Sums-of-squares formulas over the integers have been studied extensively using their equivalence to consistently signed intercalate matrices. This representation, combined with combinatorial arguments, has been used to produce sums-of-squares formulas and to show that formulas of certain types cannot exist. In this paper, we introduce an algorithm that produces consistently signed intercalate matrices, or proves their nonexistence, extending previous methods beyond what is computationally feasible by hand.