Logic Problem of the Day/Week/Month

Proving once again that Futurama is the best adult-oriented cartoon on TV. Family Guy has never required more than two brain cells to enjoy. Actually, the more you have, the less enjoyable it is.

This problem comes from last week's episode of Futurama, which is actually one of my favorite episodes of all time. Professor Farnsworth has invented a body swapping device that swaps two peoples minds into each other's bodies. He and Amy Wong initially use it, so that Amy can eat as much as she wants without ruining her figure, and the Professor can enjoy a youthful body.

Neither of them is satisfied with the swap, however. Farnsworth's aging body's colon and teeth don't satisfy Amy's gluttony, and Amy's body is just too weak for the Professor. They try to swap back, but as they discover, once two bodies have been swapped, those same two bodies can never be swapped with each other again. Bender, who needs accomplices to steal the crown jewels from the Robo-Hungarian Prince, talks the Professor (who is occupying Amy's body) into body-swapping with him.

Meanwhile, Leela sees Amy in Farnsworth's body complaining about her inability to enjoy food. Leela complains about expensive movie tickets, so Amy convinces Leela to swap, so Leela can get a senior discount at the theater.

Meanwhile, Bender uses Amy's body to attempt to seduce the guards of the crown jewels. When it doesn't work, and he finds himself face-to-face with the Robo-Hungarian Prince, he convinces the Prince that he's actually just a lowly bending robot. The Prince, tired of his pampered life, thinks it would be all in good fun to switch bodies with a lowly commoner robot for a day. Bender rushes him back to Planet Express to swap into his Bending body, so he can have the Prince's body, rich life, and (most importantly) hot robo-fiancée.

When Bender arrives back at Planet Express, he discovered that Farnsworth has left with Bender's body to join the circus and live life to the fullest. Needing to find a robot body to swap with the Prince, he comes upon Scruffy the janitor's mop bucket which, in the 31^st century, is a robot. Bender (in Amy's body) swaps with the Bucket, then again from the Bucket to the Prince.

(The Bucket is now in Amy's body and there is an odd side-plot about the Bucket being in love with Scruffy and trying to seduce him.)

Meanwhile, Leela (who is in Farnsworth's old and unattractive body) complains that Fry is unattracted to her in the ugly body, and that he doesn't love her, only her attractive normal body. Fry denies it (but secretly admits it), so to get payback on Leela, body swaps into Zoidberg's ugly body, to prove to Leela that she's just as unattracted to him when he's ugly.

Lastly, Hermes discovers that Amy (in Leela's body) has been gorging herself on food. To put an end to the destruction of Leela's body, Hermes body-swaps into it, and surrender's his own body, already destroyed by twenty years of the munchies, to Amy.

So at this point in the episode:
Amy is in Hermes' body;
Hermes is in Leela's body;
Leela is in Professor Farnsworth's body;
Professor Farnsworth is in Bender's body;
Bender is in the Robo-Hungarian Prince's body;
The Robo-Hungarian Prince is in the Mop Bucket's body;
The Mop Bucket is in Amy's body;
Fry is in Zoidberg's body; and
Zoidberg is in Fry's body.

Your mission, should you choose to accept it, is to devise a system of body-swapping that will return everyone into his original body. Remember that once two bodies have been swapped, they can never be swapped with each other again. That means at this point:

Amy's body cannot be swapped with Farnsworth's, Bender's, or the Bucket's;
Farnsworth's body cannot be swapped with Leela's;
Leela's body cannot be swapped with Hermes';
The Bucket's body cannot be swapped with the Prince's; and
Fry's body cannot be swapped with Zoidberg's.

Got it?

Here's a hint if you need help (highlight to read).
You may use an arbitrary number of Harlem Globetrotters as intermediate bodies during the swapping. What is the minimum number of Globetrotters required?

The answer given in the episode (although there are other allowable sequences). Highlight to read:
They used two additional Globetrotters—Ethan "Bubblegum" Tate and "Sweet" Clyde Dixon—to restore everyone to their original bodies. The sequence they used was as follows:


Zoidberg (in Fry's body) is swapped with Clyde Dixon;
Fry (in Zoidberg's body) is swapped with "Bubblegum" Tate;
Zoidberg (in Clyde Dixon's body) is swapped with Tate (in Zoidberg's body);
Fry (in "Bubblegum" Tate's body) is swapped with Clyde Dixon (in Fry's body);
(Zoidberg and Fry are now in the correct bodies, and Dixon and Tate are swapped;)


Leela (in Farnsworth's body) is swapped with Tate (in Dixon's body);
The Prince (in the Bucket's body) is swapped with Dixon (in Tate's body);
Leela (in Dixon's body) is swapped with Hermes (in Leela's body);
The Prince (in Tate's body) is swapped with Bender (in the Prince's body);
(Leela and the Prince are now in the correct bodies; Tate is in Farnsworth's body; Dixon is in the Bucket's body; Hermes is in Dixon's body; and Bender is in Tate's body;)


Amy (in Hermes' body) is swapped with Hermes (in Dixon's body);
Farnsworth (in Bender's body) is swapped with Bender (in Tate's body);
(Hermes and Bender are now in their correct bodies; Amy is now in Dixon's body; and Farnsworth is now in Tate's body;)


Amy (in Dixon's body) is swapped with the Bucket (in Amy's body);
Farnsworth (in Tate's body) is swapped with Tate (in Farnsworth's body);
(Amy, Farnsworth, and Tate are now in the correct bodies; the Bucket is now in Dixon's body;)


Dixon (in the Bucket's body) is swapped with the Bucket (in Dixon's body);
Everyone is now in their correct bodies.

Got that?

(Aside: for the reason why it's impossible (in the 2+2=5 sense of impossible, not in the traveling faster than the  speed of light sense of impossible) to put a human mind in a robot, see Michael Flynn's explanation of J.R. Lucas' proof.)