A square which can be dissected into a number of smaller squares with no two equal is called a perfect square dissection (or a squared square). Square dissections in which the squares need not be different sizes are called Mrs. Perkins's quilts. If no subset of thesquares forms a rectangle, then the perfect square is called "simple."
Perfect square dissections correspond to square numbers which are the sum of square numbers. The simplest place to look for such a square might therefore be thought to be square pyramidal numbers. However, there are only two such numbers: 1 and 4900, and even though
it turns out not to be possible to arrange the 24 squares to form a ￼ square.
A rectangle which cannot be built up of squares all of different sizes is called an imperfect rectangle. Arectangle which can be built up of squares all of different sizes is called perfect. The number of perfect rectangles of orders 8, 9, 10, ... are 0, 2, 6, 22, 67, 213, 744, 2609, ... and the corresponding numbers of imperfect rectangles are 0, 1, 0, 0, 9, 34, 103, 283, ...