A difficult question on representing a square by a sum of 4 (or 5 ) squares The question above is not about the decomposition of a square $$m^2 = a^2 + b^2 + c^2 + d^2 $$ as a sum of 4 or more squares. We know it is always possible and we also have algorithms to do that.  
My question is about going backward, that is if we only have a partial decomposition of $$m^2$$ as a sum of 4 (or 5) squares , can we rebuild $$m^2$$ If we have say 3 out of 4 or 4 out of 5 squares, can we reconstruct the square?

Not uniquely.
Since for instance, given $a=3, b=5, c=11$
$18^2= 3^2+5^2+11^2+13^2$
$78^2= 3^2+5^2+11^2+77^2$
As we can write $(m+d)(m-d) = a^2+b^2+c^2$
there are as many solutions for $m,d$ as there factorisations of $a^2+b^2+c^2$.
