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MatheMUSEments
Dots and Boxes
By Ivars Peterson
Muse, July/August 2001, p. 36.
The familiar game of Dots and Boxes seems simple. But
it's really a lot trickier than it looks.
Mathematician Elwyn Berlekamp of the University of
California at Berkeley first learned to play the game
when he was in grade school. He has been studying it
ever since and has even written a book about strategies
for playing it.
The playing field is a rectangular or square grid
of dots. You and your opponent take turns joining two
dots with a line. When a player adds the fourth line
that completes a box, he or she claims the box by
marking it with an initial, then takes an extra turn.
If the same move happens to close two boxes, the player
claims both boxes but still gets only one bonus move.
The game ends when all boxes are taken. The player who
closed more boxes wins.
Beginners tend to connect dots at random to see
what happens. It usually doesn't take long to figure
out that it's a good idea to avoid adding the third
side to a square. If both players avoid third sides,
they end up with a kind of maze made up of several
chains of connected boxes.
That's when things get interesting. Your aim is
to force your opponent to be the first to add a third
line to a square that belongs to a chain. Then you can
claim all the boxes in the chain. But it pays not to
be greedy. It's better to make a final move that leaves
two boxes of a long chain unclaimed. The other player
can claim the "gift" boxes, but then must add a bonus
line somewhere else--and if all that is left are other
chains, this lets you capture another one.
Berlekamp has worked out strategies that allow an
alert player starting second to always win a game
played on a three-by-three grid of squares. No such
guarantee is possible if you play on a five-by-five
grid of squares. Amazingly, there are so many possible
moves that no person or computer can try all of them
to find the perfect set of moves.
So if you want to win, try the three-by-three grid.
But if you want to experiment, the five-by-five grid
is the ideal playing field.
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