Unshuffling a deck of cards

TL;DR

This paper explores the mathematical structure of unshuffles, a specific card shuffle, providing solutions to classical problems and characterizing the permutation groups generated by unshuffles for decks of size 2n.

Contribution

It introduces the concept of unshuffles, analyzes the permutation groups they generate, and relates these groups to perfect shuffles, extending understanding of card shuffle mathematics.

Findings

Solution to a generalized Elmsley's Problem using unshuffles.

Characterization of the permutation group generated by unshuffles.

Identification of conditions under which the group differs from the perfect shuffle group.

Abstract

We investigate the mathematics behind unshuffles, a type of card shuffle closely related to classical perfect shuffles. To perform an unshuffle, deal all the cards alternately into two piles and then stack the one pile on top of the other. There are two ways this stacking can be done (left stack on top or right stack on top), giving rise to the terms left shuffle ($L$) and right shuffle ($R$), respectively. We give a solution to 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 $\langle L, R \rangle$ for a deck of $2n$ cards for all values of $n$. We prove that the group coincides with the perfect shuffle group unless $n\equiv 3 \pmod 4$, in which case the group $\langle L, R \rangle$ is equal to $B_n$, the group of centrally symmetric permutations of $2n$ elements, while the perfect shuffle group is an index 2 subgroup of $B_n$.