Rupert Von Thal hacks his way through the undergrowth of the prehistoric jungle and calls out to his companion, Mrs Beatrice Mapp. He warns her not to stray too far alone, but she says she’s perfectly capable of taking care of herself. He reminds her that she’s no longer in the literary salons of Bloomsbury and this is an exceedingly hostile environment, but she says his time would be better spent cutting more firewood for them. As she pulls at a nearby branch, there’s an enormous roar and a gigantic 14-foot high insect charges towards them. Rupert defends them with his machete, but it has no effect until he follows Beatrice’s advice and stabs the beast in the eye. Even then, his machete is snapped by the mantis’s huge pincers. Rupert tells Beatrice he’s explored the most inaccessible corners of the world and experienced the most terrible things, but he’s never seen anything like this before.

The sky is starting to turn mauve, so Beatrice suggests they rejoin the others before night falls. Rupert wonder whether the Doctor was right when he said they’d all gone back in time and Beatrice thinks it’s time they asked him some stiff questions! Beatrice says she always knew his experiment would succeed, from the very first moment they all assembled in her sitting room. It seemed like just a lark at first, but now they want to know how he brought them here and why…

It was some time ago when the Doctor and Nyssa addressed their specially invited audience and thanked Beatrice for letting them use her home. Beatrice said they were all intrigued by what he hoped to achieve, but others in the group were just getting impatient for it to begin. Nyssa asked Professor Quandry to be patient, but he mistook her for a servant and refused to be lectured, even when the Doctor introduced her as his research associate. The group, which included some of the finest minds in the Empire, were here to lend their minds to a unique experiment. Rupert said he was just a humble explorer and was flattered to be invited, but Nyssa admitted they needed 12 people and because they were two short, he’d been invited just to make up the numbers. The final member of the group was to be Brenda, Beatrice’s maid, who arrived just in time with the tea trolley. The group was asked to sit around a table and join hands, making them believe they were to take part in a séance, but the Doctor explained that it was actually an experiment in transference. He gave each of the 12 members a different series of binary numbers which they were to keep repeating out loud, then he asked them to go into a trance.

The experiment began and the room filled with the sound of numbers being repeated out loud in various different sequences. Before long, the sound became a hum of background noise and Nyssa knew instantly that something wasn’t right as Block Transfer Computation didn’t usually work this quickly. The Doctor urged everyone to stop, but then the air itself started to flicker and the room dissolved around them…

Nyssa hears an elderly man’s voice coming from all around her. The voice welcomes her to his City and tells her he’s a very old friend. She asks him what he wants, but the man appears to know her and when he states her name, she gasps in shock and opens her eyes - to find the Doctor watching over her, a look of concern on his face. At first he wonders if she was just having a bad dream, but when she tells him there’s someone out there in the jungle waiting for them, he assumes it must have been an after-echo of the Computations. She assures him she was actually inside a City down in the nearby valley and she’s now certain it wasn’t the TARDIS that drew them here. The Doctor agrees and says he sensed during the séance that someone or something had knocked them off-course. The campfire is getting low and Nyssa wonders where the others have got to. She asks the Doctor if he’s planning to tell Beatrice and Rupert that the experiment went wrong, but he says from their point of view it was a success and he’s given them a glimpse of their prehistoric past.

In the nearby valley, a huge warrior scorpion named Kranlee approaches Madam Teegarna and warns them of new arrivals, who he describes as foul, soft-bodied things. Teegarna is unhappy and says they threaten the equilibrium and disturb their precious Calculations. Kranlee suggests destroying them before they cause further upset, but Teegarna orders him to take a patrol and bring the creatures here.

Rupert and Beatrice return to the camp and place more wood on the fire. They ask the Doctor what he proposes to do about getting them home again, but he admits that it might be a problem. When Rupert asks him why he can’t just do his “fancy magic” again, Nyssa explains about Block Transfer Computation, a way of making real things happen in the world through the power of mathematics. She explains that the Universe is fashioned out of raw probability and the numbers run very deeply into the fabric of reality, so by altering the numbers, they can alter reality. Beatrice can understand the logic, but Rupert is less convinced. Nyssa tells them about the TARDIS being stolen by the street urchin Thomas Brewster and the Doctor says he was trying to use the computations to access the space-time continuum where he hoped to pick up a trace of their ship. The experiment was more successful than he anticipated and somehow the computations actually brought them here, but unfortunately it took the combined concentration of 12 people to transport them and they can’t go back with just the four of them here. The others in the séance who broke the circle out of fear or panic are either still in Bloomsbury or somewhere adrift in time and space. Rupert is shocked when he realises this means they’re stuck here forever!

Madam Teegarna visits Lohkaar, another member of the scorpion Hierarchy, and assures him that Kranlee’s soldiers will not fail in their mission. The older scorpion admits that he’s intrigued by what he’s heard of the new arrivals. Teegarna clearly holds little respect for Lohkaar and accuses him of scuttling off to see His Excellency. She always thought it odd that they chose a member of another species to rule over them and suggests it was only because they believed him to be unique, but they now know this isn’t true as the new arrivals also walk on two legs. Lohkaar warns her against speaking treason and says it isn‘t their job to be curious, but to serve their King and maintain the Calculations. Teegarna says the Hierarchy are stuck in their ways and think nothing will ever change, but she promises it will and says they won’t be ready for it. Lohkaar believes the Calculations will maintain the equilibrium perfectly and they’ll remain in their state of perpetual excellence. He believes Teegarna is insubordinate and he warns her to be mindful of her place, but once he’s left she remarks that she’s very well aware of her position in the Hierarchy.

