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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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Why is the induction proof insufficient to demonstrate why it is true? –  Arkamis Nov 28 '12 at 17:48
    
Good point. I revised the question to say I am looking for a physical explanation of why it is true. –  Andrew Liu Nov 28 '12 at 17:51
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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 –  Amzoti Nov 28 '12 at 18:32
    
Also, see Theorem 7 - very nice here: www.ejpam.com/index.php/ejpam/article/download/546/96 –  Amzoti Nov 28 '12 at 18:43
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1 Answer

up vote 3 down vote accepted

See my question here. In particular, follow the link to the pictures of wooden blocks.

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Thanks! For future reference, here is how I found the picture: (1) Go to mathoverflow.net/questions/8846/proofs-without-words/8847#8847 (2) Search for "Man-Keung Siu" –  Andrew Liu Nov 28 '12 at 17:57
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@AndrewLiu I've seen that picture, but I prefer the one at ckrao.wordpress.com/2012/03/14/… because there is no halving, and six whole pieces are used rather than three with one of them shaved. –  alex.jordan Nov 28 '12 at 18:03
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