Navigation Fritz Dooley Home Mancala Home Mancala Game Rules Mancala Resources Mancala Game Theory Thesis Introduction Terminology Structural Analysis of Mancala Human versus Computer Mancala Mancala Strategies Best Opening Move in Mancala Conclusion Exhibit 1: Rules of the Game Exhibit 2: Number of Game States email me: fritz@fritzdooley.com 
Terminology



L 
K 
J 
I 
H 
G 


Y 

4 
4 
4 
4 
4 
4 

M 
4 
4 
4 
4 
4 
4 



A 
B 
C 
D 
E 
F 


The numbers indicate the number of pebbles in each bin. The letters are labels for each bin. An opening in which Mi first plays her pebbles from Bin C, then makes another move from Bin E, followed by Yo playing her pebbles in Bin H, then making another move from Bin G, is written simply as CEHG. Following this opening, the board looks like this:


L 
K 
J 
I 
H 
G 


Y 
1 
6 
6 
6 
7 
1 

2 
M 
4 
4 

5 
0 
6 



A 
B 
C 
D 
E 
F 


We have experimented with different ways of measuring the strength or payoff of a given move. The measure we have found most useful is the change in the difference in the number of pebbles in the two mancalas. We call this the “payoff” of a move or turn. It is expressed as a nonnegative number from the perspective of the current player (the current player can never put pebbles in the other’s mancala, or remove pebbles from her own). A player’s “advantage” is the absolute difference in the number of pebbles in each mancala, expressed as positive or negative from the perspective of the current player.
Thus, in the above example, the CE play had payoff of 2 and gave Mi an advantage of 2. Yo’s subsequent HG play had payoff of 1 and resulted in an advantage (to Yo) of –1.
A pebble is “in play” if it has not yet landed in a mancala.
Though the formal game rules define the winner as the person with the most pebbles in her mancala when either player runs out of pebbles to play (and the other player places the remaining pebbles from her side of the board into her mancala), it follows logically that the winner of the game is the first player to have in her mancala one half of all the pebbles plus one. A variation of the rules suggests keeping count of the total number of pebbles captured in each game, and summing the totals for tournament play. In this way, some wins are better than others. For purposes of this study, we looked at winning as a binary event (with the exception of an occasional draw), and thus considered the game over as soon as a player had captured more than half of the pebbles (at least 25 in a game starting with four pebbles per bin).
A “state” is a snapshot of the board at a given point in a game, plus information about whose turn is next. The same state can potentially be reached by many different paths through the game tree. Each fork in the game tree is a “node.” A node consists of a state, and a path (series of moves) traveled to reach that state.
A state is a “winning state” if there exists a strategy by which a hard player taking the next turn can be assured of a win. It is a losing state if it offers no such strategy, and for every strategy available, results in a winning state for the next player. It is a drawn state if it offers no winning strategy, but offers at least one strategy whereby the next player can be assured of at least a draw.
The character of a state is its property of being a winning, losing, or drawn state.
A state is “indeterminate” if it is neither winning, nor losing, nor drawn. I.e., the game could “go either way,” even with both players playing perfectly. This is not the same as a state whose character is simply unknown; it is one whose character is positively undefinable as “winning” or “losing.” I will later show that this theoretical class of state does not exist.
The first state in a game in which one mancala contains at least one half of all the pebbles, plus one, or in which each mancala contains exactly half the pebbles. (A state in which a player has no more pebbles on her side of the board automatically meets one of these criteria, as at that point, all the remaining pebbles are swept into the other player’s mancala.) At this point the game has been won or drawn, and there is either no need or no possibility of further play.
(c) 2008 by Fritz Dooley