A recent thread on Hacker News that I started with a flippant comment turned into a little mathematical puzzle. What's the sum of the square of the first n whole numbers? It's well known that the sum of the first n whole numbers is n(n+1)/2 . But what's the value of sum(i=1..n) n^2 ? (I'll call this number S for the remainder of this post). It turns out that it's easy to prove that S = n^3/3 + n^2/2 + n/6 by induction. But how is the formula derived? To help with reasoning here's a little picture of the first 4 squares stacked up one on top of the other: If we fill in the blank squares to make a rectangle we have the basis of a derivation of the formula: Looking at the formerly blank squares (that I've numbered to assist with the thinking) we can see that the columns have 1 then 1+2 then 1+2+3 and finally 1+2+3+4 squares. Thus the columns are sums of consecutive whole numbers (for which we already have the n(n+1)/2 formula. Now th