Summary

The square pyramidal numbers are the sums of the first n squares: 1^2 + 2^2 + ... + n^2 = n (n+1)(2n+1)/6. This puzzle illustrates a nice geometric proof of this sum formula. The original sum is represented by a step pyramid where the (square) levels contain 1^2, 2^2, 3^2, ... , 5^2 blocks. The challenge is to assemble six of these pyramids into a 5 x 6 x 11 box.

Instructions

To build the puzzle, print six copies of the file. The print file is sized in mm, so you'll have to scale by a factor of 10 if you're using MakerWare, which expects the units to be cm. If you want the pieces to have a little help staying together, they print with holes for 3mm (diameter) x 3 mm (height) cylindrical rare earth magnets. I recommend printing the magnet insertion tools to help get the magnets seated in the holes.

There's an instruction sheet in the source files showing which way to insert the magnets in the various holes. I used a little JB Weld epoxy to keep the magnets in the holes. The six pieces can be built into 2 almost-cubes. The magnets will hold the cubes a little bit apart so that you can see the split between the two halves of the box, but you can push the cubes together to make a 'solid' rectangle and they will stay put.

This project is supported by the National Science Foundation under DUE-12-45540. More information on the ``Motivating First-Year Calculus" project is on my webpage.

I learned about this construction from Okay Arik's really cool Wolfram Demonstrations Project demonstration of this proof. I don't know who invented the proof!

Square Pyramidal Numbers Puzzle Manipulative