TITLE: Intuition behind the formula for $\sum i^{2}$
QUESTION [10 upvotes]: I've been trying to figure out the intuition behind the closed formula:
$$\sum i^{2} = \frac{(n)(n+1)(2n+1)}{6}$$
This is not hard to prove via induction, so I'm not interested in the proof that this is true. Rather, I'm more interested in why somebody would even make this guess for an inductive step. What's so confusing to me is that  $\sum i = \frac{(n)(n+1)}{2}$ makes sense using the Gaussian trick of taking the number line and folding it in half and realizing that $(1+n) = (2+(n-1)) = (3 +(n-2)) ... $ so that you end up with this form.
But, what's the intuition for $i^{2}$? Looking at these two formulas, one realizes very quickly that
$$\sum i^{2} = \sum i * \frac{(2n+2)}{3}$$
But, why is that true intuitively? What's the intuition for this?

REPLY [6 votes]: $$\sum_{k=1}^n k^2 = \frac 13n(n+1)(n+\frac 12)$$

(source: scienceblogs.de) 
Note that I've obtained this from this Mathoverflow question. The original author is Man-Keung Siu.