The paper reviews the controversy on the shape of the fcc-based phase diagram near the triple point among the disordered (A1), L1z and L1o phases. The discrepancy was originally between the cluster variation method (CVM) results and Monte Carlo (MC) simulation results, but the contro­ versy developed into how to interpret MC results. The paper examines historically how the controversy started, how it developed, and how it is being solved. The key point is that the MC treatment allows antiphase boundaries (APB) to appear in the simulation, and APB's make the interpretation of the results difficult. The CVM is straightforward since it presupposes the stability of the ordered phase being treated, but can miss important physics unless it is used cautiously.