The episode is based on a body swap scenario in which no pair of bodies can swap minds more than once. The proof demonstrates that after any sequence of mind switches, each mind can be returned to its original body by using two additional individuals who have not yet swapped minds with anyone.

Wikipedia article with context and full proof

There’s lots of legit physics and maths in the show. I believe some of the writers are maths professors and shit.

That’s really cool.

I know there are other instances of this scenario occurring in books and TV. I recall the episode of Stargate SG-1 (“Holiday”, S2:E17) where they body swap after visiting a planet and meeting Ma’chello, but had to figure out that they could not swap back directly with a previously switched pair.