There are a number of interesting results related to the tiling of squares.
The total number of squares having sides parallel to the axes that are contained in an 
grid is the square pyramidal number
 |
whose values are given for 
, 2, ... by 1, 5, 14, 30, 55, 91, ... (OEIS A000330).
The total number of squares in any orientation that are contained in an 
grid is the 4-dimensional pyramidal number
 |
whose values are given for 
, 2, ... by 1, 6, 20, 50, 105, 196, ... (OEIS A002415).
Laczkovich (1990) proved that there are exactly three shapes of non-right triangles that tile the square using similar copies,
corresnamely those with angles 
, 
, and 
(Stein and Szabó 1994), i.e., 
,
, and
.
The proof of Laczkovich (1990) involved thousands of 
triangles, but included no
explicit illustration. A construction involving 32 
triangles (14 of which are
distint), a number believed to be minimal and which is illustrated above, was subsequently
made by Tom Sirgedas (Pegg 2025).
Right triangles with edge lengths in the proportion 1:2:
and having no two such triangles the same size also tile the square. The best known
solution has 8 triangles (Berlekamp 1999).
-
-
Square Dissection."