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Abstract: In this talk I will quickly survey some results about energy minimizing configurations of points on the sphere, and then explain how to use the computer to solve Thomson's 5-electron problem by showing that the triangular bi-pyramid is the configuration of electrons on the sphere which minimizes the electrostatic potential. (Yes.) I'll also explain how to use the computer to rigorously establish the more general result that the triangular bi-pyramid is the energy minimizer with respect to all power law potentials up to about exponent 15.05, at which point the answer changes to a family of pyramids with square base. The team of Melnyk, Knopf, and Smith discovered this phase transition in the late 1970s. As in the other talks, I'll demonstrate the software I wrote to help solve these problems. I'll also discuss the advantages and shortcomings of a computer-assisted approach like this.