Dots and Boxes Strategy: Chains and Control

The rules

1. Players take turns drawing one line between two adjacent dots, across or down.
2. Whoever draws the fourth side of a box claims it and must draw another line immediately. Claiming several boxes in a row keeps the turn going.
3. When every line is drawn, the player with more boxes wins. Equal boxes is a draw.

Phase 1: safe moves

Early on, draw lines that do not give a box away, which means never drawing the third side of any box. Both players do this until no safe line remains. The skill in this phase is not in the lines themselves but in what shape they leave: by the end of the safe phase the board is divided into chains (strings of boxes each missing two sides, joined end to end) and cycles (closed loops of such boxes), and whoever must draw next has to open one.

Phase 2: who opens the first chain

When the safe lines are gone, the player to move must draw the third side of a box somewhere, opening a chain. The opponent takes the boxes of that chain... and then must open the next one. So taking a chain hands the move back, and the player forced to open the long chains loses most of the boxes. The winner is the player who makes the opponent open the long chains: that player has control.

The double-cross: keeping control

When your opponent opens a chain of three or more boxes, do not take all of them. Take all but the last two, then draw the line that leaves those two boxes as a single domino with one side missing (a "double-cross" position). Your opponent takes the two boxes, but now they must open the next chain for you. You gave up two boxes to keep control, and the next chain is worth more than two.

Do this at every long chain except the last one on the board, which you take entirely. For cycles the sacrifice is four boxes instead of two.

Counting chains: the long chain rule

Control can be planned before the safe moves run out. On a grid with an even or odd number of dots, who gets control depends on how many long chains (three boxes or more) will exist when the safe moves end. The rule, in its simplest form for the rectangular grids here: count the dots in the grid; the first player wants the number of long chains plus the number of dots to be even, and the second player wants it odd. During the safe phase, each player tries to create or merge chains, or split them, to make the count come out right. Cycles do not change the count.

You do not need to apply the rule precisely to improve: just count the long chains forming during the safe phase and ask whether the current parity favours you. If not, look for a safe move that splits a long chain into two short ones (changing the count by one) or joins two chains into one.

Short chains

Chains of one or two boxes are different: they are given away freely in the fight for control. When you must open something, open the shortest chain, since those boxes are lost anyway and the opponent cannot double-cross a chain of one or two profitably. Expect to lose the short chains and win the long ones if you hold control.

Grid sizes

On the 4 × 4 dot grid (nine boxes) games are short and the chain count is easy to see. On 7 × 7 (36 boxes) the safe phase is long and the chain structure is rich; one or two long chains usually decide the game, and the parity fight is real. The Hard robot plays the ending exactly and the safe phase with chain counting, so against it you must win the count.

Common mistakes

1. Drawing a third side early. It gives a box away for nothing.
2. Taking every box in a chain. You lose control and open the next chain yourself.
3. Opening a long chain when a short one exists. Give the smallest.
4. Ignoring the chain count in the safe phase. That is when control is decided.
5. Double-crossing the last chain. Take the last one completely.

Glossary

• Chain: boxes joined end to end, each missing two sides, so that opening one gives all of them.
• Cycle: a chain that closes on itself.
• Control: being the player who never has to open a long chain.
• Double-cross: leaving the last two boxes of a chain for the opponent to keep control.
• Safe move: a line that does not create a box with three sides.