The Doctor tracks Nyssa down in the jungle after she wandered off to clear her head and admire the view. She seems surprised by the attitude of their Victorian colleagues - she thinks they should be glad he brought them somewhere so beautiful, yet all they want to do is go home. The Doctor tells her they’re scared as they like to control things and Nyssa says she’s glad to be away from that time. She says the Victorians pretend to be civilised, but at the same time they were enslaving and exploiting the rest of their world, making wars and creating inequality. The Doctor asks if she’s picked up any more signals from the City and she says she can still sense it, like a whisper in the dark. She closes her eyes and concentrates, but it’s too faint to make anything out. She thinks this place is a paradise and says it’s a shame it will eventually fill up with the ancestors of Rupert and Beatrice. The Doctor reminds her that some of his best friends have been humans and she admits that she’s never quite understood why.

Deep in the City, the elderly man’s voice calls out to Nyssa again and asks her to open up her mind to him - but then he’s interrupted by the arrival of Lohkaar and the link is broken. He rebukes the scorpion, but Lohkaar reminds him it’s feeding time for his pet. The old man points to the carcass of a mantis and asks Lohkaar to drag it over to the pit, then he cries out with delight when his pet, a gigantic spider, emerges from the shadows. He tells Lohkaar that he’s always been very fond of spiders and he has an affinity with all arachnids. The King is keen to question the strangers and when Lohkaar feigns ignorance, he reveals that he knows bipedal hominids have arrived nearby. He demands that in future he be told of all new developments and reminds Lohkaar that he still rules here, even if he is getting old. He adds that the arrivals aren’t just any creatures, they happen to include two particular individuals - Nyssa and the Doctor!

Rupert is unhappy that their group has split up and says this isn’t a good idea when you’re in the wild. Beatrice says the Doctor and Nyssa haven’t been gone long, but Rupert wonders what’ll happen if they summon up more of their “mumbo jumbo magic” and abandon them here. He returns to the campfire and turns over the crickets that are cooking, saying they’re nearly ready. He remembers Beatrice said she first met the Doctor and Nyssa in a tea-shop near the British Museum, where they were babbling on about the fifth dimension, but before he can find out more, they hear something moving in the bushes. As they look closely, they see eyes blazing in the darkness and they realise they’re surrounded. Suddenly the bushes are pushed apart and an army of giant scorpions emerge from cover and charge towards them. Beatrice grabs torches from the fire and hands one to Rupert, telling him to use it to warn the creatures off. It seems to work, but they know they can’t keep this up forever.

Just then, the Doctor and Nyssa return and shout out a warning to the scorpions, who step back and stop their attack. It’s not clear whether the creatures can understand them, but when the Doctor asks if he can speak to whoever is in charge, all the scorpions can hear are unintelligible and meaningless words. Kranlee wonders what all the gibberish is about and wonders if the man is offering himself as a sacrifice. The Doctor realises he’s not making himself understood and Nyssa says this is the final proof they need that the TARDIS isn’t in this time period, because if it was it would be telepathically translating their words. Rupert suggests they resort to brute force, but Beatrice says they don‘t know how many more scorpions may be hiding in the bushes. The Doctor agrees and tells everyone to put their torches back in the fire, then they put their hands up.

Kranlee sends a message to Madam Teegarna back in the City and tells her they’ve overcome the new arrivals and will be back in the City within the hour. She congratulates him and tells him to hurry as His Excellency is keen to see the creatures. After she switches off, she admits that she’s keen to meet them herself…

Kranlee escorts the prisoners through the jungle and eventually they arrive at the City. He addresses the other scorpions and tells them to look upon the mammalian freaks with disgust. The creatures obey and gawp upon the humans as they’re led into the main entrance. Beatrice is amazed and compares the building to St Paul’s Cathedral, but Rupert reminds her this is nothing more than a giant termite hill and there are probably millions of scorpions living in this nest. Nyssa confirms that this is what she saw in her dream. Beatrice realises they’re being shepherded along like cattle and Rupert worries they’re being led to a slaughterhouse. As they listen to the chanting more closely, they can identify the words as zeros and ones. It sounds like music made up of mathematics and they realise it’s some form of Block Transfer Computation.

Nyssa can’t believe this is really Adric, but the King insists that it’s true and says he remembers her impeccable logic. He says he’s been languishing here in the backwaters of history for years, but Nyssa knows Adric is dead because she watched the freighter burn up in the atmosphere before crashing into prehistoric Earth. The Doctor becomes angry and demands to know what Adric has done to this world, but the old man turns the argument on its head and blames the Doctor for what happened. He asks what happened to Tegan and Nyssa tells him she’s fine and was left back at Heathrow. Beatrice realises Adric holds some sort of grudge against the Doctor and assures him she and Rupert aren’t replacement companions, nor have they been kidnapped. Adric calls to Lohkaar and asks him to take the strangers to a place where they can rest while he catches up with his old friends. The Doctor and Nyssa are amazed to discover that Adric is able to talk to the scorpions and they both understand each other.

