Squaring the square

From Academic Kids

A square with sides equal to a unit length multiplied by an integer is called an integral square. The squaring-the-square problem consists of tiling one integral square using only other integral squares.

Squaring the square is a trivial task unless additional conditions are set. The most studied restriction is the "perfect" squared square, where all contained squares are of different size (see below).

Other conditions that lead to interesting results are nowhere neat squared squares and no-touch squared squares (see tiling).

Contents

Perfect squared squares

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, 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 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.

Missing image
SQSQ21.gif
Lowest-order perfect squared square

Simple squared squares

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. Duijvestijn using a computer search. His tiling uses 21 squares, and has been proved to be minimal.

Mrs. Perkins' quilt

When the constraint of all the squares being different sizes is relaxed, the resulting squared square problem is often called the "Mrs. Perkins' Quilt" problem.

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.

See also

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