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.

With the universe collapsing, the Doctor forms an uneasy alliance with the Master to save it. In case all does not go well, the Watcher takes the Doctor's companions to a safe place, but his plans for them don't stop there. The one hope for all lies back on Earth.