How to win at dots and boxes: give boxes away to take the rest

Dots and boxes looks like a game of grabbing boxes. It isn't. Almost every box on the board is decided in the last few moves, and the player who takes the most boxes is usually the one who was willing to give some away on purpose.

The game has been studied properly — there is a whole book of mathematics about it — and the useful part fits on one page. Here it is, from the rule every beginner needs to the one that wins close games between good players.

Rule one: never draw the third side

A box with three sides drawn is a gift. Whoever moves next draws the fourth side, takes the box, and — because closing a box earns another turn — keeps going.

So the opening of every game is the same: both players draw lines that don't put a third side on any box. These are the safe moves, and the first phase of the game is simply using them up.

On a board of five boxes by five there are a lot of safe moves at the start and then, surprisingly quickly, none. At that point someone has to draw a third side somewhere. The whole game is about making sure that someone is your opponent — and that what they are forced to open is small.

What the board turns into: chains and loops

When the safe moves run out, look at the board again. The undrawn lines will have split it into regions:

• A chain is a row of boxes connected end to end, where drawing any line inside it lets the next player take the whole thing box by box.
• A loop is a chain that closes on itself, like a ring.
• Short chains are one or two boxes long. Long chains are three or more.

From here on, every move opens something. The only question is what. The rule is simple: open the smallest thing on the board. Give away the single boxes and the pairs first. Hold the long chains back.

The trick that wins: take all but two

Here is the move that separates people who play dots and boxes from people who have studied it.

Your opponent opens a long chain — say five boxes. The obvious play is to take all five. The expert play is to take three, and leave the last two boxes by drawing the line at the far end of the chain instead of the one between them.

Those two boxes are now a closed pair with one line missing in the middle. Your opponent takes both with a single line — and then, because they closed a box, they must move again. And there is nothing safe left to draw. They have to open the next chain for you.

This sacrifice is called a double-cross in the literature, and it is how the player "in control" stays in control. You give up two boxes per chain to take everything else. For a loop, the equivalent is to leave four boxes, because a loop has no open end.

The arithmetic is worth doing once. Say the board ends as four long chains of 3, 3, 4 and 6 boxes, sixteen in all, and your opponent has to open the first one. Each player opens the smallest chain available.

• Greedy — you take every chain you're given, then open the next. You get the first 3, they get the second 3, you get the 4, they get the 6. You lose, 7 to 9.
• In control — take 1 and give 2, take 1 and give 2, take 2 and give 2, then take all 6. You win, 10 to 6.

Same board, same chains, and the player who gave away six boxes won by four.

Control is worth keeping as long as the chains still to come are bigger than the two boxes you hand back each time. When they are not, just take everything.

The long chain rule: who gets control

All of that assumes you are the one being handed the first long chain. Which player that is gets decided much earlier, by parity — and there is a rule for it that you can use at the table.

Count the dots on the board. A five-by-five box grid has six rows of six dots: 36. Then:

• The first player wants the number of dots plus the number of long chains to be even.
• The second player wants it odd.

With 36 dots, that means the first player wants an even number of long chains at the end, and the second player wants an odd number. (Loops count differently in the full theory, but on a small board this version is right far more often than it is wrong.)

You influence the count in the middle of the game, while safe moves still exist: joining two short regions into one long chain changes the number by one, and so does cutting a long chain in two. When you can see how the board is going to split, steer it toward your number.

You won't count perfectly, and neither will your opponent. Just knowing that the number of long chains decides the ending puts you ahead of almost everyone who plays casually.

A checklist that will beat most people

1. Never draw a third side while a safe line exists anywhere on the board.
2. When the safe lines run out, open the smallest region — singles, then pairs, never a long chain if you can help it.
3. When you're given a long chain, take all but the last two and hand those back with the end line. Leave four in a loop.
4. Stop sacrificing when it stops paying — on the last chain, or when the remaining chains are tiny, take everything.
5. In the middle game, count the long chains and nudge the total toward your parity.

Why it is a perfect game for two

Dots and boxes has no luck, no hidden information and no setup — a grid and two people is the whole game. Every position is fully visible, so a turn-based match can sit for hours between moves and you lose nothing by walking away: one look at the board and the whole plan is back in your head.

It also rewards exactly the kind of learning that makes a rivalry fun. The first few games are won by whoever avoids the third side. Then one of you discovers the all-but-two trick and wins five in a row. Then the other one learns to count chains. It is a game that keeps having a next level, and you find it together.