Ancient Board Games "Mancala" and "Go" and Their Mathematical Proofs

Board games have been played for at least 4,000 years, and there is some knowledge of games played in ancient Egypt, Mesopotamia, Assyria, Cyprus, and Crete. The rules of most ancient games remain unknown. The first systematic description of games and their rules is a report ordered by King Alfonso X of Castilein 1283. Various sources over the next 100 years indicate that both board games and card games were widespread enough in Europe to prompt governments to try to control them.

Mancala is one of the oldest board games in the world. One theory is that mancala is an African invention, perhaps from North Africa. Mancala refers to a type of counting game of which Wari is a specific example, dating back to ancient times, known in Egypt and throughout Africa and Asia, and probably brought to Europe by sailors. Only two mancala games have an established tradition of championship competition: warri (also known as awel, awari, oware and wars) in West Africa and the Caribbean, and bao (also known as solo, bau, and mbau) in East Africa.

The game board consists of 2n+2 bins, n playing bins per player, plus one scoring bin, mancala, per player. Any number of stones, j, are placed in each of the n playing bins with . The first player picks up all the stones in one of his n playing bins and starts to place them one at a time into consecutive bins moving counter-clockwise around the board, including his own mancala, but not his opponent's mancala. The player picks up the stones in the bin in which his last stone is placed and continues on in the same manner. If the last stone is placed in any empty bin on his side of the board then he captures all the stones in his opponent's bin directly across from that bin. All captured stones plus the capturing stone get placed in his mancala. The game ends when one of the players runs out of stones in his playing bins. When this happens, the other player gets to place any stones remaining in his bins into his mancala.

MATHEMATICAL PROOF: A mathematical proof consists of steps that lead to a conclusion. Therefore, a mathematical proof for a board game consists of the sequence of steps one needs to take to win the game.

The mathematical proof for mancala consists of the following: There are forty-eight stones on the board, so the first player needs at least twenty-five to win the game. There are many possible sequences of moves that the first player can make. Thus it can be concluded that the first player can always win if he knows the proper sequence of moves. The number of stones that may be collected on the first move and the number of possible first moves made by each player can be determined as shown in the following theorem.

Theorem 1: Given a Mancala board with n playing per player and j stones per bin, where
1.
On the first move, the player is guaranteed one stone if the player moves from bin k, k2.
On the first move, the player receives no stone if the player moves from bin k, k>j.
3.
On the first move, the player is guaranteed another turn is the player moves from bin k, k.
a.
The player is guaranteed a total of two stones if the player next moves from bin k, k.
b.
The player is guaranteed a total of one stone if the player next moves from bin k, k>.

Proof: 3 < j < n

Assume player one moves from bin k, kAlthough the game of Go originated in China, it became popular in Japan. Introduced sometime around 700AD, Go was at first popular mainly among nobles and Buddhist monks, but over the next 1,000 years it proliferated. During the centuries of clan war, playing Go was considered rigorous moral and intellectual training; in peacetime, it was a respectable recreation. It is often compared and contrasted with chess. Go is the only game left that has not been solved by the computer.

As in chess, the pieces are colored black and white, but in Go black plays first. It is usually played on a 19 X 19 board by two players who alternately place black and white stones on the intersections of the squares. The board starts blank and pieces once played are not thereafter moved except to be taken off as prisoners. Pieces are captured singly or
en masse by being surrounded so that they are not connected to any adjacent open intersection. The object of the game is to capture one's opponent's stones by surrounding them, and thereby control large areas of the board.

MATHEMATICAL PROOF: In Go, the possibilities are almost infinite. There are 32,940 opening moves, after symmetry is taken into account, 992 of which are deemed strong. Estimates of the number of possible board configurations vary but are typically on the order of 10174.

Efforts to figure out the game led to the notion of a "sum" that would numerically evaluate the strength of any move. The sum accounts for each move's effects on the match's smaller battles, or components, and assigns a value to their outcome. Knowing the value of each component tells you the value of the sum. If the sum generated by a given move is positive, it's a winning move.

Instead of trying to solve the entire problem all at once, one must break the problem into smaller pieces. In theory, you can take 100 moves and break them down into 50 games each two moves long. You just find the value of each and add them up. This kind of simplifying approach is the basis for the mathematical field of combinatorics.

Reference
Beasley, J.D. (1990). The Mathematics of Games. Oxford: Oxford University Press.