Brain Games
All games
Brain Games › Games › Boxes
Boxes
Board · Classics · plays in your browser, no download
Join two dots with a line. Close the fourth side of a square and you take it and go again. It is drawn on the back of an envelope in every school in the world, and the correct strategy is the opposite of what it looks like: late in the game, you win by giving squares away.
Where it comes from
The game was published by Édouard Lucas, the French mathematician better known for the Tower of Hanoi and for the Lucas numbers, under the name la pipopipette. He did not claim it: he credited the idea to former students of his at the École Polytechnique. The date usually given is 1889, though the text most often cited is the one in his L’arithmétique amusante of 1895.
It then spent most of a century being treated as a doodle. The serious analysis is twentieth-century, and much of it is Elwyn Berlekamp’s: the theory of long chains, of parity, and of the double-cross, which together turn the endgame from guesswork into arithmetic. Berlekamp, who also co-wrote the foundational work on combinatorial game theory, eventually gave Dots and Boxes an entire book of its own.
What the theory says, in one line, is that the game is not about taking squares. It is about who is forced to open the first long chain, and that is decided long before anyone has scored anything.
How to play
| Draw | Click or tap between two neighbouring dots to join them. |
|---|---|
| Scoring | Complete the fourth side of a square and it is yours. |
| Going again | Close a square and you must move again. Two at once means one extra turn, not two. |
| Winning | Most squares when the grid is full. |
That “must” is the hinge of the whole game. You do not get to take a square and stop. Once you are in a chain, you are obliged to eat the whole thing and then open the next one for your opponent — unless you refuse.
The move that wins games
When your opponent opens a chain of, say, six squares, the obvious play is to take all six. Do that and you must then open the next chain yourself, and they take that one.
The alternative is the double-cross: take all but the last two, then deliberately play the line that splits those two rather than closing them. You give up two squares — and your opponent is now the one obliged to open the next chain. On any board with more than a couple of long chains, trading two squares for the rest of the game is overwhelmingly worth it.
Three things that help
Early on, never draw the third side of a square. That is simply a gift. Fill in safe lines — ones that leave every nearby square with at most two sides — for as long as they exist.
Count the chains before you are forced to move. When the safe moves run out, somebody has to open up. Knowing whether that will be you is the entire middlegame.
Give two away to take ten. The double-cross feels wrong every single time and is right almost every single time.