I just read about, and watched, the episode where ken keeler invents and then proves a theorem for mind and body switching, but I don’t understand the ins and outs of it. Is it possible to explain this theorem simply?

In: 9

Assumption: once a pair of people have been put into the brain-swap machine, that particular pair can’t do it ever again. Also since you don’t know exactly what swaps have happened, you must bring in additional people who have never been on the brain swap machine and they must always be used as one of the two people used.

For each person, imagine every person facing the body they want to be in. Ie: whoever has Phillip J Fry’s brain, that person faces Fry’s body. What you’ll end up with is a collection of little circles of people, possibly just 2 people facing each other but that’ll be a “circle”. We process each “circle” one at a time.

With your new players, you can start performing the brain swaps. Swap Fry’s mind with the mind of one of your new players, but just one. Now that new player body also immediately swaps with whom Fry was looking at, thus putting Fry back where he belongs. Repeat that and you’re fixing the whole circle until you come back around to the start of the circle, and use the 2nd new player to deal with closing up the loop. If you’ve done it right, the circle is fixed, but the 2 new players have effectively swapped brains even though they have not been on the machine together yet.

Repeat for each loop. Every time you complete the loop, the new players have swapped minds again.

If at the end of fixing all the circles the 2 new players are swapped, they can go into the machine themselves to fix themselves.

Problem solved. Also, Farnsworth, get rid of the machine. It’s dangerous.

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That’s a really complicated theorem and it’s hard to explain simply. I suggest looking up some more information on the theorem and watching some more videos to get a better understanding.