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.