<s>[INST] Are two sequences equal if the sums and sums of squares are equal? Are two sequences $(x_i)_{i=1,\ldots,n}$ and $(y_i)_{i=1,\ldots,n}$ equal if $\sum_{i=1}^nx_i=\sum_{i=1}^ny_i$ and $\sum_{i=1}^nx_i^2=\sum_{i=1}^ny_i^2$?
 [/INST] Something interesting. On all $3 \times 3$ magic squares, the sums of the top and bottom rows is the same (of course) but, also, the sums of the squares of the top and bottom rows is the same. For example
\begin{array}{|c|c|c|}
\hline
   8 & 3 & 4 \\
\hline
   1 & 5 & 9 \\
\hline
   6 & 7 & 2 \\
\hline
\end{array}
$8 + 3 + 4 = 6 + 7 + 2 = 15$
$8^2 + 3^2 + 4^2 = 6^2 + 7^2 + 2^2 = 89$
Also
$8 + 1 + 6 = 4 + 9 + 2 = 15$
$8^2 + 1^2 + 6^2 = 4^2 + 9^2 + 2^2 = 101$
This seems to work for some families of larger order magic squares, but I haven't been able to prove it.
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