Maddenation

Spherical geometry

Marilyn published another good puzzle Sunday. In case you missed it, I’ll repeat it here.

Say that you randomly choose three points on a sphere. What are the chances that all of them lie on the same hemisphere?

It was submitted by Josh Wilnewski, of Detroit, MI

DadPuzzles11/07/05 1 comments

Comments

Dad • 11/11/05 10:49 PM:

OK. The answer is…there is a 100% chance that all three points lie on the same hemisphere. It becomes obvious once you realize that the problem didn’t say you have to pick the hemisphere before choosing the points. Once the points are selected, it’s always possible to slice the sphere so that all three points are on one side.

If you pick the hemisphere first, then the chance that three randomly selected points will lie in that hemisphere is 1 in 8.

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