Conjecture: the minimum number of obtuse angles over all 6-point sets in
3 dimensions is 2.
This has been established experimentally to a high degree of probability
by creating 28 million (so far) random 6-point sets and counting the
obtuse angles in each set. I do not know how to prove it, and I see no
way to find a counterexample except by trying additional millions of
point sets, which are not likely to turn up any.
I think this is a new problem, and I have other similar conjectures.
Steve Gray