James Madison University

Title: Dynamical Systems and Games

Directors:

Anthony Tongen
Roger Thelwell

Email:

Dates of Program: May 11 - June 19, 2009

Summary: The focus of the research project is Mancala, an ancient family of board games popular in Africa and Asia. While there are many possible rule variants, this "sowing" type game is based on moving stone "seeds" from one container to others according to prescribed deterministic rules. Play can be surprisingly involved, with a large number of legal moves possible each turn. Surprisingly, there has been little published mathematical research of this very interesting game. John Conway developed his own variant, "Sowing," which led to some simple mathematical language and structure which could be further developed.

The primary research question for this project is "Is there an optimal strategy, against which no other competing strategy can win?" Mancala has been played for more than ten thousand years, suggesting that no obvious optimal strategy exists. However, with M³ mathematicians, we propose to do the following:

Change the number of containers and number of seeds to see when an optimal strategy exists. For instance, with four total containers and one seed initially in each container, the first player has an optimal strategy, against which the other player cannot win.
Explore and implement various rule sets and strategies numerically to build intuition.
Use a combinatorial approach to discuss the number of possible moves and strategies.
Consider the game as a discrete dynamical system. What type of analysis is possible using this abstraction?

Student Researchers Supported by MAA:

Rex Ford
David Melendez
Juan Carlos Ortega
Zurisadai Pena
Melinda Vergara

More Information: www.math.jmu.edu/~tongen/NREUP/

Program Contacts:

Bill Hawkins
MAA SUMMA
bhawkins@maa.org
202-319-8473

Michael Pearson
MAA Programs & Services
pearson@maa.org
202-319-8470