The Mathematics of Shuffling Cards

The book under review is an interesting read that explores the mathematics associated with the many different ways to rearrange a deck of cards. Prior to reading this book, I had only used the standard riffle shuffle where the cards are split roughly in half and combined together again in uneven packets first from one half and then the other. But there are also overhand shuffles and smoosh shuffles along with variations such as the face up-face down riffle shuffle or the over-under shuffle. The mathematical representations of shuffles is also quite varied. Using Markov chains, hyperplane arrangements, the signed symmetric group, P-partitions and others to describe shuffles allows the authors to apply results from these areas to shuffles and vice versa.

 

I was surprised at the breadth of this book, but as the authors note, these ideas have been developed over the course of two mathematicians’ careers. The writing is clear with the authors defining terms as needed and specifically showing connections between shuffling and the different disciplines in mathematics, including combinatorics, probability, and representation theory. The proofs are clear and leave little to the imagination. The appendices within chapters are a nice touch, making it more self-contained within the chapter and easy to find additional information. The book is well-sourced, with each chapter containing discussions of related works and citations for further reading. However, I would prefer that citation numbers always appear with the author names as it can be tiresome to flip back to the bibliography.

 

The book is somewhat technical, requiring upper-level algebra and probability knowledge, and is not recommended for casual readers. The audience of card enthusiasts or magicians and mathematicians may be narrow, but those interested in the surprising interplay between mathematical disciplines will find this book particularly engaging. The book is structured in such a way that readers can skip ahead to the chapters they are most interested in, although reading the first few sections of each chapter provides a broad look at all the mathematical concepts that card shuffling could address. Overall, this book is a great resource for those with a strong background in mathematics who are interested in the deeper mechanics of the familiar card shuffle.

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Tricia Muldoon Brown (tmbrown@georgiasouthern.edu) is a Professor of Mathematics at Georgia Southern University with interests in combinatorics, recreational mathematics, and sports.