Download this article
 Download this article For screen
For printing
Recent Issues

Volume 17
Issue 4, 543–722
Issue 3, 363–541
Issue 2, 183–362
Issue 1, 1–182

Volume 16, 5 issues

Volume 15, 5 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 8 issues

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Editors' interests
 
Subscriptions
 
ISSN 1944-4184 (online)
ISSN 1944-4176 (print)
 
Author index
To appear
 
Other MSP journals
Unshuffling a deck of cards

Cornelia A. Van Cott and Katie Wang

Vol. 17 (2024), No. 4, 669–687
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 2k cards. We also find the structure of the permutation groups L,R for decks of 2n cards for all n. The group coincides with the perfect shuffle group unless n 3(mod4), in which case the group L,R is equal to Bn, the group of centrally symmetric permutations of 2n elements, while the perfect shuffle group is an index-2 subgroup of Bn.

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