Ripped from The Prisoner of Brenda with apologies to Matt Groening, Ken Keeler et al.
Professor Farnsworth: "There. This time I'm sure I've fixed the Mind-switcher." An invention in which two people (or a robot and an intelligent wash bucket or whatever) can sit and have their minds swapped. After a brief discussion as to why anyone would want to do such a thing, Amy Wong concludes "We are just the people this mind-switcher was made for by us!" They switch minds but soon realize that it wasn't such a good idea. Getting back into the Mind-switcher to undo their mistake, nothing happens! Farnsworth (now in Amy's body): "I failed to take into account the cerebral immune response. Once two bodies have switched minds, they can never [directly] switch back. However, perhaps they can [indirectly]. Maybe we can swap back using a third [or more] body for temporary storage space."
Show how Amy and Farnsworth can swap their minds back using one or more "temporary storage space" bodies so that in the end everyone involved is back to normal. Or prove that it's impossible.