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A Better Way to Deal the Cards

by Mark A. Conger (University of Michigan) and Jason Howald (SUNY Potsdam)

Award: Lester R. Ford

Year of Award: 2011

Publication Information: American Mathematical Monthly, vol. 117, October, 2010, pp. 686-700.

Summary

Most card games deal out cards cyclically one at a time, but some games traditionally deal out three cards at a time, which supposedly improves the hands. Although a well shuffled deck would imply that the dealing method cannot matter; in practice, decks are not fully shuffled. The authors ask if the dealing method makes a difference. A discussion of some standard combinatorial tools augmented with numerical simulations shows that cyclically dealing cards one at a time definitely helps randomize the outcome. The authors next ask if cyclical dealing is the best strategy. A lattice path argument indicates why cyclical dealing cannot be the optimal dealing strategy; the authors conclude by giving numerical evidence for the value of back-and-forth dealing.

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About the Authors (From the MathFest 2011 Prizes and Awards Booklet)

Mark Conger received his B.A. from Williams College in 1989, where he worked with Frank Morgan on minimal networks. A student presentation at an MAA meeting in 1988 gave him an opportunity to see Persi Diaconis speak about shaved dice. After a stint as a computer programmer and another studying physics, Mark rediscovered that mathematics was where his heart lies.

As a student at the University of Michigan in 2002, he was given a choice by his future advisor, Divakar Viswanath, between reading a paper about physics or a paper by Diaconis about card shuffling. Influenced, perhaps, by the talk 14 years earlier, he opted for card shuffling and the pattern was set for the next 5 years. In 2007 he received what he believes to be the second known Ph.D. in card shuffling.

Mark now teaches at the University of Michigan. He enjoys woodworking and taking things apart.

Jason Howald grew up in Franklin, Indiana (and, a bit, in Cardwell, Montana) and graduated from Miami University of Ohio and the University of Michigan. Jason's Ph.D. work in Algebraic Geometry focused on multiplier ideals, but Mark Conger looked past all that and introduced him to the mathematics of card shuffling, which he has savored for a few years. Besides mathematics, Jason enjoys studying physics, logic, philosophy, and software design. Jason and his wife Cornelia Yuen have settled down at positions at SUNY Potsdam in (way) upstate New York, where they enjoy teaching, researching mathematics, juggling, creating origami, and baking.

Subject classification(s): Discrete Mathematics | Combinatorics
Publication Date: 
Monday, August 22, 2011

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