  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.
