The set of all Nash equilibria of a non-cooperative game with more than two players is defined by equations and inequalities between nonlinear polynomials, which makes it challenging to compute. This paper presents an algorithm that computes this set for the simplest game with more than two players with arbitrary (possibly non-generic) payoffs, which has not been done before. We give new elegant formulas for completely mixed equilibria, and compute visual representations of the best-response correspondences and their intersections, which define the Nash equilibrium set. These have been implemented in Python and will be part of a public web-based software for automated equilibrium analysis. For small games, which are often studied in economic models, a complete Nash equilibrium analysis is desirable and should be feasible. This project demonstrates the complexity of this task and offers pathways for extensions to larger games.