Intuitive/Visual proof that $(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$

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I think this image is due to Anders Kaseorg.

\sum_{i=1}^n i^3 – \sum_{i=1}^{n-1} i^3 = n^3

\left(\sum_{i=1}^n i\right)^2-\left(\sum_{i=1}^{n-1} i\right)^2&=\left(\sum_{i=1}^n i-\sum_{i=1}^{n-1} i\right)\left(\sum_{i=1}^n i+\sum_{i=1}^{n-1} i\right)\\
&=n\left(\sum_{i=1}^n i + \sum_{i=1}^{n-1} (n-i)\right)\\
&=n\left(n + \sum_{i=1}^{n-1} n\right)\\
&=n\left(n + n(n-1)\right)\\
&=n\cdot n^2 = n^3

(This is to show that it makes mathematically intuitive sense)