Unshuffling a deck of cards

Van Cott, Cornelia A.; Wang, Katie

doi:10.2140/involve.2024.17.669

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Unshuffling a deck of cards

Cornelia A. Van Cott and Katie Wang

Vol. 17 (2024), No. 4, 669–687

DOI: 10.2140/involve.2024.17.669

Abstract

We investigate the mathematics behind unshuffles, a card shuffling technique closely related to classical perfect shuffles. To perform an unshuffle, deal all cards alternately into two piles and then stack one pile on top of the other. There are two ways this can be done (left pile on top or right pile on top), giving rise to the terms left shuffle ( $L$ ) and right shuffle ( $R$ ), respectively. We solve a generalization of Elmsley’s problem (a classic mathematical card trick) using unshuffles for decks with $2^{k}$ cards. We also find the structure of the permutation groups $⟨ L, R ⟩$ for decks of $2 n$ cards for all $n$ . The group coincides with the perfect shuffle group unless $n \equiv 3 (\mod 4)$ , in which case the group $⟨ L, R ⟩$ is equal to $B_{n}$ , the group of centrally symmetric permutations of $2 n$ elements, while the perfect shuffle group is an index-2 subgroup of $B_{n}$ .

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Keywords

perfect shuffle, permutation group, symmetric group

Mathematical Subject Classification

Primary: 20B35

Milestones

Received: 17 February 2023

Revised: 24 May 2023

Accepted: 6 June 2023

Published: 2 October 2024

Communicated by Vadim Ponomarenko

Authors

Cornelia A. Van Cott
Department of Mathematics and Statistics University of San Francisco San Francisco, CA United States

Katie Wang
University of California Berkeley, CA United States

Vol. 17, No. 4, 2024