Sum of Squares

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

S_n = 1^2 + 2^2 + 3^2 + \ldots + n^2 = \frac{n(n+1)(2n + 1)}{6}.

Sum of Squares

The sum of the first 6 cubes represented as a solid.


Imagine the sum S_n  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 n^2 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  n, n + 1, and 2n + 1. 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.

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