Mathematics often reveals itself in the most surprising of ways - even in something as everyday as shuffling a deck of cards. During graduate school I tackled a particular card problem that revealed amazing formulas and theorems. You can find that project on my GitHub page.
Prof. Jason Fulman, a mathematician at USC Dornsife College, delved into the intriguing world of card shuffling in his upcoming book, “The Mathematics of Shuffling Cards” co-authored with Persi Diaconis.
The Numbers Behind the Shuffle
Fulman’s exploration reveals some fascinating insights. Depending on the type of shuffle, the number of iterations needed to thoroughly mix a deck of 52 cards differs. Here’s what he found:
- Riffle Shuffle: The most efficient method, it takes only seven shuffles for a good mix.
- Smooshing: Spreading the cards randomly over each other requires between 30 and 60 seconds.
- Overhand Method: This method requires a staggering 10,000 repetitions!
It’s interesting to note that the end-use of the cards also plays a role. For instance, in blackjack, since card suits don’t matter, just four or five riffle shuffles is enough.
Fair Play and Card Dealing
Fairness in card games is primarily ensured through the dealing process. Two common methods - the cyclic and the back-and-forth - are typically used. While the cyclic method follows a repeating sequence, the back-and-forth alternates the direction, thereby speeding up the process and increasing randomness.
Beyond the Card Table
The relevance of card shuffling extends beyond card games and magic tricks. Computer scientists use insights from shuffling to determine optimal distribution of files in databases. Biologists study shuffle mixing time to understand the order of genes and estimate evolutionary distances between organisms.
As Fulman points out, studying “patience sorting,” can even help in understanding passenger airline boarding and traffic flow improvement.
The Future of Card Shuffling Studies
But there are many questions that mathematicians like Fulman continue to grapple with. For example, the exact number of shuffles required for the almost perfect shuffle used by Las Vegas casino dealers, or the optimal guessing strategy after a series of riffle shuffles, remain open problems.
It’s clear that given enough time, mathematicians will continue to delve into and decode these card-shuffling conundrums.
Noteworthy Links
New Book Release - “The Mathematics of Shuffling Cards” by Jason Fulman and Persi Diaconis is due to release on June 28th. It is available for pre-order at the American Mathematical Society, Amazon, and Barnes & Noble.
I have no affiliation with the authors of the book or these articles. I just love the subject and have experience writing my own proofs to arrive at similar conclusions.