Magnetic domain walls can move when driven by an applied magnetic field. This motion is important for numerous devices, including magnetic recording read/write heads, transformers, and magnetic sensors. A magnetic film with a sawtooth profile localizes magnetic domain walls in discrete positions at the narrowest parts of the film. We propose a controllable way to move these domain walls between these discrete locations by applying magnetic-field pulses. A similarly patterned magnetic film attached to a larger magnetic element at one end of the film can operate as an XOR logic gate.