Keeler's theorem and products of distinct transpositions

Abstract

An episode of Futurama features a two-body mind-switching machine which will not work more than once on the same pair of bodies. After the Futurama community engages in a mind-switching spree, the question is asked, "Can the switching be undone so as to restore all minds to their original bodies?" Ken Keeler found an algorithm that undoes any mind-scrambling permutation with the aid of two "outsiders." We refine Keeler's result by providing a more efficient algorithm that uses the smallest possible number of switches. We also present best possible algorithms for undoing two natural sequences of switches, each sequence effecting a cyclic mind-scrambling permutation in the symmetric group S_n. Finally, we give necessary and sufficient conditions on m and n for the identity permutation to be expressible as a product of m distinct transpositions in S_n.