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In Futurama's episode 6x10, the professor invents a device which allows to switch mind between two bodies. He then switches his mind with Amy. It later turns out that the a given pair of bodies can not complete this procedure twice with each other.

 

In this situation, is it possible to undo what is done using other bodies to swap minds between?

 

What if initially not 2, but N minds has been switched between N bodies? Consider the worst case scenario that each pair of N bodies has been switched already.

For example, if they use 1 additional body:
Let the Professor be $P$ and Amy be $E$. Upper case letters represent a body, lower case letters - minds. They start out with $Pp,Ee$ then after one switch, they have $Pe, Ep$. If they use third person $A$, then they can do $Pe, Ea, Ap$ and $Pp, Ea, Ae$, but $E$ and $A$ has already switched once so they would face the same problem again.

In Futurama's episode 6x10, the professor invents a device which allows to switch mind between two bodies. He then switches his mind with Amy. It later turns out that the a given pair of bodies can not complete this procedure twice with each other.

In this situation, is it possible to undo what is done using other bodies to swap minds between?

What if initially not 2, but N minds has been switched between N bodies? Consider the worst case scenario that each pair of N bodies has been switched already.

For example, if they use 1 additional body:
Let the Professor be $P$ and Amy be $E$. Upper case letters represent a body, lower case letters - minds. They start out with $Pp,Ee$ then after one switch, they have $Pe, Ep$. If they use third person $A$, then they can do $Pe, Ea, Ap$ and $Pp, Ea, Ae$, but $E$ and $A$ has already switched once so they would face the same problem again.

In Futurama's episode 6x10, the professor invents a device which allows to switch mind between two bodies. He then switches his mind with Amy. It later turns out that the a given pair of bodies can not complete this procedure twice with each other.

Example: Let the Professor be $P$ and Amy be $E$, then let $b$ represent a body and $m$ represent a mind. They start out with $PbPm, EbEm$ then after one switch, they have $PbEm, EbPm$.

In this situation, is it possible to undo what is done using other bodies to swap minds between?

What if initially not 2, but N minds has been switched between N bodies (randomly)?

In Futurama's episode 6x10, the professor invents a device which allows to switch mind between two bodies. He then switches his mind with Amy. It later turns out that the a given pair of bodies can not complete this procedure twice with each other.

In this situation, is it possible to undo what is done using other bodies to swap minds between?

What if initially not 2, but N minds has been switched between N bodies? Consider the worst case scenario that each pair of N bodies has been switched already.

For example, if they use 1 additional body:
Let the Professor be $P$ and Amy be $E$. Upper case letters represent a body, lower case letters - minds. They start out with $Pp,Ee$ then after one switch, they have $Pe, Ep$. If they use third person $A$, then they can do $Pe, Ea, Ap$ and $Pp, Ea, Ae$, but $E$ and $A$ has already switched once so they would face the same problem again.

In Futurama's episode 6x10, the professor invents a device which allows to switch mind between two bodies. He then switches his mind with Emmy. It later turns out that the a given pair of bodies can not complete this procedure twice with each other.

Example: Let the Professor be $P$ and Emmy be $E$, then let $b$ represent a body and $m$ represent a mind. They start out with $PbPm, EbEm$ then after one switch, they have $PbEm, EbPm$.

In this situation, is it possible to undo what is done using other bodies to swap minds between?

What if initially not 2, but N minds has been switched between N bodies (randomly)?

In Futurama's episode 6x10, the professor invents a device which allows to switch mind between two bodies. He then switches his mind with Amy. It later turns out that the a given pair of bodies can not complete this procedure twice with each other.

Example: Let the Professor be $P$ and Amy be $E$, then let $b$ represent a body and $m$ represent a mind. They start out with $PbPm, EbEm$ then after one switch, they have $PbEm, EbPm$.

In this situation, is it possible to undo what is done using other bodies to swap minds between?

What if initially not 2, but N minds has been switched between N bodies (randomly)?