The Dots and Boxes chain rule, counted out loud

What a chain is in Dots and Boxes

Before we count anything, let's agree on what we're counting. A chain is a row of boxes that each already have two sides drawn, linked to each other through their open sides. Draw any line inside it and one box gets its third side. Your opponent closes that box, which earns another line, which closes the next box, and down the row it goes.

A chain of one or two boxes is a short chain. Three or more makes a long chain, and long chains are the only ones the rule cares about. A loop is a chain that bends around and meets itself, so it has no open end; the smallest possible loop is four boxes in a square. Loops stay out of the count, too.

Counting long chains on a real board

Here's our board: 3×3 boxes, so four dots along each side. Every horizontal line has been drawn and none of the vertical ones. Have a look before reading on, and see if you can guess who's winning.

First finger: dots. Four rows of four is sixteen. Now the long chains. The top row is three boxes with both ends open, so that's one. The middle row, two. The bottom row, three. Sixteen plus three makes nineteen.

First player: wants dots + long chains to be even. Second player: wants it odd.

Nineteen is odd, and odd belongs to the second player. Every line left on this board opens a chain, and it's the first player's turn, so they have to start handing them out. With perfect play from here, the second player finishes ahead 5 boxes to 4. Press the button under the board and try to prove me wrong.

How one line changes the long chain count

Let's rewind and change one decision. Suppose nobody ever drew the line under the box in the top-right corner, and the right-hand edges of the top two rows were drawn instead.

Count again. Still sixteen dots. But the top and middle rows now link up around the right-hand side into one long chain of six, and the bottom row is still a chain of three. Two long chains. Sixteen plus two is eighteen, which is even, so now the first player is the one in control. With best play they collect 7 of the 9 boxes.

Same board, same nine boxes, and a narrow loss became a big win. That's why strong players fight over the chain count while the board is still half empty and safe lines are around to play with.

Ways to change the count while you still can

1. Join two areas into one chain by leaving the wall between them open. Two long chains become one.
2. Split a long area with a line across it, so it ends up as two chains, or one long and one short.
3. Trim a chain down to two boxes, which drops it out of the count completely.
4. Bend a chain round until its ends meet and it becomes a loop. Loops don't count either.

All of this has to happen before the safe lines run out. Once no safe line is left anywhere, the count is locked in, and the result usually follows it.

Why the chain count matters: control

The rule works because of what happens when somebody opens a long chain. The player receiving it takes all but two boxes, then hands that pair back with a double cross, leaving the other player to open the next chain. Being the one who receives every chain is called having control.

Control costs two boxes per long chain, or four per loop, which is why margins on small boards are so tight. On a big board full of long chains, it's worth a great deal more.

If the count is against you and the chains are already set, limit the damage. Open the smallest chains first. Offer loops early, since keeping control of one costs your opponent four boxes. And when you have to give away a pair, draw the line between the two boxes, not at the end, so the pair can't be handed back to you.

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