I was watching episodes of the new season of Futurama recently. In one particular episode Professor Farnsworth invented a device to switch minds of two people. But as anyone who has ever watched the show might guess, there was a little problem. Once two people went through the body switcheroo, they could not go through the switching process again. There are many body switches and peoples minds end up all over the place, and no obvious way of putting everyone back in their own bodies. Luckily, the Globetrotter Scientists figure out that no matter how scrambled people’s minds and bodies get, they can always be placed correctly as long as you introduce two extra, untouched, people. My aim here is to explain why.
This is going to be a pretty short post for me as this is actually a pretty easy thing to prove, once you look at it the right way. Basically what it comes down to is two extra people give you a way to switch any two people, even two people who have switched before, but only one time. To see why, just look at the following diagram.
So, you see, A and B can be switched, and since no switch involves a swith between A and B directly, it can be done on two people that have already been switched. You’ll notice that S1 and S2 are also switched. This doesn’t really matter since this switching process never involves a trade between S1 and S2 they can always switch back after fixing everyone.
But during the process, the bodies of A and B are involved in switches with both the bodies of S1 and S2, meaning this sort of switch can only be done once on two people. Still, since the original body switches were done in a way that no bodies could have their minds switched twice, they can be reversed without having to switch the minds of the same pair of bodies twice.
There are still, however, other interesting questions to answer about this scenario. For example, what if you didn’t have any two extra people? When is it still possible to put people back to normal? And come to think of it, what’s the most efficient way of doing it? These are questions for another time, so that will be all for now.