## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

## June 30, 2007

### Count and Capture

You're probably familiar with board games such as Monopoly, Candy Land, or Clue. These games are old enough that your parents likely enjoyed them when they were young, too. There are other games, which people have enjoyed across generations, that have been around even longer—some for thousands of years. One such game is awari, which originated in Africa. If you haven't tried it, you might be surprised at how tricky and involving this seemingly simple game can be.

Awari is an example of a "count-and-capture" strategy game. It belongs to a family of board games called mancala games. In its traditional form, the awari game "board" consists of two rows of six hollows, with four seeds in each hollow, or cup.

Two players sit across from each other, with six cups belonging to one player and six to the other. Each player aims to capture the most seeds.

On each turn, the first player takes all the seeds from one of the six cups on her side and, moving counterclockwise, adds one seed to each succeeding cup, until all the seeds are used up. The second player then takes the seeds from any one of the six cups on his side and does the same.

On any given turn, when a player drops her last seed in a cup on her opponent's side and that cup contains only one or two seeds (making a total of two or three seeds), she removes all the seeds from this cup, taking them out of the game. She also takes any seeds in cups immediately before the emptied cup if those cups now also total two or three. Players can take seeds only from their opponent's side. The game ends when one player has no seeds left on his side, and so he cannot move any seeds. The winner is the player who has captured the most seeds.

The awari board after player #1 takes her first turn (above).

The awari board after player #2 takes his first turn (above).

But how does this all come about? How do you win such a game?

A few years ago, computer scientists in the Netherlands turned to computers to find the answer. They calculated the best move and eventual outcome for all 889,063,398,406 positions that can possibly occur in the game—from having four seeds in every cup to having all 48 seeds in one cup. They found that, if you play perfectly, the game always ends in a tie.

Now, there's no mystery left. But, the good news is: Unless you're playing against a computer that can store and retrieve every possible move, you can still have a lot of fun playing the game and developing your own strategies for winning.

You can play awari against a computer at awari.cs.vu.nl, a Web site created by the computer scientists who solved the game.

Muse, July/August 2006, p. 28-29.