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The relation

$$ \sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k\right)^2 $$

baffled me when I first found out (i.e. yesterday on a train trip). Writing an inductive proof is easy and I know that there is a recursive way to obtain a general formula for

$$ \sum_{k=1}^n k^j $$

for any $j \in \mathbb{N}$, but I feel like this relationship between the sum of the first n cubes and the sum of the first n integers should have some nice geometrical proof. The closest thing I found was the first answer to this question, but I still don't find it intuitively clear. Maybe I am asking for too much here.

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    $\begingroup$ math.stackexchange.com/a/728239/119285 $\endgroup$ – b00n heT Nov 11 '16 at 10:44
  • $\begingroup$ Hmm. I see how this picture would prove it if we know that it can always be drawn, but I am still confused as to why it should always be possible to fill an $n \times n$ grid like that. $\endgroup$ – Bib-lost Nov 11 '16 at 12:45
  • $\begingroup$ It's not an $n\times n$ grid, it's a $(1+2+3+\dots+n) \times (1+2+3+\dots+n)$ grid. $\endgroup$ – Hans Lundmark Nov 11 '16 at 12:55