A flow-Mach-number-induced hysteresis phenomenon, in the shock-on-shock interaction of conical shock waves, is investigated numerically, by solving the Euler equations, using a W-modification of the non-stationary Godunov method with second-order accuracy both in space and time. The investigation reveals a multi-path hysteresis loop. It is shown that there are flow Mach number ranges in which three different wave configurations can be obtained for identical flow conditions. This study complements an earlier study by Ben-Dor et al. (2001) in which an angle-of-incidence-induced hysteresis was investigated both numerically and experimentally over a similar geometry. Based on the experimental findings of Ben-Dor et al.'s (2001) study, it is hypothesized that, in fact, four different wave configurations, three inviscid and one viscous, can be obtained for identical flow conditions. Since the geometry under investigation resembles supersonic intakes, this finding is relevant to their performance in supersonic/hypersonic flight.