Squaring the square

A square with integral sidelength is called an integral square. The squaring-the-square problem consists of tiling a square with integral squares.

Of course squaring the square is a trivial task unless additional conditions are set. The most studied restriction is the "perfect" squared square (see below). Other possible conditions that lead to interesting results are nowhere neat[?] squared squares and no-touch[?] squared squares (see => tiling).

Perfect Squared Square:

A "perfect" squared square is such a square such that each of the smaller squares has a different size. The name was coined in humorous analogy with squaring the circle.

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, and then applied Kirchhoff's laws and circuit decomposition[?] techniques to that circuit.

The first perfect squared square was found by Roland Sprague[?] in 1939.

If we take such a tiling and enlarge it so that the formerly smallest tile now has the size of the square S we started out from, then we see that we obtain from this a tiling of the plane with integral squares, each having a different size.

It is still an unsolved problem, however, whether the plane can be tiled with a set of integral tiles such that each natural number is used exactly once as size of a square tile.

Martin Gardner has written an extensive article about the early history of squaring the square.

Lowest order perfect square

A "simple" squared square is one where no subset of the squares forms a rectangle. The smallest simple perfect squared square was discovered by A. J. W. Duijvestin using a computer search. His tiling uses 21 squares, and has been proved to be minimal.

References:

Brooks, R. L.; Smith, C. A. B.; Stone, A. H.; and Tutte, W. T. The Dissection of Rectangles into Squares, Duke Math. J. 7, 312-340, 1940
Martin Gardner, "Squaring the square," in The 2nd Scientific American Book of Mathematical Puzzles and Diversions.
C. J. Bouwkamp and A. J. W. Duijvestijn, Catalogue of Simple Perfect Squared Squares of Orders 21 Through 25, Eindhoven Univ. Technology, Dept. of Math., Report 92-WSK-03, Nov. 1992.
C.J.Bouwkamp and A.J.W.Duijvestijn, Album of Simple Perfect Squared Squares of order 26, Eindhoven University of Technology, Faculty of Mathematics and Computing Science, EUT Report 94-WSK-02, December 1994.

External links:

(Perfect Squares):

(Nowhere-neat Squared Squares):