Alfred is a guy who really likes solid shapes. He also really likes to keep his money in his wallet.
What makes him the happiest? Getting a boxful of shapes at the Cheap Solids Store!
The store has a new promotion: Get a sphere-shaped container for $20 and fill it up with tetrahedrons from the Tetrahedron Bin™. Each tetrahedron must have a corner on the center of the sphere. As long as nothing is sticking out, you may take the whole sphere home.
The Tetrahedron Bin™ in the corner is full of these funny things:
Each tetrahedron in the bin has side length 1, and the sphere has radius 1.
As Alfred is a very stingy guy, he wants to know in advance how many tetrahedrons he can get for his $20.
TL;DR: Given a sphere of radius 1 and tetrahedrons of side length 1, what is the maximum number of tetrahedrons you can pack into the sphere, such that each tetrahedron has a vertex on the center?
Note: Notice there is no lateral-thinking tag on this one. That means you are not allowed to deform the tetrahedrons or the sphere, or feed them into a shrinking machine, or give them to an alien. Everything is subject to the normal laws of physics.
Edit: I am aware that this is an open problem, with no proof found as of now, hence the title, "The two million dollar question," given by my math professor. I just thought the puzzlng community might want to see this and think about it over lunch.