Compound Perfect Squared Squares (CPSS's) have a rectangular inclusion in their tiling. Despite this seeming flaw in compounds, these squared squares are more scarce than simple perfect squared squares. A great deal of thought and ingenuity has gone into the devising of methods of construction for compound squared squares. The order 24 CPSS by T.H. Willcocks was found many years before computers were able to verify it was indeed the lowest order example of its kind.
Compound squared squares can be created from existing simple squared rectangles by selecting a squared rectangle from the catalogue and designating one of the square elements as a rectangle of indeterminate width and height, then making the height and width of the squared rectangle equal (to make it square), then recalculating both the sizes of the squares and the sides of the included rectangle using elementary algebra. The solution will be unique and with luck another perfect squared rectangle may be found in the catalogue which is able to fit into the included rectangle without any repeated elements.
Every CPSS is one of at least 4 isomers (tilings with the same set of tiles). In pdf's and drop down menus, isomer number is shown with square id and discoverer id. The first few numbers of isomers of which we have examples are 4, 7, 8, 16, 32, 48 and 64. We speculate that many other numbers of isomers below 100 are possible but that instances are likely to be of high order and hard to find.
The people listed below have published or have been credited with the discovery of compound squared squares.