The high-amplitude magnetic anomalies observed by the Mars Global Surveyor imply the presence of a large intensity of magnetization in the Martian crust. We investigate the mathematical question of determining the distribution of magnetization that has the smallest possible intensity, without any assumptions about the direction of magnetization. The greatest lower bound on intensity found in this way depends on an assumed layer thickness. An analytical expression is discovered for the optimal magnetization, and numerical methods are described for solving the equations that determine the distribution. Some relatively small scale numerical calculations illustrate the theory. These calculations enable us to conclude, for example, that if the magnetization of Mars is confined to a 50-km thick layer, it must be magnetized with an intensity of at least 4.76 A/m.