James Madison University
Title: Mancala-Type Games and Numerical Solution of ODEs
- Anthony Tongen
- Laura Taalman
- Paul Warne
- tongenal (at) jmu (dot) edu
- taal (at) math (dot) jmu (dot) edu
- warnepg (at) jmu (dot) edu
Dates of Program: May 17 - June 25, 2010
About 45 years ago, Fehlberg (1964) speculated that the power series method could be used to solve initial valued ordinary differential equation (IV ODE) problems with a level of accuracy that could not be achieved by other methods. About 13 years ago, a team of researchers at JMU took the first steps toward demonstrating that this is, indeed, true. Specifically, Parker and Sochacki (1996) demonstrated that power series can be used in a modified version of Picard's method and that this Parker-Sochacki/Power Series (PS) method is a practical, efficient and accurate algorithm for solving IV ODE problems.
The PS method, in addition to solving IV ODEs, has been used to solve integral-differential equations, find roots, and solve numerous application-oriented problems. Nonetheless, to date, the full potential of the PS method has not been realized because of the lack of a software tool that is accessible to the broader research community interested in these kinds of problems. Co-PI Warne has been a part of the JMU team using the PS method for the last ten years and he is at the forefront of innovation with the method. The second research group proposes to do the following:
• The primary objective is to develop a software tool that can be used to easily formulate IV ODE models and efficiently solve them using the PS method.
Single player Mancala-type games
Mancala is an ancient family of board games popular in Africa and Asia. While there are many 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; in fact, last summer's M3 students created an annotated bibliography of only 11 articles dealing with Mancala-type games, including computer science articles which were exhaustive in more than one way. Therefore, with M3mathematicians, we propose to do the following:
- Examine the single-player game called Tchoukaillon.
- We will investigate and probe the interesting mathematical patterns involved in Tchoukaillon boards.
- We will modify the rules of Tchoukaillon to include periodic boards, which will allow us to better relate Tchoukaillon to the next game to be studied, Tchuka Ruma.
- We will examine patterns of the modified-Tchoukaillon boards.
- Examine the single-player game called Tchuka Ruma.
- Last summer we discovered that multiple sequences of moves terminated in the same board state. Our goal is to quantify this particular 'equivalence class' and extend the results to other boards.
- A large problem with attempting to quantify sowing games is that the number of possible moves grows very quickly. However, if we are only interested in winning boards, we can prune the game tree by starting at the winning game board and working backwards, while at the same time starting from the opening board and moving forward. We can then join these two trees to eliminate numerous extraneous board states.
Student Researchers Supported by MAA:
- Anthony Chieco
- Brittany Dyson
- Reginald Ford II
- Durrell Lewis
- Fiearra Mason
More Information: www.math.jmu.edu/~tongen/NREUP/
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