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Encyclopedia > Dots and Boxes

Dots and Boxes (also known as Boxes, Squares, Paddocks, Square-it, Dots and Dashes, Dots, or, simply, the Dot Game) is a pencil and paper game for two players (or sometimes, more than two). In Mexico dots and boxes is called Timbiriche. In Guatemala, it is called Totito Chino (Chinese Tic-tac-toe). In Australia, it is commonly known as Paddocks. Games that can be played with only pencil and paper: Battleship was played as a pencil and paper game, long before Hasbro came out with a board game version. ...

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 earns one point and takes another turn. (The points are typically recorded by placing in the box an identifying mark of the player, such as an initial). The game ends when no more lines can be placed. The winner of the game is the player with the most points. Example game of dots and boxes on the 2-by-2 board. ... Example game of dots and boxes on the 2-by-2 board. ...

The board may be of any size. When short on time, 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. However, B must now add another line, and connects the center dot to the center-right dot, causing the remaining boxes to be joined together in a chain as shown at the end of move 8. With A's next move, A gets them all, winning 3–1.

## Contents

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, they have to open the next (and larger) chain, and the novice loses the game, 4–5. The double-cross strategy in dots and boxes. ... The double-cross strategy in dots and boxes. ...

An experienced player faced with position 1 instead plays the double-cross strategy, taking all but 2 of the boxes in the chain, leaving 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 the player will certainly win. Therefore, when played by 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. Combinatorial game theory (CGT) is a mathematical theory that only studies two-player games which have a position which the players take turns changing in defined ways or moves to achieve a defined winning condition. ... In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. ... In combinatorial game theory, the Spragueâ€“Grundy theorem states that every impartial game is equivalent to a nimber. ...

## 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. Gâ€²is the dual graph of G Dual graph is a term used in the mathematical study of graphs. ...

A variant played in Poland allows a player to claim a region of several squares as soon as its boundary is completed.

A Kansas City variant places the dots at random, giving players control as to the structure they assume so long as every completed shape have no more or less than four sides.

## References

Elwyn Ralph Berlekamp is professor of mathematics at University of California, Berkeley. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

Results from FactBites:

 Dots and Boxes - Wikipedia, the free encyclopedia (732 words) Dots and Boxes (also known as Boxes, Squares, Paddocks, Square-it, Dots and Dashes, Dots, or, simply, the Dot Game) is a pencil and paper game for two players (or sometimes, more than two). However, B must now add another line, and connects the center dot to the center-right dot, causing the remaining boxes to be joined together in a chain as shown at the end of move 8. Therefore, when played by 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.
 Dots and Boxes (323 words) If on their turn a player is able to complete the fourth side of a 1 x 1 box, the player places their initial in the box and places another line, which itself may complete another box. These last 2 are declined by making the move to complete the two box surround and leave the 2 box dividing line undrawn. When played between experts, boxes is a battle for control - by playing to force your opponent to start the first long chain, perhaps at the sacrifice of several short chains.
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