As soon as they’re alone, Nyssa tells Adric that he’s changed as he was never this cold before, but he tells her the time he’s spent in this place has made him bitter. He admits that he’s thinking of revenge and the Doctor realises that’s why they’ve been brought here. Adric says he has something to show the Doctor and invites him over to the pit, but when he asks the Doctor to look down over the edge, he pushes him in and the Doctor tumbles headfirst down into the darkness. Nyssa shouts angrily at Adric, but the Doctor calls up to her and assures her he’s alright. Adric says he’s delighted as his pet wouldn’t want her meal to be spoilt. Nyssa sees the creature first and warns the Doctor, who turns to find himself facing an enormous spider which crawls from the shadows towards him…

Rupert and Beatrice are led into a rather small cave and are pleased to have a chance to put their feet up for a while. Madam Teegarna is desperate to speak to them as she wants to know where they came from and how her King knows so much about them, but all her attempts to communicate fail. Frustrated, she leaves and starts to pull a giant boulder across to seal the prisoners inside. Rupert is determined not to be locked up, so he and Beatrice rush forward and tread on the giant scorpion’s tale. Teegarna struggles back, but without the use of her sting there isn’t much she can do. It screeches at them and Beatrice is amazed when she think she heard it say “please”. Just then, more scorpions arrive and the two humans are forced to retreat from the reinforcements.

As the Doctor backs nervously away from the giant spider, Adric tells him not to run around too much as the more calories his pet burns off, the hungrier she’ll get. Nyssa pleads with Adric to get the Doctor out of there, but he tells her it was only a bit of fun and he just wanted to scare him a little. Eventually he calls to Lohkaar to lower a rope so the Doctor can escape, then Nyssa turns furiously on Adric and demands to know how he could do this after everything the Doctor had done for him. Adric tells her it’s been a long time since he had any fun, but when the Doctor tries to reason with him, he orders Lohkaar to remove him. As the Doctor’s dragged away, Adric turns to Nyssa and says he’s pleased the two of them are now alone.

Outside the cave, Teegarna assures Kranlee that she hasn’t been hurt and that she didn’t think the male mammal wanted to kill her. She believes the prospect of enclosure made him panic and when Kranlee dismisses them as freaks, she corrects him. She says they’re reasonable civilised beings and the way they communicate with each other shows how similar they are to His Excellency.

Adric leads Nyssa to a balcony and they look down over the City of Excellence. She admits that the view is impressive, then the two of them make their way across a strengthened rope bridge made from spider silk. She asks him about his pet spider, but he tells her all the creatures here are his servants. They even built this City to his own design, which makes Nyssa wonder how long he’s been here as that must have taken decades. Adric talks about being left for dead, but Nyssa tells him he mustn’t blame the Doctor as he’d explained to her and Tegan why they couldn’t go back in time to save him. Adric recalls being very close to escaping from the freighter and says he was already working on the last of the Cybermen’s logic traps. He only needed a few more moments to release the ship from their control, but a Cyberman blew the console to smithereens and there was nothing more he could do. Nyssa remembers watching the ship burn up on the scanner and Adric notes bitterly how they watched him die from the safety of the TARDIS. He’s convinced the Doctor could have gone back over his own timeline if he really wanted to and he suspects Nyssa only believes the Doctor because she hasn’t been betrayed by him…yet.

The scorpions hurl the Doctor into the cell alongside Rupert and Beatrice, who rush to check that he’s alright. They ask about Nyssa and he tells them she’s still with Adric, but he hopes she’s safe. He explains that Adric was an orphan who stowed aboard his TARDIS so he took the boy under his wing and they became a team of four. He was headstrong and argumentative like any other teenager, but he was also clever and loyal. Beatrice asks why he left, but the Doctor tells her he died, at least that’s what they thought. He doesn’t understand what happened, but if this really is Adric then that makes him a glitch in time, an abomination. Rupert suggest they do something about their captivity, so he urges the others to help him shift the boulder.

Adric tells Nyssa he is no longer alone and that he has millions at his command. They look down and watch rows and rows of scorpions working in what Adric calls ’counting houses’. Nyssa realises they’re like beads on a vast abacus and she likens it to what they saw on Logopolis. Adric says the Doctor taught him how vital numbers were and how important his badge for mathematical excellence was. He says mathematics is god here and the hymn of creation is a never ending series of zeros and ones. They listen to the numbers being chanted and Adric calls it the endless, reassuring music of the calculations. He says it’s what holds this reality together. Nyssa tells him she wants to go back to the others, but Adric stops her and says he knows he used to be a brat and wonders if she was secretly pleased when he died. She’s horrified by the suggestion and says she was devastated, but Adric grabs her arm and holds her in a painful grip. He tells her he’s been so lonely here. The scorpions worship him and bring him everything he needs, but he’s craved attention from others of his own kind. She pleads with him to let her go, but he’s unhappy because she never got the chance to kiss him goodbye.

The group in the cave are doing everything they can to heave the huge boulder aside, but then suddenly it starts to move as though being dragged from the other side. They stand back and watch as Madam Teegarna appears in the entrance. She urges them not to be alarmed and says her people are scared that things will change here because of their arrival. None of the group are able to understand what she’s saying, but Beatrice realises the scorpion is concerned about something. The Doctor promises Teegarna that everything will be alright and although they can’t understand each other, she pleads with him not to take His Excellency away. Rupert points out that the door to their cell is now open, but the Doctor doesn’t think it would be safe to leave. He ignores the advice but immediately sets off an alarm, which means the others scorpions will soon be coming for them. Teegarna makes a signal with one of her pincers and Beatrice believes she’s asking them to follow her.

