The winner is the player who pockets the most coins.
When a cut leaves a coin with no strings, the player "pockets" the coin and takes another turn. This game is played on a network of coins (vertices) joined by strings (edges). ĭots and boxes has a dual graph form called "Strings-and-Coins". Unusual grids and variants ĭots and Boxes need not be played on a rectangular grid – it can be played on a triangular grid or a hexagonal grid. 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. In combinatorial game theory, Dots and Boxes is an impartial game and many positions can be analyzed using Sprague–Grundy theory. If the other player also sacrifices, the expert has to additionally manipulate the number of available sacrifices through earlier play. 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 them the first chain long enough to ensure a 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. If the chains are long enough, then this player will win. 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. The opponent will take these two boxes and then be forced to open the next chain. Ī 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. But with their last move, they have to open the next, larger chain, and the novice loses the game. 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. At this point, players typically take all available boxes, then open the smallest available chain to their opponent. 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. An experienced player would create position 3 and win.įor 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. The "double-cross" strategy: faced with position 1, a novice player would create position 2 and lose. With A's next move, A gets all three of them and ends the game, winning 3–1. However, B must now add another line, and so B connects the center dot to the center-right dot, causing the remaining unscored boxes to be joined together in a chain (shown at the end of move 8). But the first player ("A") makes a sacrifice at move 7 and B accepts the sacrifice, getting one box. The second player ("B") plays a rotated mirror image of the first player's moves, hoping to divide the board into two pieces and tie the game.
Strategy Example game of Dots and Boxes on a 2×2 square board. A 5×5 board, on the other hand, is good for experts. When short on time, or to learn the game, a 2×2 board (3×3 dots) is suitable. The winner is the player with the most points. The game ends when no more lines can be placed. A point is typically recorded by placing a mark that identifies the player in the box, such as an initial. A player who completes the fourth side of a 1×1 box earns one point and takes another turn. Usually two players take turns adding a single horizontal or vertical line between two unjoined adjacent dots. The game starts with an empty grid of dots. It has gone by many other names, including the dots and dashes, game of dots, dot to dot grid, boxes, and pigs in a pen. It was first published in the 19th century by French mathematician Édouard Lucas, who called it la pipopipette. 2 player paper and pencil game A game of dots and boxesĭots and Boxes is a pencil-and-paper game for two players (sometimes more).
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