White the number of techniques for numerically approximating steady-state heat transfer problems continues to grow ( e.g., domain methods and variants such us finite differences and finite elements, boundary integral methods, and collocation methods), there still is a need for a procedure that develops exact solutions to simplified domain configurations to provide a family of test problems for use in evaluating the merits of a particular technique. In this paper an analytic solution to a family of mixed boundary value problems of the two-dimensional Laplace equation over the unit square is developed. Associated with this family of solutions is an error bound based on a lemma by Hopf. The mathematical development enables the numerical analyst to test the accuracy of other numerical techniques and evaluate the merits of a particular new development or variant of the base method.