# squaring the square

The problem of squaring the square is how to tile a square with **integral squares** (squares of integral side-length). Of course squaring the square is a trivial
task unless additional conditions are set. The most studied restriction
is the **perfect squared square**: a square such that each
of the smaller squares has a different size. The name was coined in humorous
analogy with squaring the circle and is first recorded as being studied by R. L. Brooks, C. A. B. Smith,
A. H. Stone, and W. T. Tutte at Cambridge University. The first perfect
squared square was found by Roland Sprague in 1939. If such a tiling is enlarged so that the formerly smallest tile becomes as big as the original
square, it becomes clear that whole plane can be tiled 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 square tiles such
that each natural number is used exactly once as the size of a tile. 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. Other possible conditions that lead to interesting results
are nowhere-neat squared squares and no-touch squared squares. Developments
leading to squaring the square can be traced back to 1902 and the first
appearance of Henry Dudeney's Lady Isabel's
Casket, later published as problem #40 in *The Canterbury Puzzles*.