Player’s guide

Dots and Boxes Strategy Guide

Most people learn Dots and Boxes on paper as children and play it the same way: take every box offered, and hope for luck at the end. There is a lot more to it. One trick, giving away two boxes on purpose, decides most games between experienced players. This guide covers where the game came from, how chains work, and how to get a game going here.

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01 · Dots and Boxes

A game from a French mathematician

Dots and Boxes was first described in print in 1889 by Édouard Lucas, the French mathematician also known for the Tower of Hanoi puzzle. He called it la pipopipette. It spread as a pencil-and-paper game because it needs nothing but a grid of dots.

Mathematicians kept coming back to it. Elwyn Berlekamp, who studied the mathematics of games, wrote a whole book on it in 2000, and much of what follows comes from his analysis.

02 · Dots and Boxes

The safe phase: do not draw a third side

At the start of a game, almost every edge is safe. The rule is simple: do not draw the third side of a box, because your opponent will take it and move again. Both players draw safe edges until there are none left, and then someone has to open up boxes.

By the time the safe moves run out, the board has turned into chains, runs of boxes connected so that taking one lets you take the next, and loops, chains that join back on themselves.

03 · Dots and Boxes

Chains and the double-cross

When your opponent opens a long chain, you could take every box in it. But then you must move again, and you will have to open the next chain for them. The better play is often to take all but the last two boxes, then draw the edge that leaves those two as a pair. Your opponent takes the two boxes, and then they have to open the next chain.

Giving away two boxes to keep control is called the double-cross. In a game with several long chains, the player in control can give away two boxes in each chain but the last and still win comfortably.

With a loop, the same idea costs four boxes instead of two: take all but four, and leave them as two pairs.

04 · Dots and Boxes

The long chain rule

Since control decides the game, the real fight in the safe phase is over who will be forced to open the first long chain, one with three or more boxes. Berlekamp’s long chain rule gives a way to aim for it: on a rectangular grid, the first player wants the number of dots plus the number of long chains to be even, and the second player wants it to be odd.

You change the number of long chains by joining or splitting regions while safe moves remain. It takes practice to see, but even a rough count, “there will be two long chains, and I want three”, changes how you play the middle of the game.

05 · Dots and Boxes

Short chains first

When you are forced to open something, open the smallest region you can. Giving away a single box or a pair costs little, and it passes the problem back to your opponent. Keep the long chains for last, when control decides how they are shared.

06 · Dots and Boxes

Playing Dots and Boxes on LarkSpark

You can start a game on the Dots and Boxes page without an account. Play against one of eight bots, from Altair at level 1 to Orion at level 8. The lowest levels draw edges at random. From level 3 the bots take every box they are offered, and the top two also avoid drawing third sides while safe moves remain. None of them plays the double-cross yet, so they are good practice for it.

To play a person, pick an online match, or sign in and open a private room to send a friend a code. Your recent results are listed on the game’s stats page.

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