Dots and Boxes strategy: the double-cross and the long-chain rule

The whole game in one sentence

Dots and Boxes is about being the player who does not have to move first when the safe lines run out. Everything else — chains, loops, double-crosses — is machinery for arranging that.

1. Safe moves are lines that give no box a third side.
2. When safe moves run out, someone must open a chain, and the opponent takes it.
3. The player who takes a chain must then open the next one — unless they use the double-cross.

Level 1 — Basic: stop giving boxes away

1. Never give a box its third side

Count the sides of both boxes touching the line you are about to draw. If either would reach three, your opponent takes it — and then keeps going.

2. Play in the middle of empty regions

Lines in wide-open space are safe for longest. Lines near existing lines are where third sides appear.

3. Count safe moves

Before the chains form, quietly count how many safe lines remain. If it is an odd number and you are to move, you will be the one forced to open — so start planning to change the count.

4. When you must open, open the smallest

If every move gives something away, give away the shortest chain. Two boxes lost is much better than eight.

Level 2 — Intermediate: the double-cross

Taking every box is a trap

If you take every box in a chain, you must then make the next move — and the only moves left will open the next chain for your opponent. Taking everything means giving everything away for the rest of the game.

The double-cross

The fix, and the single most important technique in the game: when handed a chain of three or more boxes, take all but the last two, then draw the line that splits those two — giving them to your opponent but leaving them on the move.

The arithmetic: you sacrifice two boxes to force your opponent to open the next chain. If the next chain is longer than four boxes, you profit. On any board with several long chains, the player who knows this beats the player who does not, almost every time.

Chains and loops

• A chain is a run of boxes opened from one end. The sacrifice costs two boxes.
• A loop is a closed circuit of boxes — a ring. Declining a loop costs four boxes, because you must leave two pairs.

Loops are therefore more expensive to decline, and creating one can change who wins the parity battle.

Level 3 — Advanced: chain parity and control

The long-chain rule

The central result of the theory, usable at the table:

The first player should aim for an odd number of long chains (three or more boxes); the second player should aim for an even number — on a board with an even number of dots per side. On a board with an odd number, it is the other way round.

"Aim for" means it: you can control the number of chains by how you play the opening, by joining regions together or keeping them separate. Most strong play in the first half of the game is about steering towards the right parity, not about boxes at all.

Keep control

Control is the state of being able to decline chains indefinitely — you take all but two of each chain and hand the move back, over and over. The player with control wins every long chain on the board. The whole endgame is a fight to acquire and keep it.

The corollary: if you cannot get control, your aim changes completely. You want short chains, so that declining them is not worth your opponent's while, and you want the total number of boxes in long chains kept small.

Loops as a parity weapon

Because declining a loop costs four boxes instead of two, adding or removing a loop changes the arithmetic of who can afford to keep control. Strong players build or avoid loops deliberately, as a way of tipping a close endgame.

Practical method

1. In the opening, count how the board is dividing and steer the chain count towards your parity.
2. Avoid making the third side of any box unless it serves that plan.
3. When the chains are set, work out whether you have control — and if not, whether you can get it by sacrificing early.
4. In the endgame, count boxes exactly. Nothing after this point is guesswork.

The mistakes that decide games

1. Taking every box in a chain and then being forced to open the next.
2. Never declining anything — the double-cross is not optional at a decent level.
3. Ignoring chain parity in the opening, when it is still cheap to change.
4. Treating a loop like a chain. Declining a loop costs four boxes, not two.
5. Opening the longest chain when a shorter one was available.
6. Playing the opening as though it does not matter. It decides the chain count, which decides the game.

Common questions