The Doctor takes Adric and a young air hostess named Tegan Jovanka, who has come aboard the TARDIS by accident, to the planet Logopolis, home of a race of mathematicians whose help he hopes to enlist in reconfiguring the outer shell of the TARDIS.

The mysterious, wraith-like Watcher brings Nyssa from Traken to join them and warns of impending danger - something that is borne out as the Master arrives and kills a number of the Logopolitans.
The Logopolitans' lead
er, the Monitor, reveals that the universe passed its normal point of heat death long ago and has been preserved only by his people's calculations, which - by way of a signal beamed from a perfect copy of the Pharos Project radio telescope on Earth - have kept open numerous CVEs through which the excess entropy can drain.

This process has now been halted by the Master's interference, and the Doctor is forced to join forces with his arch-enemy in order to save the universe.

Their plan is to use the real Pharos Project to transmit a copy of the Logopolitan program and thus keep open the CVEs, but the Master seizes the opportunity to blackmail the peoples of the universe by threatening them with destruction unless they agree to his demands.

In foiling this scheme the Doctor falls from the gantry of the radio telescope. As he lies injured on the ground the Watcher appears again and merges with the Doctor as he regenerates.