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.


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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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.", I. "Dots-and-Boxes for Experts." Sci. Amer. 284, 102-103, Jan. 2001.Wilson, D. "Dots-and-Boxes Analysis Index."

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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.

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