Nyssa tries to flee from Adric who tries to assure her he was only joking and that he never meant any harm. He says he’s been apart from real people for so long that his social skills are a little rusty. Nyssa demands to see the Doctor, but Adric can’t understand why she cares about him so much. Nyssa says the Doctor means well and always tries to do good, but Adric isn’t convinced. He explains that both he and the scorpions have an affinity with numbers and together they’ve played around with time in ways she wouldn’t believe, but then he starts to cry and says they’ve made such a glorious chaotic mess of things. He likens it to a sticky cobweb of time, with him at the centre. She asks him whether he built his empire for survival, power or vanity and Adric is surprised by her words. He thought they both had a lot in common - they were both young orphans who were dependent on the Doctor. He’s sure Nyssa loved back him then and she agrees that she loved the awkward, frustrating pig-headed boy he was then. He grabs her roughly again and tells her being worshipped by the insects isn’t enough. She pleads with him not to touch her and says she cares for him a great deal, but he says that’s not enough and he needs her to love him. He reveals that this is why he brought her here. He created this world not just to survive, but to lure her here so they could be together. He wants her to be his Empress…and his bride!

The Doctor leads the group down a warren of tunnels, but Rupert wonders if he really has any idea where they’re going. It seems that this is as far as Madam Teegarna is prepared to go and they’ll be on their own from now on. Beatrice realises the scorpion is afraid, but they’re sure it can’t if it’s because of the dark or the fact that the tunnel leads deeper underground. Teegarna is trying to explain to them that the tunnel will go deeper for a while, but will eventually emerge into the fern forest outside, but the journey will be very dangerous and they must decide for themselves whether to enter. Rupert is happy for the Doctor to lead the way, but once they’re both safe, the Doctor says he’ll return for Nyssa and then see about getting them both back to their own time. Beatrice asks him if it’s always like this - is it always dangerous for the people who spend any time with him? The Doctor agrees that sometimes terrible things happen to the people he travels with, but he insists that won’t happen today. They race off into the darkness, accompanied by a very reluctant Teegarna…

Adric tells Nyssa there’s one last thing he wants to show her and he takes her to the peak of his Tower of Excellence. He tells her to look up and she sees a massive object that Adric calls Star . He says this is how he knew she’d arrived in his world and it was through Star that he was able to reach into her mind. Nyssa says it looks metallic and asks if it’s a meteorite, but he says it’s much more than that. Star is what makes his ideas manifest and it’s why his people worship him and how he’s able to communicate with them. He tells her she’ll need to see all his secret things if she’s to marry him and stay here forever, but she insists she won’t do that. He becomes hostile and tells her that’s the reason she’s here, but she says she was fond of him, but nothing more. Adric blames the Doctor again and says that neither the Doctor nor Nyssa will ever leave here. Nyssa becomes frightened, but then an alarm sounds and reluctantly Adric goes to attend to it. As he leads her back inside, he tells her he’s sick of scorpions as they make cold and nervous companions.

The Doctor, Rupert and Beatrice soon become lost in the tunnels and it’s possible they may even have been going round in circles. Teegarna stops for a moment and they hear a strange noise coming from the darkness, making them wonder what else could be down here with them. Teegarna tries to explain that some of her ancestors sleep down in these catacombs, where they eke out a half-life, and this is why none of the others ever come here. The Doctor examines the walls and notices they’re formed from some kind of man-made polymer. They may be encrusted with dirt and rock, but beneath that they’re definitely made of high-density and heat-shielded metal. The truth slowly dawns on him - they’re in the ruins of a burned spaceship - the freighter that Adric was aboard when it crashed on Earth!

Adric races back to his throne room and orders Lohkaar to shut off the alarms. He asks what the emergency is and Lohkaar tells him the prisoners have escaped and taken Madam Teegarna with them into the catacombs. Adric approves of the Doctor’s logical thinking, but says it’s a shame it’s so hopelessly misguided. He orders Lohkaar to wake “them” up and when Nyssa demands to know what he’s doing, he admits that the Doctor was right all along. It’s revenge that he’s after and the Grandparents will see that he gets it…

The Doctor realises the new creatures are the forebears of the scorpions in the City. Teegarna continues to tempt her Grandparents with the fresh meat, but they need no encouragement as they haven’t eaten anything warm-blooded since they feasted on the last of the reptilians. They also warn Teegarna that she has entered forbidden territory and if the meal proves not to be satisfying, they’ll turn on her instead. The Doctor is frustrated that he can’t communicate with the scorpions and is curious to know how Adric is able to do it. He theorises that his former friend must have a telepathic frequency enhancer, like a sort of psychic magnifying glass, and wonders if they can tap into it themselves. He holds hands with Beatrice and Rupert and tells them to open their minds.

They concentrate and eventually the Doctor identifies the correct frequency. The language of the scorpions starts to become clear and they hear Teegarna pleading with her Grandparents to spare her life. She offers them the lives of the mammals in return for safe passage back to the City, but then to their surprise, the Doctor interrupts their conversation. He introduces himself and notes, with interest, that Teegarna was given her name by Adric, who said it translates as “mouth on legs”. She tells them it would have been better if they’d never come to the City as she believes they’re a danger to her kind. The Grandparents have heard enough and give orders for the humans to be cocooned in readiness for the feast. The Doctor tells Beatrice and Rupert to cover their eyes, then the scorpions envelop them all in a fast-setting web.

