Put points in 3D space. What is the maximal number of unit lengths? For four points, it is obvious the maximal number is six unit lengths, using a tetrahedron. With five points, two joined tetrahedra give nine unit lengths.
This Demonstration gives the following best-known solutions up to 20 points.
If you can find missed solutions matching or surpassing the best known maximum values, please send them to the author.