Possible Duplicate:
Intuitive explanation for the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$
There is the interesting identity:
$$\left ( \sum_{i=1}^n i \right )^2 = \left ( \sum_{i=1}^n i^3 \right ) $$ which holds for any positive integer $n$.
I know several was of proving this (finite differences, induction, algebraic tricks etc..), but even so I still find it "weird" that it is even true.
Is there a very nice intuitive way to prove this using some kind of combinatorial argument? (Like the why the sum of the volume of the first $n$ cubes should be the area of ... not sure here?)
If you have any pretty different proof that could be enlightening I would love to see it.
Thanks a lot!
