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.