With his ship's Cloister Bell tolling of impending doom, the Doctor decides to repair the TARDIS's chameleon circuit and so goes to Earth to obtain the exact external dimensions of a real police box which can be used correctly to reconfigure the ship. While he is there, an air hostess, Tegan Jovanka, en route to her new job, enters the TARDIS thinking it to be a real police box.

The Master has devised a complex trap for the Doctor and kills Tegan's aunt Vanessa. The Doctor is apparently warned of an impending ordeal by a mysterious figure in white, the Watcher, who appears to him several times. The Doctor goes to Logopolis, a planet of mathematicians, but the Master in his TARDIS has hitched a ride in the Doctor's TARDIS and now starts killing the Logopolitans. The Watcher appears again, having brought Nyssa from Traken to Logopolis.

The Master's meddling causes the Logopolitans' complex block transfer computations to go wrong, culminating in his use of a sound suppressor to prevent the calculations from being spoken at all - the means by which the Logopolitans run their programs.

The Monitor on Logopolis explains that it is his people's calculations that are keeping the universe from destruction. It passed its normal heat death years ago, and the Logopolitans' programs are holding open numerous CVEs through which the excess entropy from our universe is draining. With the Master's interference, the CVEs will close and the Universe will be destroyed. Watching on the TARDIS scanner, Nyssa sees Traken destroyed as the entropy begins to take effect.

The Doctor and the Master join forces to prevent the destruction of the universe and hurry to the Pharos Project in Cambridge on Earth with a copy of the Logopolitan program. They intend to use a radio telescope to transmit it and thus keep the CVEs open. The Master sees an opportunity to seize power and promptly blackmails the peoples of the Universe with destruction unless they agree to his demands. The Doctor tries to prevent the Master from carrying out his threat and succeeds, only to fall from the gantry of the radio telescope.

As he lies under the telescope, Adric, Nyssa and Tegan rush to join him. The Watcher appears once more and merges with the Doctor as he regenerates into a younger form.

_{Synopsis from Doctor Who: The Fourth Doctor Handbook by David J. Howe, Mark Stammers and Stephen James Walker, reprinted with permission; further reproduction is not permitted. Available from Telos}