There are a number of interesting results related to the tiling of squares. For example, M. Laczkovich has shown that there are exactly three shapes of non-right triangles
that tile the square with similar copies, corresponding to angles , , and (Stein and Szabó 1994). In particular,
given triangles of shape
with no two the same size, tile the square. The best known solution has 8 triangles
(Berlekamp 1999).

The total number of squares contained in a grid of unit square is the square
pyramidal number