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I know that

$$1^2+2^2+3^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$$

and I can prove it using the Principle of Mathematical Induction. Now, I am trying to gain a physical explanation of why it is true, but am having trouble.

I am assuming that such an understanding is valuable. If you think this is a waste of time, please let me know why you think so.

I tried to visualize the sum of squares using blocks and see it that would contain $\frac{1}{6}$ the number of blocks as a large cube of blocks with dimension $n\times(n+1)\times(2n+1)$. Using this method, I noticed that

$$1^2+2^2+3^2+\cdots+n^2=n\cdot1+(n-1)\cdot3+(n-2)\cdot5+\cdots+1\cdot(2n-1)$$

but that didn't help me understand the original equation.

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    $\begingroup$ Why is the induction proof insufficient to demonstrate why it is true? $\endgroup$ – Emily Nov 28 '12 at 17:48
  • $\begingroup$ Good point. I revised the question to say I am looking for a physical explanation of why it is true. $\endgroup$ – Andrew Liu Nov 28 '12 at 17:51
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    $\begingroup$ Look at the beautiful color slide in this presentation (I haven't looked at the references below to know if they are the same): math.berkeley.edu/~rbayer/09su-55/handouts/ProofByPicture.pdf $\endgroup$ – Amzoti Nov 28 '12 at 18:32
  • $\begingroup$ Also, see Theorem 7 - very nice here: www.ejpam.com/index.php/ejpam/article/download/546/96 $\endgroup$ – Amzoti Nov 28 '12 at 18:43
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See my question here. In particular, follow the link to the pictures of wooden blocks.

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