Dots and Boxes

How to play

Take turns drawing one line between two neighbouring dots. Whoever draws the fourth side of a box claims it and immediately goes again, so a single move can cascade into a long chain of boxes. The game ends when every line is drawn and the player with more boxes wins. It looks like a doodle and is really a counting game: the interesting part is not taking boxes, it is deciding who is forced to open the next chain.

Tips

• Avoid drawing the third side of any box. That hands the box to your opponent for free.
• The endgame is about chains. Count how many chains are left and who has to open the first one.
• Give up two boxes on purpose to keep control. Taking every box you can often loses the long chain.
• An even number of long chains usually favours whoever did not open the first one.
• Early on, play lines that touch as few boxes as possible and let your opponent run out of safe moves.

Why the endgame is a counting game

Dots and Boxes was described by the French mathematician Edouard Lucas in 1889, and it has attracted serious mathematical attention ever since, most famously from Elwyn Berlekamp, who wrote an entire book on it. The reason it rewards study is that the naive strategy, take every box you can, loses to a player who understands chains.

Late in the game the board decomposes into chains, runs of boxes where completing one opens the next. Whoever is forced to open a chain gives all of it away. The winning technique is control: when handed a chain, take all but the last two boxes, then give those two away with a single move that forces your opponent to open the next chain. The player who keeps control usually wins even while giving boxes away.

Playing beyond the obvious

• Never draw the third side of a box unless you must. It gives the box to your opponent.
• Count the chains before the safe moves run out. An even number of long chains favours the player who did not open the first one.
• When given a chain, consider leaving the last two boxes. Sacrificing two to keep control beats taking all and opening the next chain yourself.
• Early moves should touch as few boxes as possible, stretching the phase where safe moves still exist.
• The computer on Hard picks the cheapest sacrifice when it has to open something. Make its cheapest option expensive.

Common questions

Why should I ever give boxes away?

Because opening the next chain costs more. Leaving two boxes and forcing your opponent to open a longer chain is usually a net gain.

What is a chain?

A run of boxes where completing one opens the next, so a single move can claim all of them in sequence.

Do I get another turn after completing a box?

Yes, always. That rule is what makes chains cascade.