Using a Card Trick to Illustrate Fixed Points and Stability

Abstract

Mathematical ideas from number theory, group theory, dynamical systems, and computer science have often been used to explain card tricks. Conversely, playing cards have been often used to illustrate the mathematical concepts of probability distributions and group theory. In this paper, we describe how the 21-card trick may be used to illustrate concepts such as a discrete dynamical system, a fixed point, and the stability of a fixed point. The 21-card trick is a way of dealing cards in order to predict a card that is selected by a volunteer, within three moves. The 21-card trick and its two generalizations are examples of piece-wise linear, non-homogeneous, discrete dynamical systems, all of which have a global stable fixed point.

Keywords:

• 21-card trick
• discrete dynamical system
• fixed point
• stable
• ceiling function

Additional information

Funding

Work on this project was partially supported by Assigned Release Time (ART) that William Paterson University awarded to both authors.

Notes on contributors

Jyoti Champanerkar

Jyoti Champanerkar is Associate Professor of Mathematics at William Paterson University, Wayne, NJ. She received her Ph.D. in Applied Mathematics from the New Jersey Institute of Technology. Her research interests are dynamical systems and mathematical biology. She is keenly interested in imparting quality mathematics education, especially to future mathematics teachers of the country. She has presented at many conferences around the country, and in several professional development workshops to mathematics high school teachers in NJ and VA.

Mahendra Jani

Mahendra Jani is Professor of Mathematics at William Paterson University, Wayne, NJ. He received his Ph.D. in Mathematics from City University of New York. He has published research papers in algebraic topology and in enumerative combinatorics. He has presented his research work in national and international conferences in the US and abroad. He has received grants to give workshops to teachers to improve mathematics education in high schools and middle schools. He guides undergraduate mathematics majors in their capstone projects in order to provide them some research experience.