The Prisoner of Benda
Leela is worried that her eye is too big, but when Fry tries to console her by telling her she's the most beautiful woman he's ever known, she snaps at him for only caring about her looks. The news reports on the visit of Nicolai, the playboy robo-Hungarian emperor... whose crown Bender plans to steal. Nobody wants to be his accomplice. Meanwhile, the Professor builds a brain-switching device. He regrets the youth he wasted playing it safe. He and Amy agree to switch bodies so he can live life to the full, while she can indulge her passion for food. After wiping out on a jet-powered snowboard, the Professor says that Amy's body isn't up to the task. It turns out they can't switch back due to the cerebral immune response. Two bodies that have switched minds can never switch back... unless they use a third body for temporary storage. Bender volunteers, switching with Amy... and then he takes off to go steal Nicolai's crown. Amy could switch with the Bender-Professor, but then although the Professor would have his own body back, Amy would be stuck in Bender's body, unable to switch with him when he brings her body back because those bodies have already done a swap.
Bender (in Amy's body) tries to seduce Basil (the Emperor's cousin and chief of the Royal Guard), and simply knocks him out when he's not interested. Amy's unhappy with the Professor's body, which is too aged and feeble for gorging on food. She switches with Leela, who wants to get senior discounts. As Amy uses Leela's body to chow down, Fry is repulsed to find Leela in the Professor's body. "You DO only like me for my body!" she accuses him. Fry, annoyed, decides to swap bodies with Zoidberg to give Leela a taste of her own medicine. The Professor, in Bender's body, decides to run away with his invincible robot body and join the robot circus, where he becomes "Nonchalanto," a daredevil. Back on the royal boat, Bender is caught, and convinces Emperor Nicolai that he's really a robot. Nicolai has tired of his life of luxury and beautiful robo-women, and he and Bender agree to swap bodies. Nicolai doesn't want Amy's body, but is interested in swapping into the body of a common bending-robot for a day, so long as Bender will give the speech he's supposed to give at the UN the next day. When Bender finds that the Professor has run away with his robot body, he swaps places with Scruffy's washbucket-robot instead, and then swaps with the Emperor. He wants to meet his human roommate, and Bender points him towards Fry... who now has Zoidberg's mind. "It is I, Bender!" says Nicolai-as-washbucket: "Do you not recognize your best friend?" Zoidberg, happy that the guy whose body he's using has a friend, happily goes home with him to Fry and Bender's apartment.
Leela tries to hide her revulsion at Fry in Zoidberg's body, and is also upset to find that Amy has made Leela incredibly fat. Still, Fry makes the mistake of agreeing with her, restarting their argument. Fry challenges her to a romantic dinner at Elzar's. Meanwhile, Hermes finds Amy wolfing down whole sticks of butter, horrified at herself. He recommends they swap bodies right away, since his body is already ruined. At the circus the Professor meets Big Bertha, a Hungarian robo-cannon who shoots robots through the air. Bender's body is too big: she's so old, he'd shatter her. He offers to put her in a new body, but she proudly refuses: "My body may be old, but it is mine, and every crack a memory of heavy things shot a long way!" On Nicolai's boat, Bender is enjoying the royal lifestyle... until the robo-woman he thought would be his lover pulls a gun on him. She's been having an affair with Basil, who plans to murder Nicolai and claim his throne. Bender flees. At Elzar's, Fry and Leela gross each other out with their borrowed bodies, embarrassing each other in public while claiming not to be. A passionate argument ends with them making out on the table... while Amy, at the next table in Hermes' body, looks on in disgust. Meanwhile, in Scruffy's quarters, washbucket-robot (in Amy's body), decked out in lingerie, tries to seduce Scruffy: "I love you. Washbucket has always loved you." Scruffy knows it would be sweet for a while, but it wouldn't work. He sends her away... and then lies down on his cot and sobs. Bender, still pursued by Basil, manages to make it to the UN to deliver Nicolai's speech. Nicolai corners him in front of the crowd. The event, televised live, is seen by the Hungarian robots of the robo-circus. To save their emperor, Bertha volunteers to shoot the Professor to the UN. She does so, and expires. He crashes through the roof of the UN building, and does battle with Basil. Meanwhile, Leela and Fry have just had sex in the Professor and Zoidberg's bodies. Content, they turn on the TV... and see the battle in progress, which the Professor wins when a legion of armed robo-carnies leaps out of his chest.
Everybody gathers back at Planet Express, where Bender gives the Prince (still a washbucket) his crown back... but it's a fake crown; the real one is secreted in the storage space of Nicolai's body. Hermes has slimmed down Leela's body to its original state. He was inspired by Amy, who has gotten Hermes in shape. She says she was eating a pork sundae, when suddenly she just lost her appetite… forever. The Globetrotters use the power of math to figure out how to restore everybody's minds by just using two extra players. A quick body-swapping montage later, and everybody's back to normal. "My body may be old," says the Professor, "but it is mine." As Nicolai leaves, fake crown on his head, Bender gloats that he's left with the real jewels... and then realizes he left them in Nicolai's body… oops: "All right. I'll need accomplices..."