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 
[`wikipedia`](http://en.wikipedia.org/wiki/Squaring_the_square), [`wolfram`](http://mathworld.wolfram.com/PerfectSquareDissection.html), and [`squaring.net`](http://www.squaring.net/sq/ss/ss.html). 

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](http://www.squaring.net/history_theory/sprague.html). 
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](http://www.squaring.net/sq/ss/spss/spss.html) who 
related the problem to electrical networks (graphs). 

Another post about squared squares is 
[link](https://mathoverflow.net/questions/145356/whats-the-best-way-to-characterise-the-distribution-of-prime-elements-in-simple) 
it has some related numerical data.