeared in several books, but it defied solution for over 30 years.

In ''Tilings and patterns'', published in 1987, Branko Grünbaum

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