Dots and Boxes - Strategy

Strategy

At the start of a game, play is more or less random, the only strategy is to avoid adding the third side to any box. This continues until all the remaining (potential) boxes are joined together into chains – groups of one or more adjacent boxes in which any move gives all the boxes in the chain to the opponent. A novice player faced with a situation like position 1 in the diagram on the left, in which some boxes can be captured, takes all the boxes in the chain, resulting in position 2. But with their last move, they have to open the next (and larger) chain, and the novice loses the game.

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.

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 start the first long chain. Against a player who doesn't understand the concept of a sacrifice, the expert simply has to make the correct number of sacrifices to encourage the opponent to hand him the first chain long enough to ensure a win. If the other player also knows to offer sacrifices, the expert also has to manipulate the number of available sacrifices through earlier play.

There is never any reason not to accept a sacrifice, as if it is refused, the player who offered it can always take it without penalty. Thus, the impact of refusing a sacrifice need not be considered in your strategy.

Experienced players can avoid the chaining phenomenon by making early moves to split the board. A board split into 4×4 squares is ideal. Dividing limits the size of chains- in the case of 4×4 squares, the longest possible chain is four, filling the larger square. A board with an even number of spaces will end in a draw (as the number of 4×4 squares will be equal for each player); an odd numbered board will lead to the winner winning by one square (the 4×4 squares and 2×1 half-squares will fall evenly, with one box not incorporated into the pattern falling to the winner).

A common alternate ruleset is to require all available boxes be claimed on your turn. This eliminates the double cross strategy, forcing even the experienced player to take all the boxes, and give his opponent the win.

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.