Geometrical interpretation of $(\sum_{k=1}^n k)^2=\sum_{k=1}^n k^3$

summation induction

Solution 1:

The clearest proof I've seen is this one, which comes from here: you just have to stare at it for a few seconds to see how it works. (It's a variant of the other proofs, of course, but actually has cubes in it.)

Animation

Solution 2:

There is this picture. Here, they represent $x^3$ as $x$ squares of side length $x$. The big square is the sum of all numbers up to $x$.

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