This is a geometric puzzle though it might conceivably also define a special class of Pythagorean triples.
A perfect squared square PSS is a square (as a plane figure)
partitioned into smaller squares, each of a different size.
There are other types of squared squares or
squared rectangles that have been studied (e.g. simple SPSS, vs compound CPSS), see
Question. Is there a perfect squared square that can be split into two perfect squared squares? That is, could we use the building blocks (smaller squares) that form the given perfect squared square to form two smaller perfect squared squares?
Of course if the given perfect squared square has side $c$ and the two smaller ones have sides $a$ and $b$ respectively, then the numbers $a$, $b$, $c$ would form a Pythagorean triple (since the areas of the two smaller squared squares sum up to the area of the given bigger squared square).
Question. Which Pythagorean triples (if any) could be represented in the above form?
For some Pythagorean triples $(a,b,c)$ the numbers $(a^2,b^2,c^2)$ seem to sometimes appear as the sides of neighboring smaller squares forming the partition of a perfect squared square. For example, for the Pythagorean triple $(3,4,5)$ the squared numbers are $(9,16,25)$ and these appear as the sides of three neighboring squares from the partition of the Lowest-order perfect squared square (i.e. formed by only 21 squares which is smallest possible for SPSS, same links as above). Could one say anything more about this (an explanation, or a description when it occurs, for which Pythagorean triples $(a,b,c)$)?
Interestingly, a simple geometric argument shows that the analogue in three or more dimensions of squaring the square has no solutions, e.g. one cannot partition a cube (as a three-dimensional geometric figure) into smaller cubes, no two of which are congruent (see first link). One is tempted to make a wild guess that this might have something to do with Fermat's last theorem (however obvious it seems that there could be no actual relation).
Incomplete history: Roland Sprague published in 1940 the first simple squared square link. He used squared rectangles found earlier by Zbigniew Moroń, plus additional squares. Another important early work was by R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T.Tutte link who related the problem to electrical networks (graphs).
Another post about squared squares is link it has some related numerical data.
Hereabbreviations SPSS and CPSS are used for Simple Perfect Squared Squares and Compound Perfect Squared Squares, seems a standard terminology. So my question is about PSS (without specifying SPSS or CPSS, but assuming, as the following discussion shows, that CPSS are fine). $\endgroup$