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Title: M-cubed: Mentoring for Minorities in Mathematics

Directors:

• Anthony Tongen
• John Johnson

Email:

• tongenal (at) jmu (dot) edu
• johnsojh (at) jmu (dot) edu

Dates of Program: May 16 - June 24, 2011

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 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:

• Amanda Fernandez
• Tyesha Hall
• Mikias Kidane
• Spencer Simms

Program Contacts:

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

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