The chessboard is an example of a square dissected into smaller squares all of the same size.
A somewhat more challenging problem is to dissect a square into a finite number of squares that are
all of different sizes. This was for a long time considered impossible, until the discovery in 1925
by Zbigniew Moron of a dissection of a nearly square rectangle, 32×33, into 9 differently sized
squares, the fewest possible. This led R. Sprague (1939) to find a 55-square solution, and
R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte (1940) to find a 26-square solution,
and to develop a general method for constructing such diagrams with 39 or more squares.
However the unique minimal solution, showing 21 squares in a square 112×112 was not found until
1978 when it was published by A. J. W. Duijvestijn in the *Journal of Combinatorial Theory*.

There is a now website devoted to the subject of squaring the square: http://www.squaring.net/ Which relieves me of the need to publish further details here. The minimal solution is diagrammed on this page about Duijvestijn. The dimensions of the composite squares involved are often very large, which makes showing them in true-to-scale diagrams difficult.

An article by T. H. Willcocks was published online in *The Games and Puzzles Journal* and is now here:
http://www.mayhematics.com/j/gpj35.htm
This allows two squares of the same size, but not placed side by side, and describes a method of approach
to solving this type of problem.