Tetrahedral Pyramid

Its so quiet in this section so I thought I should put in a fun little problem that I have been working on lately.

Its a math problem really but you need a computer program to calculate the solutions. Part of the challenge is to find a fast enough algorithm.

The problem is to find two pyramids that consist of spheres so that when you combine all the spheres you can build one larger pyramid using all the spheres. Got it?

A tetrahedral pyramid of spheres has one bottom and three sides. So the first pyramid consists of four spheres (if you don't count one sphere as a pyramid), the next consists of ten spheres, the next has twenty, etc.

So the first solution is 10 + 10 = 20

Can anyone give me the 100:th solution? Or perhaps the 500:th solution?

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100:th solution:

56410524040 + 792107977164 = 848518501204

500:th solution:

174481731261016816 + 218675214798554020 = 393156946059570836

 

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I was actuallly having a bit of a problem with your explaination.

When I was trying it I was adding 1 sphere to every edge, and then adding spheres in the centre. There would have been a mathematical process that bit was easy, but the whole two pyramids making one that was a little offsetting.

 

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