With the merciless Melkur in control of The Source, the Doctor and his allies have only a slim chance of deposing him, which also puts the peace of the Traken Union in severe jeopardy.

With the threat of doom ever-present, the Doctor and Adric press on for Logopolis, where the muttered mathematical incantations of its people can completely overhaul the TARDIS. En route they discover an accidental yet demanding passenger on board while, all the while, the Master secretively journeys with them to ensnare the Doctor in a diabolical trap.