# Dots and boxes strategy: chains and the double-cross

> Why the third side of a box is a gift, how chains decide the game, and the double-cross move that lets you keep control near the end.

Dots and boxes looks like a game of quick grabs, but the end of a game is usually decided by a little counting. Once you know three ideas, the third side, the chain and the double-cross, you will see the same patterns on every board. Print a few grids from the [dots and boxes maker](https://paperoom.yunusemrevurgun.com/dots-and-boxes/) and try each idea as you read.

## The rules in one breath

Players take turns drawing a line between two neighbouring dots. When your line completes the fourth side of a box, you write your initial in it and must draw again. When no lines are left, the player with more boxes wins.

## Never draw the third side

A box with three sides drawn is a present for the next player: one line finishes it. Early in the game, the safe moves are the lines that do not make a third side anywhere. Draw those for as long as you can.

*Two sides of a box are safe. Drawing the third side lets your opponent finish the box.*
Each player avoids third sides, so the board slowly fills with boxes that have exactly two sides drawn. Eventually there are no safe moves left, and someone has to give something away. This is where the game is really decided.

## Chains

When safe moves run out, the boxes with two sides usually join up into corridors called chains. Once anyone draws a line in a chain, the next player can take every box in it, one after another, because each line they draw completes a box and earns another turn.

*Your opponent opens a chain of four boxes. Each of your lines completes a box, so you take all four and keep drawing.*
That means the end of the game is a series of chains being handed over. The player who has to open the chains is often the one who loses, because the other player collects them. The players are, in effect, playing for control: the right to be the one who receives each chain.

Short chains of one or two boxes behave differently from long chains of three or more, and good players count the long chains on the board before the safe moves run out.

## The double-cross

Here is the move that keeps control. When your opponent opens a long chain, you could take every box. But then you must draw again, and if every remaining move opens another chain, you would be handing the next chain straight back.

Instead, take all but the last two boxes, then draw the line at the far end of the chain. This leaves the last two boxes as a pair with one line missing between them. Your opponent takes both with that single line, and now they must move again, so they have to open the next chain for you.

*You took the first two boxes. Instead of taking the last two, you close the far end. Your opponent gets both boxes with one line, then has to open the next chain.*
Giving away two boxes to win a longer chain is often a good trade. In a closed loop the same idea works with four boxes: take all but the last four and leave them for your opponent.

## Counting for control

Whether you can force your opponent to open the first long chain depends on the number of moves left before the safe moves run out, which is closely linked to how many long chains the board will have. Mathematicians and strong players have written whole books on the rule of thumb for this, often called the long chain rule.

For a friendly game, a simpler habit goes a long way. Before the last safe moves are played, count the long chains and think about who will be forced to open the first one. If it looks like you, look for a move that changes the count, such as splitting a long chain into two short ones or joining two short ones.

## A short history

Dots and boxes was published by the French mathematician Édouard Lucas in 1889 as *la pipopipette*. You can read more about its history in the post on [dots and boxes and la pipopipette](https://paperoom.yunusemrevurgun.com/blog/dots-and-boxes-la-pipopipette/).

> Tip: Play your first games on a 4 × 4 grid of dots. With only nine boxes, you can see every chain forming and try the double-cross a few times in one sitting.

Paperoom's boards range from small grids to larger ones, with optional bonus boxes and a score table, so you can keep a tally across several games.

## Games in this post

- [Dots and boxes](https://paperoom.yunusemrevurgun.com/dots-and-boxes/): Square dot grids for two to four players: draw lines, claim boxes and add up the score on a printed table.

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