Theorem 1. Let n ∈ N, n ≥ 2. The inverse of any permutation in Sn can be written as a product of distinct transpositions in Sn+2 \ Sn.
Mathematically inclined aficionados of the cult animation series Futurama will no doubt recognise the theorem above – it was first posited by Ken Keeler in the 2010 Writers Guild* Award-winning episode ‘The Prisoner of Benda’.
“Professor Farnsworth and Amy build a machine that can swap the brains of any two people. The two use the machine to swap brains with each other, but then discover that once two people have swapped with each other, the machine does not swap them back. More characters get involved until the group is thoroughly mixed up, and they start looking for ways to return to their own heads.”
“Clearly, the problem of undoing what the machine has done may be studied using permutations; each brain swap can be described by a transposition in Sn, where n is the number of characters involved in the brain-swapping.”
As a result of their permutational analyses, the team conclude that :
“ – as long as the brain-swapping machine Professor Farnsworth and Amy build swaps a prime number of brains cyclically, they can always fix the chaos created by incorporating enough extra characters to the mix.”
* Note: The Writers Guild is officially apostrophe-less.