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After an encounter with the Master, an airline stewardess named Tegan Jovanka becomes an unwitting stowaway aboard the TARDIS as it travels to the planet Logopolis. There, the Doctor discovers that the Master's interference with the Logopolitans' advanced mathematics has unleashed a wave of entropy which threatens to consume the entire universe.
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The Doctor and Adric learn from the wizened Keeper of Traken that a great evil has come to his planet in the form of a Melkur - a calcified statue. The Keeper of Traken is nearing the end of his reign and seeks the Doctor's help in preventing the evil from taking control of the bioelectronic source that is the keystone of the Traken Union's civilisation.
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On the planet Alzarius in the alternate universe of E-Space, the Doctor, Romana and K9 find survivors of a crashed starliner, strange creatures lurking in the marshes and a genetic riddle which stretches back centuries.
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On the lush planet Tigella, two opposing factions are divided over the usage of the Dodecahedron, an ancient and powerful artefact which provides the entire planet's energy. The Doctor is summoned to arbitrate the conflict, but the power-crazed Meglos intercepts the call and impersonates him in order to steal the Dodecahedron.
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The newly-regenerated Doctor helps UNIT battle a sentient robot being manipulated by a corrupt scientific organization.
