Jackeea:

*smashes head off computer* Just solve mine already.

Wikipedia:

In an interview, David X. Cohen revealed that the episode writer Ken Keeler, a PhD mathematician, penned and proved a theorem based on group theory, and then used it to explain the plot twist in this episode.[1] However, Keeler does not feel it carries enough importance to be designated a theorem, and prefers to call it a proof.[2] Cut-the-Knot, an educational math website created by Alexander Bogomolny, refers to this proof as the "Futurama Theorem."[3]
The episode is based on an irreversible body swap scenario. The proof demonstrates that, no matter how many minds have been swapped, each can be returned to its original body by using only two additional individuals who have yet to swap minds with anyone, and without swapping between the same pair of bodies more than once.
A formal statement is as follows:
Let A be a finite set, and let x and y be distinct objects that do not belong to A. Any permutation of A can be reduced to the identity permutation by applying a sequence of distinct transpositions of A?{x,y}, each of which includes just one of x, y.[3]
The proof appears on the blackboard in the episode. The proof reduces to treating individual cycles separately, since all permutations can also be represented as products of disjoint cycles.[3] So first let ? be some k-cycle on [n]={1 ... n}. Without loss of generality, write:

Introduce the two new symbols x and y, and write:

Let (a b) be the transposition that interchanges a and b. For any i ? {1 ... k-1} let ? be the permutation obtained as the (left to right) composition:

Note that these are distinct transpositions, each of which exchanges an element of [n] with one of x,y. By routine verification:

That is, ? reverts the original k-cycle to the identity and leaves x and y switched (without performing (xy)).
Next let ? be an arbitrary permutation on [n]. It consists of disjoint (nontrivial) cycles and each can be inverted as above in sequence after which x and y can be switched if necessary via (xy), as was desired.