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

April 25, 2007

Dots and Boxes

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.

Muse, July/August 2001, p. 36.

No comments: