Is it possible for two different $n$-element sets, each of which consists of $n$ unique positive integers (they can appear in both sets, though) to have the same sum when the squares of their elements are added?


Edit: For obvious reasons, I'm not considering the case $n=1$.



Question on Sums of Squares
Yes. One way to see this is that there are more $n$-element subsets with terms up to $N$ than there are possible sums of squares, giving an answer by the pigeonhole principle.


A more beautiful answer was given by Prouhet in the 1850's, who exhibited for each $n$
an explicitly-defined pair of sets $A$ and $B$ of size $2^n$ such that 
$$\sum\_{a\in A}a^k=\sum\_{b\in B}b^k\text{ for each $1\le k\le n$}.
$$


