HackerThe hardest problem on the hardest test
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Further reading 1
Caption notes
For a tetrahedron with random vertices on a sphere, draw central lines and use fair flips to choose endpoints. The construction preserves uniform sampling while turning containment of the center into countable sign choices.
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That first line passes through the sphere at two points, so flip a coin to decide which of those two points is going to be p1.So one and only one of these eight equally likely outcomes gives us a tetrahedron that contains the center.