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Intuitive explanation for the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$

How to prove this without using mathematical induction?

$$1^3+2^3+\cdots+n^3 = (1+2+\cdots+n)^2$$

I know how to prove it by induction, is there a different way?

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Frank
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    This question is not a duplicate of the question Arjang linked to - generalizations of an identity and alternative proofs of it are different things. – Zev Chonoles Jun 27 '12 at 09:28
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    @ZevChonoles: But it is a duplicate of both http://math.stackexchange.com/questions/61482/intuitive-explanation-for-the-identity-sum-limits-k-1n-k3-left-sum-l and http://math.stackexchange.com/questions/18548/combinatorial-reasoning-for-the-identity-left-sum-i-1n-i-right-2-l I merged those two together , and perhaps will merge this one since there are useful answers here not yet brought up over there. – Eric Naslund Jun 27 '12 at 12:48

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