Dots and Boxes: Rules, Chain Strategy and Small-Board Tips
Dots and boxes is a pencil-and-paper game for two players. It looks like a simple game of luck and patience, but there is a surprising amount of strategy hidden in it, and once you know the chain trick you will beat most casual players. Here are the rules, the key strategy and tips for small boards.
Rules
- Draw a grid of dots, for example 4 by 4 dots, which makes 3 by 3 boxes.
- Decide who starts with a coin flip.
- Players take turns drawing one line between two dots that are next to each other, across or down.
- If your line completes the fourth side of a box, write your initial in it and take another turn. You must keep drawing as long as you complete boxes.
- When every line is drawn, the player with more boxes wins.
Board Sizes
A board of m by n boxes has (m+1)(n+1) dots and m(n+1)+n(m+1) lines. The last column shows how many long chains the first player wants by the end of the game, from the long chain rule explained below:
| Boxes | Dots | Lines | Chains |
|---|---|---|---|
| 2 by 2 | 9 | 12 | Odd |
| 3 by 3 | 16 | 24 | Even |
| 4 by 4 | 25 | 40 | Odd |
| 5 by 5 | 36 | 60 | Even |
The Opening: Do Not Give Away Boxes
Early on, the only rule is: never draw the third side of a box, because your opponent will simply take it. Both players add “safe” lines until there are none left. At that point every move gives something away, and the board has broken up into chains (rows of boxes that can be taken one after another) and loops.
Chains and the Double-Cross
A long chain is a chain of three or more boxes. When your opponent opens a long chain for you, the greedy move is to take every box. The clever move is to take all but the last two, then draw the line that leaves those two as a pair for your opponent. This is called a double-cross.
Your opponent gets two boxes, but then has to move again and open the next chain for you. By giving away two boxes in each long chain, you keep control and take the bigger chains at the end. For a loop, you give away the last four boxes instead of two.
The Long Chain Rule
The mathematician Elwyn Berlekamp popularised a simple counting rule: the first player wants the number of dots plus the number of long chains at the end to be even, and the second player wants it to be odd. On the usual 4 by 4 dot board (16 dots), that means the first player wants an even number of long chains.
In practice, you count the likely chains as the safe moves run out, and try to join or split chains so the count suits you. It takes practice, but even a rough count helps.
Tips for Small Boards
- Start on 3 by 3 dots (2 by 2 boxes) to learn the chain trick; games take a minute.
- Count safe lines as the board fills up: whoever runs out of safe moves first usually has to open a chain.
- When you must give boxes away, give the shortest chain first.
- Do not double-cross a short chain of one or two boxes; just take them.
- Near the end, add up the boxes in each chain before you decide whether to keep control or grab everything.
Variations
- Bigger boards: 6 by 6 dots or more makes long games for older players.
- Triangle grids: claim triangles instead of squares.
- Misère: the player with fewer boxes wins.
- Team play: two pairs take turns, with partners alternating.
Where the Game Comes From
The French mathematician Édouard Lucas wrote about the game in 1889 under the name La Pipopipette. It has been a classroom and travel game ever since. If you enjoy games with hidden maths, read our tic tac toe strategy guide or try Chopsticks.
For a game with an even simpler winning trick, try Nim, played with a few piles of coins or sweets.
FAQ
How do you play dots and boxes?
Players take turns drawing a line between two neighbouring dots. Completing the fourth side of a box wins it and gives you another turn. The player with more boxes at the end wins.
Do you have to take a box if you can?
In the usual rules you may choose not to, and that choice is the heart of the double-cross strategy.
What is the best strategy in dots and boxes?
Avoid drawing the third side of a box early on, then use the double-cross: take all but two boxes of a long chain and leave the last two, so your opponent must open the next chain.
What is the long chain rule?
The first player wants the number of dots plus the number of long chains to be even; the second player wants it to be odd.
What size board should beginners use?
Start with 3 by 3 or 4 by 4 dots, which gives 4 or 9 boxes and short games.