Lohkaar rushes to Adric with terrible news - there’s corruption in the Calculations! Adric is horrified and when Nyssa asks him what’s wrong, he blames the Doctor. The three of them visit the ‘counting houses’ and listen to the confusion in the voices of the scorpion mathematicians. Nyssa says that if the Doctor is behind this, then it’s all the more reason for Adric to make his peace with him. Adric ignores her and goes into a trance, then he reveals that he can hear the ugly, ancient voices of the Grandparents inside Star. He can also hear Teegarna and realises she’s betrayed him. He remembers Nyssa was tele-sensitive and invites her to tune in with him. Lohkaar suggests sending Kranlee down to the catacombs and Adric gives orders for Teegarna to be killed before she can do more damage. Nyssa is horrified, but Adric tells her the Grandparents’ minds are outside their harmonious web and if their voices poison the Calculations, the whole City could collapse.

Encased in their cocoons, the Doctor, Beatrice and Rupert find they’re unable to move. Teegarna tells them she’s sorry, but says they should never have come to this place so she will leave them to their fate - but as she turns to leave, she finds her way blocked by the two Grandparents. The elderly scorpions have decided to enjoy a crisp appetiser before their main course. Ignoring her screams, they begin to tear Teegarna’s body to pieces - but then they hear Kranlee calling from the shadows. Soldier scorpions emerge into the cavern and attack the two Grandparents. Injured, they have no choice but to retreat, but Kranlee decides not to follow them. He sees the disembowelled corpse of Teegarna and realises he was too late, but admits that it’s not important as he had orders from His Excellency to slaughter her anyway. He releases the Doctor and the others and tells them they must accompany him back to the Tower of Excellence.

Deep in the tunnels, the two Grandparents recover from the stings. Fortunately for them, the venom of the City scorpions is weak, but they regret not having the chance to eat the mammals. They’ve already devoured Teegarna’s brain and her sentience is inside them now. They link their pincers and concentrate, then they become one with the City. They decide to enter the minds of the younger, weaker and more suggestible scorpions so they can make them do their bidding…

Adric rushes round the counting houses, urging the scorpion mathematicians to continue their Calculations. Nyssa points out that they’re no longer listening to his commands and it’s not long before the scorpions begin flicking their stingers at him in irritation. She suggests they return to the Tower, but Adric insists that his will is law. Then, one by one, the scorpions turn towards him and, speaking in unison, they order him to bring them the mammals…

Kranlee hurries the human prisoners along as His Excellency isn’t used to being kept waiting. Beatrice remarks that the more she hears about Adric, the less she cares for him. They arrive at the throne room, but the Doctor decides to dispense with formalities and bursts into the room unannounced, calling Adric’s name loudly. The room is empty, but they can hear a strange noise coming from outside. Suddenly Adric rushes in, begging for Kranlee’s help. Nyssa joins them and says something is sending the drones wild. They close the heavy doors and barricade them, then the elderly Adric turns on the Doctor and blames him for giving the Grandparents access to Star and leaving him to die, then he blames Nyssa for distracting him while he should have been at work maintaining the Calculations. He shouts to Kranlee and Lohkaar to take the other prisoners away. Beatrice is worried that their scorpion escort might turn wild like all the others, but they insist the mutiny is only in the ranks and that because they come from the Hierarchy, their will is stronger than that of the drones.

Once the Doctor is alone with Adric, he warns that the drones are likely to break through any minute. Adric realises his people are reverting to what they once were now that he no longer has influence over them. The Doctor rejects the idea that Adric has been creating harmony and accuses him of being a dictator. He knows Adric ordered the death of Madam Teegarna and is shocked when Adric dismisses his concerns as trivia. He believes he has a powerful ally in his pet spider and he summons it from the pit to defend him long enough that he can climb to the top of the Tower. The doors burst open and swarms of angry scorpions flood into the room. The spider does its best to fend them off while Adric leads the Doctor up an entrance that will take them to the top of the Tower, where he says the mind of Star resides.

Beatrice and Rupert are resigned to being locked up again, but this time they have Nyssa for company. Kranlee tells them the soldiers of the Hierarchy will be defending the Inner Tower and they should be safe here. He orders Rupert to protect His Excellency’s bride, then drags the boulder across the cave opening to seal them in. Rupert is shocked to discover that Nyssa is Adric’s intended wife, but Beatrice has guessed that Nyssa is simply trying to buy them time. She tells Rupert they’re depending on him, the great explorer, to use all his experience and wisdom to concoct an escape plan, but he becomes very nervous and says he wouldn’t presume to lecture them. Beatrice believes he’s being excessively modest and Nyssa pleads with him to apply his expertise, but he finally crumples and admits the truth. He says he’s no hero and he’s never been involved in thrilling escapades or cavalier derring-do at exotic locations. In fact, he’s a fraud and nothing about his reputation is true. He’s never had any adventures beyond Folkestone and the reality is that he’s just Professor Quandry’s secretary. Beatrice disagrees, reminding him of everything they’ve faced since they’ve been here, so in her opinion he definitely is a hero.

With the aid of strengthened web lines spun by the spiders who built this City, Adric leads the Doctor up to the Tower. He begrudgingly admits that the Doctor may be the older man, but he believes he himself if the wiser of the two. He mocks the Doctor for being afraid of heights, but then suddenly he slips and has to be pulled back to safety by the Doctor. Adric says he just sensed the moment of his pet spider’s death and he actually felt the moment when the scorpions ripped her body to pieces. He begins to cry and swears vengeance on them, but the Doctor angrily tells him to grow up. He tells Adric it’s typical adolescent behaviour to think the world should jump to your command - Adric might have an old man’s body, but he’s still a boy on the inside. Adric warns the Doctor to be careful and says his will is everything here. With the help of Star, all he needs to do is concentrate on how he wants things to be, and it becomes so. The Doctor asks him what Star actually is, but Adric won’t reveal any more and tells him to carry on climbing…

In the underground tunnels, Kranlee and Lohkaar can hear the scorpion drones devouring everything that stands between them and the mammals. They believe the drones are seeking revenge on their way of life in the City and are now consumed by their own bloodlust. Lohkaar hopes His Excellency will save them, but Kranlee thinks their King plans to breed with the female mammal to create more of his own kind, so he says they should take direct action themselves. He suggests they force the King to listen using Nyssa as their prisoner.

The Doctor and Adric reach the top of the Tower and the identity of Star is finally revealed. Adric says it came from space, just as he did, and together they were able to control the insects. They made the spiders build them a City and made the scorpions serve them and recite the Calculations. The Doctor realises the foundations of the City are formed from the infrastructure of the freighter that Adric was on when it crashed, but he wonders what became of the Cybermen who were still on board. Adric says the scorpions feasted on them, prising apart their metallic bodies and eating the scraps of flesh packed inside. The Doctor now knows that Star is the alien computer that was controlling the freighter on its journey to Earth. Adric says the Cybermen never knew what they had - an untutored artificial intelligence - but he modified it and they’ve learnt from each other. The Doctor only wants to know one thing - how do they switch it off…

Nyssa thinks she has the answer to getting them out of this time period and back to the Victorian era. Beatrice and Rupert are delighted at the thought of seeing home again, and in fact Rupert is so overjoyed he can’t stop himself from kissing Beatrice. Even before they’ve heard Nyssa’s idea, Rupert thinks this deserves celebration - and he has a little announcement of his own to make. He goes down on one knee and proposes marriage to Beatrice, and although she thinks the circumstances are hardly seemly or conventional, she accepts. Before Nyssa can explain her theory, the boulder blocking the entrance slowly moves away and Kranlee enters the cave. Rupert assumes the insurrection must have been quelled, but the huge scorpion ignores them and grabs hold of Nyssa with a huge pincer. Kranlee tells them he plans to test His Excellency’s desire to mate with Nyssa. He drags Nyssa out and orders the others to follow them. They reluctantly agree, if only to take this opportunity to teach Kranlee some manners.

Adric tells the Doctor that Star can’t simply be switched off and it’ll only allow them access to its inner workings if it’s been flattered, praised and treated with respect. Adric is beginning to realise the Doctor is ashamed of him and is embarrassed by the fact that he wasn’t able to save him. The Doctor reminds him that the scorpions are running wild and Nyssa is in danger, but Adric assures him he would never abandon his friends. He says Star focuses and amplifies his thoughts so that his will becomes manifest. The Doctor is fascinated and says it’s like a crystal brain. Adric tells him to put his hand on Star and concentrate. He does so and then they both see an image of their friends climbing the web…

Nyssa and the others are now hundreds of feet above the ground, moving very slowly and doing their best not to look down. Lohkaar asks them why they came here, but Nyssa says they don’t know what guided them to this place. She loses her grip for a second and starts to fall, but Kranlee reaches out and grabs her, pulling her back to safety. Lohkaar pleads with Kranlee to take care of her as she’s all they have to bargain with.

Adric tells the Doctor that Star may be intelligent, but it’s still just a computer and it’s in need of a reboot. He switches the system off and everything goes deathly quiet, then he switches it on again and things appear to return to normal. Adric declares proudly that his control over the drones has been restored, but when the Doctor challenges him to check, he discovers things are as bad as before. Adric can’t understand why the drones aren’t obeying him, but the Doctor says the Grandparents poison has restored the scorpions’ natural instinct to kill and destroy.

Adric pleads with Star to help them, but then suddenly he and the Doctor find their minds have been transported to the freezing depths of space. They can see prehistoric Earth down below them and then the freighter on a collision course. Star is showing them what really happened all those aeons ago. The image switches to the interior of the freighter and they watch as the young Adric clutches his Outler’s belt and awaits death. The Doctor is saddened and apologises to his friend. Adric asks Star why he’s showing them this, then the image changes again to show Adric tapping away at the controls and inputting the Calculations into the alien computer. The Doctor is amazed - Adric may have been brilliant, but even he shouldn’t have been able to relocate an entire freighter using Block Transfer Computation without help from someone.

The Doctor is fresh out of ideas, so Adric suggests they get out of here, find Nyssa, get to the TARDIS, leave everything else behind and continue their travels as before. The Doctor thinks they ought to come up with a better plan…

The Grandparents arrive above ground level for the first time in many years and, finding things uncomfortably bright, they decide to close their eyes and follow their prey’s spoor. Kranlee and Lohkaar lead Nyssa and the humans further up the web, then suddenly Kranlee senses they’re being pursued. Down below, the Grandparents can see the mammals and they increase their speed. It’s soon clear that the Grandparents are catching them up, so Kranlee suggests they stop and confront them. As there are only two of them and they’re both old, he’s confident they can kill them easily. Rupert prepares to defend Nyssa and Beatrice while Kranlee orders the Grandparents to return to the ground. The elderly creatures demand that he step aside, then they lunge forward and begin stinging him. Kranlee instantly looses all the feeling in his legs and is close to dropping from the web when Rupert steps in front of him with a makeshift sling made from spider web. He swings it above his head and releases a huge rock which hits the Grandmother squarely on the head. Furiously, the Grandfather moves to attack him, but a second rock hits him directly in the eye. Rupert tells the others to continue climbing to the top while he stays behind to hold the Grandparents off. Beatrice protests, but Nyssa agrees to look after her and they start climbing again.

