The ground-state energies of systems containing up to 12 π + 's in a spatial volume V ∼ ( 2.5     fm ) 3 are computed in dynamical, mixed-action lattice QCD at a lattice spacing of ∼ 0.125     fm for four different values of the light-quark masses. Clean signals are seen for each ground-state, allowing for a precise extraction of both the π + π + scattering length and π + π + π + interaction from a correlated analysis of systems containing different numbers of π + 's. This extraction of the π + π + scattering length is consistent with that from the π + π + system alone. The large number of systems studied here significantly strengthens the arguments presented in our earlier work and unambiguously demonstrates the presence of a low energy π + π + π + interaction. The equation of state of a π + gas is investigated using our numerical results and the density dependence of the isospin chemical potential for these systems agrees well with the theoretical expectations of leading order chiral perturbation theory. The chemical potential is found to receive a substantial contribution from the π + π + π + interaction at the lighter pion masses. An important technical aspect of this work is the demonstration of the necessity of performing propagator contractions in greater than double precision to extract the correct results.