This week I learned a wonderful thing: If you assemble spheres to make hexagons of different sizes (up to N spheres on a side), and then stack up those hexagons to make a pyramid, you get exactly the same number of spheres as if you made a cube with N spheres on a side.
I was so delighted by this that I wrote a little computer program to visualize it.
Below you can see the cases where the hexagons contain up to two, three or four spheres on a side, respectively. Next to each picture you see spheres arranged into a cube. In each case (and in fact in every possible case), the number of spheres is the same on the left and on the right.
How cool is that!
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