A rectangle which cannot be built up of squares all of different sizes is called an imperfect rectangle. A rectangle 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, ... (OEIS A002839) and the corresponding numbers of imperfect rectangles are 0, 1, 0, 0, 9, 34, 103, 283, ... (OEIS A002881). Anderson maintains an online database of perfect rectangles at http://www.squaring.net/.

Mrs. Perkins's quilt | no-touch dissection | nowhere-neat dissection | perfect square dissection | rectangle tiling