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Dots and Boxes


Dots and boxes is a two-person game based on a rectangular lattice of points. Each player, in turn, draws a horizontal or vertical line connecting two adjacent points. Whenever placement of a line complete a single 1×1 square, the square is colored in, the player scores one point, and the player having completed the square receives an additional move.

Edge-drawing

In the first part of the game, the players will avoid adding the third side of a square.

Opening a chain

As soon as this is no longer possible, the player (A) who makes the next move will enable his opponent B to form a sequence of squares, called a chain. Player B will end his move by adding a segment which does not close any square and possibly opens a new chain to A.

A wins

The best move is not necessarily the one closing as many squares as possible and opening the shortest chain to the opponent.

Forming a domino

Sometimes it is more convenient to force the other player to open a chain that it is longer than the one currently available. This can be achieved, for example, by completing all but two squares of the available chain and forming a rectangle (a so-called domino) out of the last two.

The winning strategies of this game are very complex and have not yet been completely classified. In 2001, David Wilson showed the 3×5 game is a win for the second player.


This entry contributed by Margherita Barile

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References

Berlekamp, E. The Dots and Boxes Game: Sophisticated Child's Play. Wellesley, MA: A K Peters, 2000.King, D. "Dots and Boxes--Dots and Hexagons." http://www.drking.worldonline.co.uk/hexagons/dots/.Stewart, I. "Dots-and-Boxes for Experts." Sci. Amer. 284, 102-103, Jan. 2001.Wilson, D. "Dots-and-Boxes Analysis Index." http://www.cae.wisc.edu/~dwilson/boxes/.

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Dots and Boxes

Cite this as:

Barile, Margherita. "Dots and Boxes." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/DotsandBoxes.html

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