A guardian sphere bounces through the Village and everyone ducks to the side and freezes as it passes. Number Six observes as one man, walking with a cane, walks by the guardian sphere without it paying him any attention or vice versa. Intrigued, Number Six follows the man as he goes to a lawn-sized chessboard with volunteers standing in as chess pieces. The Man With Stick invites Number Six to join them and the Queen asks him to be her Pawn. On a balcony overlooking the board, the players make their moves as Number Six asks the Queen who the Man With Stick is. She doesn't know who he is but has heard the rumors. His ancestors played chess and decapitated the player-pieces as they were removed. Number Two observes the game from the Control Room as Number Six continues to ask probing questions. He finally suggests that they escape and she offers to help. She says she can help and Number Six, distracted, forgets to make his move. He finally responds and Number Two, observing, notes he seems placid enough. Finally, the White Queen's Rook moves illegally, declaring check against the opposing King. The authorities immediately take him in for treatment and a substitute is called. The Queen speaks of the Cult of the Individual and says the Rook will be well looked after.
The game proceeds and White wins. The Man With Stick congratulates Number Six on his game and the two of them go for a walk. The Man With Stick says that he's learned how to tell the two sides apart through psychology: he judges by attitude. He's too old to escape but says he once had a plan. However, the Man warns that everyone has a plan and they have all failed since the prisoners are unable to distinguish black from white. Number Six leaves and the Queen, infatuated, follows him. He demands to know why and asks when he plans to escape. She offers to help but warns that no one she's helped has ever succeeded. He refuses her help and goes back to his flat.
Later, Number Two greets Number Six in the street and offers to let him see how the Rook is coming along. They watch as the man undergoes Pavlovian treatment. He is dehydrated and then put in a room with four water dispensers. A voice warns him to leave the water alone and the handles are electrified. The voice directs him to the blue dispenser and in return for his obedience, there's no electrical shock. Number Two is satisfied that the Rook will no longer be a problem. As Number Six strolls about the Village, he notes one Villager who stands up to him, and the Rook who walks away. Number Six follows him and finally catches up, and asks him why he ran. The Rook admits he doesn't know and agrees with anything Number Six says. Number Six asks him to come along with him. The Rook is convinced Number Six is one of the warders and explains he invented a new electronic defense system that he thought would insure peace for all nations. Number Six claims he's a prisoner too but the Rook doesn't believe him. Number Six says they'll talk again and the Rook wonders.
The Supervisor informs Number Two that Number Six is associating with the Rook and Number Two watches them at a chess game. Number Six explains that the prisoners are subservient while the warders are authoritative. Number Two checks with the psychiatrist about the success of the Rook's rehabilitation and she asserts that he's been brought to heel. Number Two discontinues surveillance and Number Six explains that the warders among the villagers are never subservient, so they know whom they can rely on as a first step in any escape plan. Number Six says they need to find other reliable men and challenges a gardener, who defies him. They then challenge a painter and a shopkeeper who are subservient to them.
Number Six and the Rook begin to assemble people they believe they can trust, and Number Two starts to notice. They're unable to get video so Number Two has Number Six brought in for testing. The psychiatrists find nothing significant and then bring on the Queen. They demonstrate to Number Two that the Queen has been programmed to fixate on Number Six and betray him to save him if necessary. They then give her a locket, claiming it's from Number Six, which will monitor her emotions and let them track her. If she sees Number Six escapes, she'll become excited and they'll sense the disturbance
The Queen follows Number Six about the Village, while the psychiatrists advise brain surgery to adjust Number Six's aggressive tendencies. Number Two decides not to given Number Six's value. Meanwhile, Number Six takes a buggy and the Queen follows in another one. Number Two lets it continue to determine how effective the Queen's tracking is. He loses her and picks up the Rook, and they sabotage a surveillance camera and steal it. The Rook steals a telephone and they go the crossroads where an electrics truck has gone to replace the stolen camera. As the repairman works on the cameras, they steal electronic parts from his buggy. As they drive back, the Queen spots them. Number Six sends the Rook ahead to hide the stolen goods and Number Six lets the Queen pick him up. She declares her love for him and breaks into tears but he's unimpressed.
That night, Number Six is preparing for bed when he finds the Queen in his flat making hot chocolate. She fawns over him and insists if he only care a little about her, she'd be happy. However, when she refuses to tell him who put her up to it, he yells at her ordering her out and she bursts into tears. He lies and admits he likes her a little and she offers to make him more hot chocolate and asks to see him again. She runs off, satisfied. The next day on the beach, Number Two approaches the Rook and inquires about his mental health. The Rook then goes into a changing hut and Number Six meets him there. They're assembling a radio from the separate components but the Rook notes they need more transistors. The Queen is on the beach and invites Number Six to sit with him, and he reluctantly agrees. She accuses him of not loving her any more and when he says he doesn't, she shows him the locket. He asks to see it and she turns it over, and he realizes it has a transistor inside. He quickly leaves with it and gives it to the Rook, who explains what it does. It's the last component that the Rook needs, and they'll be ready to escape that night. In the Control Room, they lose the signal from the locket and find the Queen on the beach. They realize the locket is gone and suspect it came off in the water. They finally locate Number Six down the beach and figure he hasn't escaped yet. Number Six then notifies his fellow escapees, including the Man With Stick.
That night the escapees assemble at the changing hut and Number Six sends a mayday over the radio. They finally contact a ship and identify themselves as a transoceanic flight in distress. The Control Room picks up the signal and Number Two says to leave the apparent crash to the ship. Number Six and the Rook put the radio on a raft and the Rook paddles out to sea, and Number Six tells the Rook to wait for him or the ship. Number Six then goes to the Stone Boat while the Control Room tries to pinpoint the radio signal. They have to keep the guardians from locating them until they're ready. They go to a tower and knock out the guardians manning a searchlight. Number Two is meditating in the Green Dome when he is informed the searchlight crew is attacked. He prepares to leave but Number Six and the others capture him and take him hostage. The signal cuts off but Number Six warns that it's too soon. He tells the others to stay put and goes to the beach to investigate.
The raft is on the shore but there's no sign of the Rook. He sees a flashing light off shore. Number Six takes the raft out to the boat and a sailor hails him. He goes aboard and asks to talk to the skipper, but discovers Number Two on a monitor screen, safe and secure in the Green Dome. The Rook is with him and Number Six believes he's the traitor. The Rook believes Number Six is a guardian: Number Six was too authoritative and the others believed he was working with Number Two. Number Two says that the ship is theirs. Number Six attacks the crew but Number Two sends a guardian sphere and immobilizes the ship's controls. The guardian sphere escorts the ship back to the Village while in the Green Dome, the Butler returns a white pawn to the chessboard.