6 points in R3 conjecture

[ou, janis]

Take 6 points in 3D space in general position: no 3 in a line, no 3
forming a right angle, no 4 in a plane. Draw all 15 lines between them.
This will form 20 distinct triangles. (For each of the six points there
are ten pairs of other points, but the apparent 60 triangles are each
counted three times.) Each triangle can have at most one obtuse angle,
so the maximum number of obtuse angles in the whole figure, over all
such point sets, is 20.

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 http://www.usedconecrusher.com

[Mary Krimmel]

This topic sounds interesting but I do not entirely understand it. How do know that the maximum is 20? (Intuition suggests that it is less.} Is there an example of a set with 20? Is it possible to find some helpful background in R2? Do you have any results for 4-point or 5-point sets?

Thank you (or anyone else) for some response. Your post has inspired me to investigate.

[Mary Krimmel]

I now see that 20 is indeed the maximum, that an example is easy to find, that my intuition was wrong. Also I see that there are at least two mistakes in my previous post (a typo and an omission); nevertheless,I think the meaning is clear.

Is it possible that a system of inequalities could solve the problem algebraically or contribute to its solution?

I assume that you found each number in [2,20] in the count of obtuse angles in the millions of 6-point sets you investigated. Have you any record of their distribution?

[Mary Krimmel]

> I now see that 20 is indeed the maximum, that an
> example is easy to find, that my intuition was wrong.

Whooops! This time my mistake is serious. I do not have an example of 20 obtuse angles. Again I go back to intuition and suppose that the real maximum is lower than 20. If you did observe results in addition to looking for a minimum, what is the greatest number you have found?

Thanks again.