# Dots and Boxes

**Dots-and-Boxes** (also known as

**Boxes** or

**Dots**) is a

pencil and paper game for two players.

Starting with an empty grid of dots (of any size, but often 10x10), each player takes turns to add a single line between any two adjacent dots, along the grid lines (i.e. not diagonally). 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. The player's turn continues until he places a line which does not complete a box.

The game ends when no more lines can be placed. The winner of the game is the player with the most boxes.

In Mexico this is called Timbiriche.

In game theory literature, analysis of Dots-and-Boxes uses the concepts developed in the game Nim.

At its surface, you would think that there isn't much strategy to this game. In the opening phase of the game, no boxes are claimed. Eventually one player is forced to give their opponent a chance to take a box - or a series of boxes. In beginners' play the chains are taken by alternate players with the longest chain taken last.

If a *long chain* is defined as one with 4 boxes, and if there are many long chains present on the board, a winning strategy exists. In this strategy, when presented with a chain, you claim the entire chain except for the last 2 boxes. These last 2 are declined by making the move to complete the two box surround and leave the 2 box dividing line undrawn. This ends your turn. Your opponent then draws the dividing line to take the 2 boxes and has to start another long chain. You win provided there are enough long chains.

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*.

Dots and boxes need not be played on a rectangular grid. It can be played on a triangular grid or a hexagonal grid. It can be played on a grid with selected vertices or edges deleted. It can be played on a grid that corresponds to a non-Euclidean geometry.