Dots and Boxes

Example game of Dots and Boxes on a 2 square × 2 square board.

Dots and Boxes is a pencil-and-paper game for two players (sometimes more). It was first published in the 19th century by Édouard Lucas, who called it la pipopipette.[1] It has gone by many other names,[2] including the game of dots,[3] boxes,[4] dot to dot grid,[5] and pigs in a pen.[6]

Starting with an empty grid of dots, two players take turns adding a single horizontal or vertical line between two unjoined adjacent dots. The player who completes the fourth side of a 1×1 box earns one point and takes another turn. (A point is typically recorded by placing a mark that identifies the player in the box, such as an initial.) The game ends when no more lines can be placed. The winner is the player with the most points.[2][7] The board may be of any size. When short on time, a 2×2 board (a square of 9 dots) is good for beginners.[8] A 5×5 is good for experts.[9]

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. But the first player (A) makes a sacrifice at move 7 and 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 (shown at the end of move 8). With A's next move, player A gets them all and wins 3–1.

Strategy

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.

For most novice players, the game begins with a phase of more-or-less randomly connecting dots, where 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. At this point, players typically take all available boxes, then open the smallest available chain to their opponent. For example, a novice player faced with a situation like position 1 in the diagram on the right, in which some boxes can be captured, may take all the boxes in the chain, resulting in position 2. But, with their last move, they have to open the next, larger chain, and the novice loses the game.[2][10]

A more experienced player faced with position 1 will instead play the double-cross strategy, taking all but 2 of the boxes in the chain and leaving position 3. The opponent will take these two boxes and then be forced to open the next chain. By achieving position 3, player A wins. The same double-cross strategy applies no matter how many long chains there are: a player using this strategy will take all but two boxes in each chain and take all the boxes in the last chain. If the chains are long enough, then this player will win.

The next level of strategic complexity, between experts who would both use the double-cross strategy (if they were allowed to), is a battle for control: An expert player tries to force their opponent to open the first long chain, because the player who first opens a long chain usually loses.[2][10] Against a player who does not 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 sacrifices, the expert has to additionally manipulate the number of available sacrifices through earlier play.

In combinatorial game theory, dots and boxes is an impartial game and many positions can be analyzed using Sprague–Grundy theory. However, Dots and Boxes lacks the normal play convention of most impartial games (where the last player to move wins), which complicates the analysis considerably.[2][10]