Does that work with any other powers other than ^2 ? Or does it work in other bases?
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The equality doesn't care about digits, so the base doesn't matter. You can also see it because the proof is geometric.
End of conversation
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Another w/o words: n² = (n+1)²-(n-1)²+...+(n+k)²-(n-k)² = (n+1+n-1)(n+1-n+1)+...+(n+k+n-k)(n+k-n+k) = 4n•1+...+4n•k = 4n sum{k} => n = 4sumk
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