Feeling slightly disorientated, the Doctor and Adric bring their minds back out from inside the Star. Adric asks what was wrong with his plan to get back to the TARDIS and the Doctor is forced to admit that they’ve temporarily lost the ship after it was stolen back in the Victorian era by a boy - an orphan and a petty thief, just like Adric when they first met. He says he’s been trying to get it back ever since, which is why he was using Block Transfer Computations. Just then they hear Rupert crying out from far below…

The Grandparents are ’playing with their food’ because Rupert won’t stop fidgeting, so the Grandfather urges his partner to deliver the killing blow. Rupert starts to recite the Lord’s Prayer, but then suddenly the Grandmother is struck by a perfectly thrown cricket ball. She screams and drops from the web, falling hundreds of feet to the ground and dying instantly. Further up the web, Nyssa congratulates the Doctor on his bowling. Beatrice suggests they lower a knotted rope made from strips of her skirt down to Rupert and slowly they manage to haul him up to safety. The Grandfather bellows angrily to the fleeing mammals, but before long they’re all out of his reach.

The Doctor turns back to Adric, who suddenly seems very weak. He tells the Doctor that his body is starting to fail and the past few hours have been too much for him. The Doctor suggests he rest while they wait for the others to return, then they can put their heads together to find a solution to their problem. Adric wants to use Star to try to mend the Calculations, but the Doctor knows he’s not strong enough and says it could kill him.

Nyssa helps to bandage a nasty wound in Rupert’s leg, then Beatrice offers him a shoulder to lean on. Kranlee is also injured, but they don’t have time to hang about and decide to continue climbing. Nyssa realises the scorpions are after Star and Lohkaar confirms that if they destroy it, there’ll be no more Calculations and their race will revert to base savagery. There’s only one Grandparent left now, but it’s twice as angry. Lohkaar suggests he and Kranlee cut the web lines to prevent it from reaching His Excellency, and as they climb, Beatrice asks Nyssa to elaborate on her idea for getting them home. She tells them that Adric was boasting about how Star amplified his will. It took twelve people at the séance to get them here, but if they can use Star to amplify their mental powers it may only need four of them to return home.

Adric oversees his City and comments that it’s virtually dead now. He‘s convinced they‘ll all die here too, but his mood is improved when Lohkaar and Kranlee arrive, bringing Nyssa and the two humans with them. The Doctor congratulates them, but then they hear the Grandfather approaching and realise he must have found another way up. Adric says he’s brought them all to the last safe place in his City and this is where they must make their final stand. Nyssa realises the Calculations have now stopped completely and everything is silent. Everyone turns to Adric for advice on what to do next, but Beatrice has already decided she wants to go home. Nyssa explains her idea, but it will require Adric to do the sums while Star amplifies them, and Rupert is unhappy that it means they’ll have to take Adric back to Victorian London with them. The Doctor thinks the idea is brilliant, but Lohkaar wonders how this will save the City. Adric admits that it won’t - it’ll save the mammals, but no one else. Sadly, he tells Lohkaar that he’s going now and leaving this world behind. The two scorpions protest, so the Doctor explains that Adric should never have come here and everything that happened was just an accident. The only way to put everything right is for him to leave.

Kranlee assumes they’ll return to their primitive state, but Adric thinks they deserve to hear the truth and reveals that when he goes, this whole world will wink out of existence. The scorpions refuse to let Adric leave and grab hold of him, accusing him of betraying them. Adric calls to the Doctor and Nyssa and tells them to place their hands on Star urgently. The Doctor immediately gets Star to vibrate at the correct frequency and then the group begin reciting the numbers just as they did back at the séance. Kranlee and Lohkaar prepare to kill Adric for his betrayal, but the Doctor distracts them by twisting the wound in Kranlee’s leg. The scorpions release Adric and he quickly rejoins the others. Everyone holds hands, but Star begins to fracture and they realise they don’t have much time. Rupert asks for Beatrice’s forgiveness, then he leans forward and kisses her. Then, he turns and uses the last remaining seconds to fend off the approaching scorpions. Moments later there’s an enormous flash and the Doctor, Nyssa, Adric and Beatrice disappear completely. Kranlee and Lohkaar turn on Rupert and tell him he’s brought about their destruction by allowing Adric to escape and for that, he will be killed. Rupert screams in agony as he is struck by multiple stings from the two angry scorpions…

The Doctor’s group find themselves floating in a strange limbo. He tries to tell Beatrice how sorry he is, but she says Rupert always wanted to be a hero and although he’d never quite managed it before, he has now. Adric says they have to move on to the next sequence, so he instructs each of the group to recite their own series of numbers. Their voices fill the air and they soon become one united voice…

Kranlee and Lohkaar stand alone in a dark, windswept Tower. The Star has finally faded away into nothingness along with their King. He’s gone from their world, forever. The Grandfather is still coming for them, but they hardly think it matters now. By the time he reaches them, they’ll have ceased to exist too. The Calculations are finished and their world is over. They don’t count any more.

