From chapter eight of Marc Chamberland’s Single Digits:
How many times should you shuffle a deck of cards so that they’re well-mixed? Gamblers know that three or four times is not sufficient and take advantage of this fact. In 1992, researchers did computer simulations and estimated that seven rough riffle shuffles is a good amount. They took their research further and figured out that further shuffling does not significantly improve the mixing. If the shuffler does a perfect riffle shuffle (a Faro shuffle), in which s/he perfectly cuts the deck and shuffles so that each card from one side alternates with each card from the other side, then a standard 52-card deck will end in the same order that it started in after it is done 8 times.
The numbers one through nine have remarkable mathematical properties and characteristics. For instance, why do eight perfect card shuffles leave a standard deck of cards unchanged? Are there really “six degrees of separation” between all pairs of people? And how can any map need only four colors to ensure that no regions of the same color touch? In Single Digits, Marc Chamberland takes readers on a fascinating exploration of small numbers, from one to nine, looking at their history, applications, and connections to various areas of mathematics, including number theory, geometry, chaos theory, numerical analysis, and mathematical physics.
Each chapter focuses on a single digit, beginning with easy concepts that become more advanced as the chapter progresses. Chamberland covers vast numerical territory, such as illustrating the ways that the number three connects to chaos theory, an unsolved problem involving Egyptian fractions, the number of guards needed to protect an art gallery, and problematic election results. He considers the role of the number seven in matrix multiplication, the Transylvania lottery, synchronizing signals, and hearing the shape of a drum. Throughout, he introduces readers to an array of puzzles, such as perfect squares, the four hats problem, Strassen multiplication, Catalan’s conjecture, and so much more. The book’s short sections can be read independently and digested in bite-sized chunks—especially good for learning about the Ham Sandwich Theorem and the Pizza Theorem.
Appealing to high school and college students, professional mathematicians, and those mesmerized by patterns, this book shows that single digits offer a plethora of possibilities that readers can count on.