Professor Sylvest is terrified by his twin sons, Romulus and Remus -- mathematical geniuses with the power to change reality on a massive scale, they don't have the maturity to accept the consequences of their actions. Sylvest and his wife leave the children alone for the night, and as the twins play at equations, a stranger who identifies himself as Professor Edgeworth materializes in their room, claiming that he has come to visit their father. But while shaking hands with the twins he administers a toxin which gives them selective amnesia. Satisfied that they don't remember where they are, he transports them to a freighter in deep space, where two birdlike aliens, Drak and Noma, are waiting for him. Romulus and Remus are locked in the crew's quarters, where, despite their amnesia, they conclude that they are prisoners and set about trying to send out a distress signal. Edgeworth, meanwhile, telepathically communicates with the sluglike Mestor, who orders him to take the twins to the safe house on Titan Three and wait for further instructions.

Peri is frightened and confused by the Doctor's regeneration, and his new persona is not reassuring; brash, egotistical and arrogant, he dismisses his past incarnation as feckless and effete, and abandons his cricketing outfit for a garish, clashing patchwork coat. While searching for his new clothing he suffers a claustrophobic fit which soon passes, but dismisses it as an insignificant side-effect of his regeneration; the attacks will soon pass. Or so he claims. While setting the co-ordinates for leisure world Presta-95, he suddenly accuses Peri of being an alien spy and tries to strangle her to death. She drives him off by showing him his furious, twisted face in a mirror, but when he snaps out of it he has no memory of what he's done. Peri's obvious terror and her recoiling at his approach soon confirm, however, that something has gone horribly wrong with his regeneration. Overcome with remorse, the Doctor decides to atone by becoming a hermit -- on the desolate world of Titan Three.

Sylvest returns home to find his sons missing -- and a dusting of zanium powder on their bedroom floor, evidence an extra-terrestrial kidnapping. He reports to the authorities, who send a squadron led by Lieutenant Hugo Lang out to search for any unauthorised vessels within range. Hugo finds an unregistered freighter transmitting an irregular signal -- the twins' attempt to attract attention -- but the freighter goes into warp drive, and when Hugo tries to follow his entire squadron is destroyed by forces unknown. The authorities reluctantly decide not to send any further ships out until they've determined who or what is responsible. Edgeworth and his allies reach the safe house on Titan Three, where the children fall asleep, exhausted. Edgeworth reports to Mestor, who orders him to remain where he is while Mestor waits to see what Earth's response will be to Mestor's destruction of their vessels. In the meantime, Edgeworth must set the children to work on the first equations, and ensure that there are no survivors from Hugo's squadron.

Hugo passes out before he can shoot the Doctor, and the terrified Doctor refuses to help him any further. Peri searches Hugo and finds his i.d. badge, proving him to be a policeman, and the Doctor reluctantly agrees to heal him once Peri has removed and hidden the power pack from his gun. Once he has finished tending to Hugo's wounds, the Doctor's personality changes once again; charged with the zestful energy that comes from having helped another living soul, he decides to investigate Hugo's earlier, delirious comment about missing children. He scans the surface of Titan Three with the TARDIS scanner and spots an artificial dome, and sets off to investigate, a reluctant Peri in tow.

Edgeworth sends Noma and Drak to ensure that their ship is spaceworthy. The surface of Titan Three is now subject to mild radioactive fallout from the energy beam which destroyed Hugo's squadron; this is interfering with the transmat and has made surface travel hazardous, so Noma and Drak go to the landing pad via the dome's service tunnels. Romulus and Remus have been put to work writing down their equations manually, as their attempt to send a distress call has shown they can't be trusted with a computer. When they throw a temper tantrum and refuse to work any further, Mestor appears to them and threatens to remove their minds from their bodies to use as he wishes. Cowed, the twins return to work, and complete an energy equation with the power to move an entire sun.

The Doctor and Peri find and enter the dome's service ducting, where the Doctor boldly strides forward until Peri points out the danger he's putting himself in. At that moment Drak and Noma arrive and hold them at gunpoint, and the Doctor flies into a blind panic and blames Peri for leading him here against his will. Noma and Drak take them back to the dome, where the Doctor recovers, apologises to Peri, and tries to convince the hostile Edgeworth that they're simple pilgrims. But as he talks, he recognizes "Edgeworth" as an old friend -- retired Time Lord Azmael, the master of the planet Jaconda. Azmael is forced to concede that this stranger must be a regenerated Doctor... but when Romulus and Remus appear the Doctor realizes that his old friend is responsible for kidnapping them.

Back at the TARDIS, Hugo recovers to find himself abandoned in an unknown ship with no power pack for his gun. Searching for a way out or for the inhabitants of the ship, he finds the wardrobe room and changes out of his burnt and torn policeman's uniform into a glittery tunic which at least fits him... and which turns out to have his power pack in its pocket. Re-armed, he returns to the console room to search for a way out.

The Doctor becomes convinced that Azmael is in some sort of trouble, and Azmael confirms that Jaconda has a new master but refuses to say any more. Neither notice when Noma, worried about the direction their conversation is taking, enters an alcove in the side of the dome and sets a timer to count down... Azmael turns down the Doctor's offer of help, and to ensure that he doesn't interfere, scrambles the dome's locking mechanism and sets the transmat to randomise once he and his companions have departed. It has become clear by now that Earth will be sending no more ships after them, so Azmael, Noma, Drak and the twins transmat to their ship and leave for Jaconda, leaving the Doctor and Peri trapped in the dome.

