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MatheMUSEments
Count
and Capture
By Ivars Peterson
Muse, July/August 2006, p. 28-29.
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.
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The board after player #1 takes her first turn. |
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The board after player #2 takes his first turn. |
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.
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