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

This question already has an answer here:

  • Proving Nicomachus's theorem without induction

    20 answers

Solutions Collecting From Web of "Intuitive/Visual proof that $(1+2+\cdots+n)^2=1^3+2^3+\cdots+n^3$"

enter image description here

I think this image is due to Anders Kaseorg.

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

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

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