The Doctor appears, out of phase with the TARDIS timeline, and must use the console to stabilise himself. He blames Peri for the accident as he was forced to guess at his co-ordinates due to her stopped watch. Hugo demands to be taken to the children, and the Doctor, theorising that Azmael has taken them to his adopted world, sets the co-ordinates for Jaconda. But Jaconda the beautiful is now a wasteland, its lush forests stripped of foliage and bark and criss-crossed with hardened slime trails. The Doctor recalls a legend from Jacondan mythology; giant gastropods, half-humanoid and half-slug, which once devastated the planet. It appears that they weren't entirely mythological after all. Horrified and frightened, the Doctor refuses to help further until Hugo holds him at gunpoint, demanding to be taken to the Jacondan palace. The Doctor reluctantly materializes in the tunnels beneath the palace, where he changes his mind once again and insists upon accompanying Hugo. Hugo warns the Doctor that if he becomes unstable once again, Hugo will have no choice but to shoot him.

As Azmael's ship approaches Jaconda, Noma informs the horrified Time Lord that he destroyed the safe house, killing both the Doctor and Peri. Remus and Romulus blame Azmael for their deaths, and childishly provoke him until he snaps and threatens to kill them both. Even as they speak, Mestor's reign of terror continues, as he uses a telekinetically-generated embolism to execute a porter who was caught stealing food from the royal hatcheries to feed his starving family.

The Doctor finds a mural on the tunnel walls depicting the myth of the gastropods; obviously they were real, and some dormant eggs survived. The Doctor hears gastropods approaching and orders Peri to turn off her torch. Once the gastropods have passed Hugo emerges from hiding, only to step into the gastropods' quickly hardening slime trail. He tries to melt himself free using his gun at its lowest setting, but the Doctor quickly grows impatient and blames Peri for switching the torch off so Hugo couldn't see where he was going. He abandons Peri and Hugo and sets off on his own to find and confront Azmael.

Azmael takes the twins to the palace laboratories, where they will complete their work; the laboratories look out over the incubator room where the next generation of gastropod eggs awaits hatching. Mestor arrives to study the twins in person, and orders Azmael to tell them the purpose of their work; once they realize they are working for a benevolent purpose they may no longer be so intransigent. Azmael demands that Mestor stop monitoring his thoughts and remove his lackeys Noma and the Chamberlain, so Azmael can concentrate on his work. Mestor suspects Azmael of plotting treason, but reluctantly concedes to his demands. Drak, who may still remain loyal to Azmael, is allowed to stay. As soon as they're alone, Azmael explains to the twins that he intends to bring the system's two outer planets into orbit around Jaconda, so they can be used to grow food for the Jacondans who are starving due to the gastropods' assault. The twins must complete the mathematical equations which will stabilise the planets' new orbits.

Noma arrives with the palace guards to take the Doctor to an audience with Mestor. There, he's pleased to see that Peri is still alive, and offers to help Azmael for the sake of the Jacondans. The Chamberlain feels that the Doctor cannot be trusted, but Mestor concludes that he has nothing to fear from this mad, arrogant Time Lord, and sends him and Peri back to the laboratory. He is, however, intrigued by the Doctor's arrival, and orders the Chamberlain to find the TARDIS and bring it to him.

Back in the laboratory, the twins complete their work while the Doctor puzzles out the contradictions in Mestor's plan. Apparently he intends to use Azmael's knowledge of time travel to shift the two outer planets into a different time phase, thus enabling them to occupy the same space as Jaconda. But the Doctor points out what Azmael has been too concerned with his adopted people to realize; the outer planets are too small to maintain a stable orbit any closer to their sun. Their orbits are doomed to decay, and the planets will inevitably fall into the sun, causing a catastrophic explosion. Convinced that the answer lies in the hatcheries, the Doctor enters and examines a gastropod egg, only to discover that it is dry, not gelatinous. He tries to cut it open with a laser, but the egg proves impervious to the cutting beam -- and stirs briefly to life, activated by the warmth. The Doctor and Azmael now realize that Mestor intends to blow up the Jacondan sun and scatter his genetically engineered eggs across the galaxy, conquering all known worlds.

The twins delete their equations from the computer; now Mestor must keep them alive. Drak suddenly drops dead to the floor, his mind burnt out; Mestor has been using him all along to spy on their conversation, and now knows that they are aware of his plan. The Doctor and Azmael set off to confront him, while Hugo and Peri take the twins to safety in the TARDIS. On the way they are confronted by Noma and the guards, but Hugo fights them off and takes Noma prisoner.

The Doctor and Azmael march into the throne room, where the Doctor tries to kill Mestor by throwing a beaker of acid at him. Mestor destroys the beaker with a single thought, and prepares to punish the Doctor by taking over his mind and body -- but first he possesses Azmael just to show the Doctor that he can. But Azmael proves stronger than Mestor had expected, and holds Mestor prisoner within his mind while the Doctor destroys the gastropod's body with a second beaker of acid. The Doctor insists that they mind-link to drive Mestor out, but Azmael knows that the Doctor is too unstable to hold him, and instead deliberately triggers his own regeneration. Since he has no further regenerations left this results in his death, and Mestor is killed along with him. Azmael hands the royal ring of Jaconda to the Doctor and thanks him for his friendship, and the Doctor mourns quietly as his old friend dies.