Unshuffling a deck of cards
type: Item
instance of: scholarly article
title: Unshuffling a deck of cards
author: Cornelia A. Van Cott
author: Katie Wang
published in: Involve
publication date: 2024-11-01T00:00:00Z
review text: Perfect shuffle is a well-known card shuffling method. Given a deck of \(2n\) cards, one separates the upper half and lower half into two piles and then interleaves the two piles one card at a time. There are two different ways to do the interleaving: the \textit{in shuffle} (denoted by \(I\)) is the one where the new bottom card comes from the upper pile and the \textit{out shuffle} (denoted by \(O\)) is the one where the new bottom card comes from the lower pile.\N\NPerfect shuffle can be used in magic tricks. However, it is difficult to master and it is natural to wonder whether there are easier shuffling techniques that can do the same job.\N\NThe unshuffle was introduced by [\textit{D. Ensley}, Math Horiz. 11, No. 3, 13--16 (2004; \url{doi:10.1080/10724117.2004.12021753})] with this motivation. Given a deck of \(2n\) cards, one deals the cards alternately into two piles with the top card going to the left pile and then stacks one of the piles onto the other. There are two different ways to do the stacking: the \textit{left shuffle} (denoted by \(L\)) is the one where the left pile gets placed at the top and the \textit{right shuffle} (denoted by \(R\)) is the one where the right pile gets placed at the top.\N\NOne can view the aforementioned shuffling methods \(I,O,L,R\) as elements of the symmetric group on \(2n\) letters. The structure of the subgroup \(\langle I,O\rangle\) generated by the perfect shuffles is already known by \textit{P. Diaconis} et al. [Adv. Appl. Math. 4, 175--196 (1983; Zbl 0521.05005)]. In Theorem 1.1 of the paper under review, the authors determined the structure of the subgroup \(\langle L,R\rangle\) generated by the unshuffles. In particular, they proved that \(\langle L,R\rangle = \langle I,O\rangle\) when \(n\not\equiv 3\) (mod \(4\)) and \([\langle L,R\rangle:\langle I,O\rangle] =2\) when \(n\equiv 3\) (mod \(4\)). Therefore, from a group-theoretic point of view, unshuffle can realize all the card arrangements arising from perfect …
reviewed by: Cindy (Sin Yi) Tsang
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 7937657
zbMATH Keywords: perfect shuffle
zbMATH Keywords: unshuffle
zbMATH Keywords: card shuffling
zbMATH Keywords: permutation group
zbMATH Keywords: symmetric group
zbMATH Keywords: Elmsley's problem
MaRDI profile type: Publication
cites work: Trailing the dovetail shuffle to its lair
cites work: The Mathematics of the Flip and Horseshoe Shuffles
cites work: Magical Mathematics
cites work: The mathematics of perfect shuffles
cites work: A look at generalized perfect shuffles
cites work: The Monge shuffle for two-power decks
cites work: Groups of Perfect Shuffles
cites work: Q4134158
cites work: Q4373814
cites work: The generalized faro shuffle
cites work: Q4906186
cites work: Moving Card i to Position j with Perfect Shuffles
DOI: 10.2140/INVOLVE.2024.17.669
description: scientific article; zbMATH DE number 7937657