# Dots and Boxes

(Redirected from Dots and boxes)

Dots and Boxes (also known as Boxes, Squares or Dots) is a pencil and paper game for two players (or sometimes, more than two). In Mexico dots and boxes is called Timbiriche.

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Dots-and-boxes.png
Game of dots and boxes on the 2×2 board.

Starting with an empty grid of dots, players takes turns adding a single horizontal or vertical line between two unjoined adjacent dots. A player who completes the fourth side of a box writes their initial in the box just completed and takes another turn. The game ends when no more lines can be placed. The winner of the game is the player with the most boxes.

The board may be of any size. 2×2 boxes is good for beginners, and 6×6 is good for experts. In games with an even number of boxes, it is conventional that if the game is tied then the win should be awarded to the second player (this offsets the advantage of going first).

The diagram on the right shows a game being played on the 2×2 board. The second player (B) plays the mirror image of the first player's move, hoping to divide the board into two pieces and tie the game. The first player (A) makes a sacrifice at move 7; B accepts the sacrifice, getting one box, but now the remaining boxes are joined together in a chain and A gets them all, winning 3–1.

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## Strategy

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Dots-and-boxes-chains.png
The double-cross strategy. Faced with position 1, a novice player would create position 2 and lose. An experienced player would create position 3 and win.

Beginners play more or less at random until all the remaining boxes are joined together into chains, whereupon any move gives away all the boxes in a chain to the opponent. A novice player faced with a situation like position 1 in the diagram on the right, in which some boxes can be captured, takes all the boxes in the chain, resulting in position 2. But with the extra move, player A has to open the next chain, and loses the game 4–5.

An experienced player faced with position 1 instead plays the double-cross strategy, taking all but 2 of the boxes in the chain: see position 3. This leaves the last two boxes in the chain for their opponent, but then the opponent has to open the next chain. By moving to position 3 player A wins 7–2.

The double-cross strategy applies however many long chains there are. Take all but two of the boxes in each chain, but take all the boxes in the last chain. If the chains are long enough then you'll win. So between experts, dots and boxes becomes a battle for control. An expert player tries to force their opponent to be the one who starts the first long chain.

In combinatorial game theory dots and boxes is very close to being an impartial game and many positions can be analyzed using Sprague-Grundy theory.

## Unusual grids

Dots and boxes need not be played on a rectangular grid. It can be played on a triangular grid or a hexagonal grid.

Dots-and-boxes has a dual form called "strings-and-coins". This game is played on a network of coins (vertices) joined by strings (edges). Players take turns to cut a string. When a cut leaves a coin with no strings, the player pockets the coin and takes another turn. The winner is the player who pockets the most coins. Strings-and-coins can be played on an arbitrary graph.

## References

• Play Dots-and-Boxes online: well.com (http://www.well.com/user/argv/java/dots.html), Yahoo (http://games.yahoo.com/games/login2?page=dt).
• Freeware Windows versions of Dots and Boxes: Dabble (http://www.ai.mit.edu/~jpg/dabble/) or ossiemanners.co.uk (http://www.ossiemanners.co.uk).

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