The sum of cubes

Here is a beautiful geometric interpretation of two important sums:


\begin{eqnarray*} 
1 + 2 + 3 + 4 + \ldots + n & = & \frac{n(n + 1)}{2} \\ 
1^3 + 2^3 + 3^3 + 4^3 + \ldots + n^3 & = & \left[\frac{n(n + 1)}{2}\right]^2 
\end{eqnarray*}

To see the first sum, the sum of the arithmetic series, compute the length of the side of the figure below in two ways: first, by walking along one edge and secondly, by walking from the lower left to upper right corner and only counting the horizontal distance.

The second sum, the sum of the cubes, is given by computing the area of the region in two ways: by adding the areas of all the smaller squares and by squaring the length of a side.

This figure appears in Knuth's The Art of Computer Programming, Volume I on page 19 where it is attributed to R.W. Floyd.