A central goal of physics education is to teach problem-solving competency, but the description of the nature of this competency is somehwat fragmentary and implicit in the literature. The present article uses recent historical scholarship on Arnold Sommerfeld and Enrico Fermi to identify and characterize two positions on the nature of physics problem-solving competency. The first, Sommerfeld’s, is a “theory first, phenomenon second” approach. Here, the relevant problems originate in one of the theories of physics and the goal of the problem-solver is to make a mathematical analysis of the relevant equation(s) and then give a qualitative analysis of the phenomenon that arise from these mathematical results. Fermi’s position is a “phenomenon first, theory second” approach, where the starting point is a physical phenomenon that is analyzed and then brought into the realm of a physics theory. The two positions are illustrated with solutions to two problems and it is shown that the two positions are reflected in problem collections of university educations in physics.