With a flash of light, the Doctor’s group reappear and they find themselves back in Beatrice’s sitting room in her house in Bloomsbury. Adric collapses and Beatrice rings for help from her maid, Brenda. The Doctor tries to help his friend and tells him they couldn’t have got here without his calculations. Brenda rushes in and is amazed to see her mistress home again. It’s been six weeks since they disappeared and Beatrice is shocked when it occurs to her that all of London will think they’re dead…although in poor Rupert’s case they’ll be right. Brenda looks down at Adric and thinks he’s an old vagabond, but Beatrice orders her to prepare a guest room for him. Adric slowly comes round, but is confused about where they are until the Doctor reassures him they‘ve arrived in Victorian London as hoped for.

Even though Beatrice tells him they can only see Tavistock Square from the window, Adric insists on taking a look, so the Doctor and Nyssa help him up. Beatrice opens the window and looks longingly at the view she thought she’d never see again. Adric, however, is upset that he’d had to abandon his people and condemn them all to death. Nyssa wonders whether the world ever really existed, but the Doctor says it did in its own bubble of time, although it’s sealed off forever now. Adric thinks he too is going to die and although Nyssa assures him he just needs rest, he warns her not to let her logical mind get clouded by sentimental nonsense. He says he should’ve died before he was 20, but the Universe found a way for him to live and he didn’t realise how lucky he was. He’s tired now and asks his friends to leave him and get on with arranging their new lives here. After all, they’ll need a home and proper employment now that they don’t have the TARDIS any more. They say goodbye to Adric and promise to look in again later.

Once he’s alone, Adric turns to the shadows and asks if his little friends hiding away in the skirting and up in the rafters have been listening? He tells them he left an empire of their forebears behind but looking at them now, he sees their brains are so tiny. He asks if they’ll let him be their ‘Excellency’ again and if they’ll sing for him. He begins to recite a sequence of numbers and asks them to join in…

Right now, all Beatrice wants to do is feel ordinary and she’s delighted just to be sitting on a normal chair in her kitchen in Bloomsbury. Nyssa wonders what Beatrice is going to tell Professor Quandry about Rupert, but before she can answer, Brenda bursts into the room and excitedly announces that Mr Adric has vanished into thin air. She says she was taking him a tray of tea, but outside the door she heard strange noises coming from the other side, like he was singing to himself. Then, when she entered the room, she found he’d gone. The Doctor and the others race to the study, but it’s just as Brenda said. Just then, the air starts to shimmer in the corner of the room and Adric starts to reappear. Nyssa asks where he’s been and he tells her it was his final journey. Excitedly he announces that he managed to find what he was looking for, but then he collapses to the floor and dies. They suspect they’ll never know what he was trying to do, but whatever it was it finally wore him out and now he‘s gone for good.

Some time later, the Doctor and Nyssa walk slowly away from the cemetery after attending Adric’s funeral. It starts to rain and Beatrice sympathises with them, saying it’s a shame their friend came so far only to fall at the final hurdle. She tells them she’s decided to travel the world as she can’t write romances from the comfort of Tavistock Square. It’s something she learned from poor Rupert and she can still hear his voice inside her head. She says goodbye and makes them promise they’ll visit again before she goes abroad. Nyssa tells the Doctor it’s time they got back to Baker Street, but suddenly he stops in his tracks and remembers this is the same location where Thomas Brewster’s mother was buried. It seems to prove that time is circular and what goes around comes around. She thinks he’s being morbid, but he can’t forget that he was supposed to look after Adric and keep him safe. He failed his friend, so perhaps it’s just as well that his travelling days are over. Nyssa asks about the Time Lords and says there must be others of his race who were exploring this time zone, but the Doctor says he’s definitely not asking Iris for any favours! She then suggests hitching a ride with a former self, but he tells he she has a lot to learn about the nature of paradox. He becomes depressed and thinks he’s made a mess of things, but at least Nyssa is still here.

He suddenly senses the particles in the air being shunted around, as if making space for something. Then, to their joy and amazement, the TARDIS materialises in front of them. They rush over before the ship disappears again and the Doctor digs in his pocket for the key, but before he can find it, the door opens and Thomas Brewster peers our and says hello. He assures them he’s fine and although his legs are like eels in jelly, he’s pleased to have found them again. The Doctor asks him how he made his way back here and Thomas says he got some help…

At first Thomas thought his saviour was some sort of phantom. He was inside the TARDIS console room, convinced that the ship was out of control, when he heard a voice speaking inside his head. He thought it was his mother again, but the voice was reciting numbers in a strange sequence. Then a mysterious old man appeared before him and called him out from his hiding place under the console. Thomas asked why he looked like a ghost and Adric explained that he was a Block Transfer Computation. He assured Thomas he was here to help - the machine was about to crash into the temporal horizon and it was clear Thomas couldn’t control it. Adric explained that he was a companion aboard this ship many years ago and he believed the TARDIS was beginning to trust him and give up its secrets. When Thomas introduced himself, Adric remembered the Doctor had spoken of him as an orphan and a petty thief. He began to re-route the power and said he was sending him and the TARDIS back. He told Thomas that if he didn’t belong in any place or time, he should stay with the Doctor, then he told him to remember his name and ask the Doctor what became of him, because it’s a good story…