Welcome to the M³ Webpage!!!

M³: Mentoring for Minorities in Mathematics is part of the National Research Experience for Undergraduates Program (NREUP) funded by the Mathematical Association of America (MAA), the National Science Foundation Division of Mathematical Sciences (NSF-DMS), the National Security Agency (NSA), and the Moody's Foundation (they ceased funding this project in 2009). We have also received internal funding from the College of Science and Math, Department of Mathematics and Statistics, and the Office for Diversity at James Madison University.

The topics for Summer 2012 are one-player and two-player sowing games. We will be in session from May 14^th to June 29^th. John Johnson and Anthony Tongen mentored the following students:

• Mabel Adubofour
  
• Tyesha Hall
• Luis Parada
  
• Ben Sebuufu (Gordon)
  
• Samuel Thoronka
• Iris Yoon (Swarthmore)
  
• Rita Zevallos (Swarthmore)
  

Project 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.
  

Final Presentation - Thursday, June 21^st, 2012 at 10:00 am in Roop 103

The topic for Summer 2011 is single player Mancala-type games. We will be in session from May 16^th to June 24^th. John Johnson and Anthony Tongen mentored the following students:

• Amanda Fernandez
• Tyesha Hall
• Mikias Kidane
• Spencer Sims (Oakwood)
  
• Elf Bauserman

Project Summary:

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 M³ mathematicians, 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.

Final Presentation - Friday, June 24^th, 2011 at 1:30 pm in Roop 103


The two topics for Summer 2010 are single player Mancala-type games and PS (Parker-Sochacki/Power Series) method. We will be in session from May 17^th to June 25^th. Drs. Warne and Tongen will mentor the first two students in developing a Matlab GUI that solves differential equations using the PS method and Drs. Taalman and Tongen will lead the next three students in studying single player Mancala-type games:

• Rex Ford
• Anthony Chieco
• Durrell Lewis
• Brittney Dyson
• Fierra Mason

Project Summaries:

PS Method
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 M³ mathematicians, 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.

Final Presentation - Friday, June 25^th, 2010 at 1:30 pm in Roop 103

The topic for Summer 2009 is Mancala-like games.  We will be in session from May 11^th to June 19^th.  Drs. Thelwell and Tongen will mentor the following four students for this research project:

• Rex Ford
• David Melendez
• Juan Carlos Ortega
• Zurisadai Pena
• Melinda Vergara

Project 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?

Final Presentation - Friday, June 19^th, 2009 at 2:30 pm  in Roop 103


The topic for Summer 2008 was Dynamical Systems and Chaos.  We will be in session from May 5^th to June 13^th.  Drs. Thelwell and Tongen will mentor the following four students  for this research project:

• Jan Herburt-Hewell (hand in pocket)
• Michael Dankwa (baseball cap)
• Lianne Louizou (long hair)
• Juan Carlos Ortega (colorful shirt)

Project Summary:
During the first two days of the summer program, the students built a chaotic waterwheel (see picture to the right).  They then proceeded to learn dynamical systems so they could better understand, mathematically, the behavior of the wheel they built.  They are next going to build the first ever choatic sandwheel (at least to our knowledge) and derive the equations that govern the behavior of the new system and answer questions like:
Do you still see chaotic behavior?
Do you still see periodic behavior?
What is the qualitative behavior of the sand wheel?
What is the quantitative behavior the sand wheel?

Final Presentation - Friday, June 13^th, 2008 at 1:30 pm  in Roop 103

You can see the experimental progress that was made by examining the following two movies: Water and Sand (you will need quicktime to view these movies).


The topic for Summer 2007 was Discrete Mathematics with applications to Biology.  We were in session from May 14^th to June 22^nd.  The following four students were the primary investigators for this research:

• Charell Wingfield (left of James Madison)
• Michael Frempong (right of James Madison
• Jan Herburt-Hewell (behind James Madison)
• Michael Dankwa (in front of  James Madison)

Project Summary:

During the first two weeks of the M³ program, the participants will be introduced to discrete equations focusing on both analysis and numerical simulation. The director will present numerous open questions and ask the students to choose a couple on which to concentrate. The entire program (participants and director) will work together to solve the open questions pertaining to two-gender population models.

During the third and fourth week of this research experience, the students will perform a biological investigation of mate choice by male Betta splendens fish using video playback of females.  In this experiment, the students will determine whether males spend more time with and direct more courtship behaviors to a female with vertical lines than to a female without vertical lines.  This experiment will be in addition to the open questions started during the first two weeks.

During the final two weeks of this research experience, the students will conclude their research along with developing a mathematical model of mate choice in Betta splendens. The students will give a 50 minute presentation of their research results on the last day of the program.  After the conclusion of the research experience, the students will disseminate their results in July at the JMU Biology REU poster session, a poster and oral presentation in October at the Shenandoah Undergraduate Mathematics and Statistics Conference, and the director will be giving an oral presentation of the results in August at the Society for Mathematical Biology's annual meeting.

Final Presentation - Friday, June 22^nd, 2007 at 1:30 pm in Roop 103

• The influence of Female-Male Interactions on Offspring Sex Selection, Michael Dankwa and Jan Herburt-Hewell
• Infectious Disease Modeling of Human Papillomavirus, Michael Fremprong and Charell Wingfield

Click here to go to Anthony Tongen's webpage.

Thanks again to MAA, NSF-DMS, NSA, and Moody for their generous support of this project!! 

edited on 5/24/07