There is a lovely formula for the sum of the first square integers. Namely

Imagine the sum as the volume of the pyramid of 1 x 1 x 1 cubes with one cube on the top layer, 4 on the next, 9 on the next and so on up to cubes on the bottom layer as seen in the figure above. From the above expression, we see that the sum is one sixth the volume of a box with dimensions , , and . So it is at least conceivable that six of these pyramids could be packed into a rectangular volume of that size. And as the pictures indicate below, there is such an arrangement.

With out new CNC router, we thought it would be fun to build a physical model to demonstrate this formula.

Read about my first experience with CNC routing the sum of squares here.