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        riddles >> putnam exam (pure math) >> Nine Points on a Sphere
        
(Message started by: James Fingas on Oct 2^nd, 2002, 6:46am)

Title: Nine Points on a Sphere
Post by James Fingas on Oct 2^nd, 2002, 6:46am

Show how to distribute nine points around the surface of a sphere in such a way that each point is equidistant from its four nearest neighbors.

// modified post to include problem statement at top - willywutang 5/21/2008

The solution to this is quite beautiful. It's like the "trees for Willywutang" question, where you just sort of stumble onto the solution. Here's a hint:

The ratio of point-distance to sphere radius is 4*sqrt(6) : 7 + sqrt(2) ... ;D

I'd also like to say that I have found distributions for the following numbers of points on a sphere (each equidistant from 4 neighbours):

6, 8, 10, 12, ...
9, 15, 21, ...

There seems to be no possible solution for 5 or 7 points, and I'm working on 11...

There are also a few distributions for 5 neighbours, but it's a MUCH harder problem in general.