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Keeler’s Theorem and Products of Distinct Transpositions - MaRDI portal

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Keeler’s Theorem and Products of Distinct Transpositions

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Publication:5419496

DOI10.4169/AMER.MATH.MONTHLY.121.02.136zbMATH Open1301.20003arXiv1204.6086OpenAlexW1605510250WikidataQ58121239 ScholiaQ58121239MaRDI QIDQ5419496

Author name not available (Why is that?)

Publication date: 10 June 2014

Published in: (Search for Journal in Brave)

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.


Full work available at URL: https://arxiv.org/abs/1204.6086



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