Dots and Boxes · Complete guide
Dots and Boxes: how to play, and why the endgame is everything
A children's doodle invented by a French mathematician in 1889 — with a rigorous theory of chains, loops and sacrifices that most players never suspect exists.
Dots and Boxes starts as a grid of dots. Draw a line between two adjacent dots; close the fourth side of a square and you claim it — and take another turn. When every line is drawn, most boxes wins.
Almost everyone plays it badly, and in exactly the same way: they take every box they can. That is the trap. Taking a whole chain leaves you on the move with nothing safe to play, so you must open the next chain for your opponent, and the next, until the board is gone.
The fix — the double-cross — is to take all but the last two boxes of a chain and then hand those two over, keeping your opponent on the move. You lose two boxes and gain control of the endgame, which is usually worth several times as much.
Underneath sits a real mathematical theory, developed by Elwyn Berlekamp: chains, loops, and a parity rule linking the number of long chains to who benefits. Once you know it, the game is decided in the opening — by the shape of the board you are steering towards, not by the boxes.
How to play
Draw a line, close a box, go again. The rules take fifteen seconds — and then the endgame turns into one of the most surprising strategy problems in any paper-and-pencil game.
Where it comes from
Invented by the mathematician Édouard Lucas in 1889, played badly by millions of schoolchildren ever since, and given a rigorous theory by Elwyn Berlekamp a century later.
- Dots and Boxes was published in 1889 by Édouard Lucas as La Pipopipette.
- Lucas also invented the Tower of Hanoi and gave his name to the Lucas numbers.
- It spread worldwide as a paper game with dozens of local names.
- Elwyn Berlekamp developed its combinatorial theory, published in 2000.
How to win
Why taking every box loses, how the double-cross sacrifice wins the endgame, and the long-chain parity rule that decides games before a single box is claimed.
- The game is decided by who is forced to open the first chain when safe moves run out.
- Never draw a line that gives a box its third side unless you have no choice.
- The double-cross — declining the last two boxes of a chain — hands the move back and wins long chains.
- Declining a loop costs four boxes rather than two, which changes the arithmetic.
Dots and Boxes strategy: the double-cross and the long-chain rule →
Common questions
Should I always take a box when I can?
No. Taking every box in a chain leaves you with only bad moves. Take all but the last two and hand the move back — that is the double-cross, and it wins games.
What is the long-chain rule?
A parity rule: on a 5×5 board the first player wants an odd number of long chains and the second an even number, and it reverses on even boards. You steer towards it in the opening.
What is the difference between a chain and a loop?
A chain opens from one end and costs two boxes to decline. A loop is a closed ring and costs four, which changes the arithmetic of who can keep control.
Can Dots and Boxes be a draw?
Only on boards with an even number of boxes, such as 4×4. With an odd number someone must come out ahead.
Does the opening matter?
Yes, though not for boxes. The opening determines how many long chains the board will produce, and that number decides the game.
Guides
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