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Keeler’s Theorem and Products of Distinct Transpositions

Ron Evans, Lihua Huang and Tuan Nguyen
The American Mathematical Monthly
Vol. 121, No. 2 (February 2014), pp. 136-144
DOI: 10.4169/amer.math.monthly.121.02.136
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Page Count: 9
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Keeler’s Theorem and Products of Distinct Transpositions
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Abstract An episode of the television series 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 Sn. 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 Sn.

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