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Shuffle groups

Banff International Research Station for Mathematical Innovation and Discovery

The crux of a card trick performed with a deck of cards usually depends on understanding how shuffles of the deck change the order of the cards. By understanding which permutations are possible, one knows if a given card may be brought into a certain position. The mathematics of shuffling a deck of 2n cards with two â perfect shufflesâ was studied thoroughly by Diaconis, Graham and Kantor in 1983. I will report on our efforts to understand a generalisation of this problem, with a so-called ``many handed dealer'' shuffling kn cards by cutting into k piles with n cards in each pile and using k! possible shuffles. A conjecture of Medvedoff and Morrison suggests that all possible permutations of the deck of cards are achieved, as long as k \ne 4 and n is not a power of k. We confirm this conjecture for three doubly infinite families of integers, and also initiate a more general study of shuffle groups, which admit an arbitrary subgroup of shuffles.

Mathematics; Group theory and generalizations; Geometry; Algebraic structures

video/mp4

Author affiliation: University of Western Australia

2020-02-26

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/73601

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/