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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.

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It is possible.

Call Amy and Farnsworth A and F. Find two more people, X and Y. Let B(C) imply that person B has person C's mind. Let "Swap B and C" imply swapping whatever minds are currently in the bodies labeled B and C.

So far, A and F swapped to get: A(F), F(A), X(X), Y(Y). Swaps Used: AF.

Swap A and X; F and Y. Get A(X), F(Y), X(F), Y(A). Swaps Used: AF, AX, FY.

Swap A and Y; F and X. Get A(A), F(F), X(Y), Y(X). Swaps Used: AF, AX, FY, AY, FX.

Finally, Swap X and Y to get A(A), F(F), X(X), Y(Y).

Everybody's back to normal! Well, as normal as they were to begin with.

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  • $\begingroup$ I bet the confusion is about what a swap means - to clarify, I am asking the physical bodies labeled A and X to swap, not the minds (wherever those might be). I'll add a comment. Note that the distinction does matter a little, since the restriction of multiple swaps is applied to bodies, not minds. $\endgroup$ – Zerris Jan 24 '16 at 3:03
  • $\begingroup$ Sorry, my bad you are totally correct and solved it super fast! $\endgroup$ – Paul Evans Jan 24 '16 at 3:09
  • $\begingroup$ I figured I'd try it for three and four people to see if it was easily doable, and if not try to understand what went wrong so as to generalize a proof - but my four person variant just worked, so I wrote it up! :p $\endgroup$ – Zerris Jan 24 '16 at 3:12
  • $\begingroup$ This episode came about because one of the writers had proved it was possible in a paper. There's a Numberphile video about it. $\endgroup$ – Arcturus Jan 24 '16 at 5:24
  • $\begingroup$ @Ampora Interesting. I was catching up on old episodes of Futurama last night and thought it'd make a good puzzler. $\endgroup$ – Paul Evans Jan 24 '16 at 19:28

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