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# 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 Squares

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

## 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. Duijvestin 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 squared being different sizes is relaxed, the resulting squared square problem is often called the "Mrs. Perkins' Quilt" problem.

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