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There is a regular tetrahedron with edge ledge of $2$ units. Your task is to put as many points within the volume occupied by the tetrahedron.

But there is a condition: there has to be at least $1$ unit distance between each points.

What is the maximum number of points you can put within the tetrahedron with the condition above?

Note that edges and vertices of tetrahedron are considered within the volume.

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I'd be surprised if you can do anything better than

choosing the points at the vertices of the tetrahedron, and halfway each edge

resulting in

10 points.

(I don't have a proof, it's just mathematical intuition, based on Sphere packing.)

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