Lesson 4 of 4
Chains
5 min

The Long Chain Rule in Dots and Boxes

The double-cross keeps control once you have it. The long chain rule tells you, before the safe lines run out, which player will get it.

Step 1 of 5

Who has control?
This is the board from the first chains lesson. No safe line is left and Red has to move. There are three long chains (the left column, the top of the right column and the arch through the middle) and two short ones (the corner box and the two boxes at the top of the middle).

The short chains just pass the turn: Red gives one away, Blue gives the other, and then Red has to open a long chain. That gives Blue control: Blue takes all but two boxes of each long chain and all of the last one, and wins 10 boxes to 6.
The long chain rule
Here the top two rows form one chain of eight boxes, and the bottom two rows another. It is Blue's move, so Blue has to open one, and Red gets control. Blue ends up with 2 boxes and Red with 14.

Who gets control comes down to a count. This is the long chain rule, for Playboard's board of 25 dots: when the safe lines run out, Blue gets control if the number of long chains is odd, and Red if it is even. Short chains don't change it, as long as each one is given away so that it can't be handed back. The rule assumes no double-cross has been played before then. Count chains, not loops: a ring of boxes with no way out to the edge of the board doesn't count.
Whose control?
No safe line is left, and Blue is to move. Count the long chains. Who gets control?
Question
Make the count odd
Now a game where a few safe lines are still left, and it is your move. Three safe lines remain, all around the top two boxes of the left column, which have one side drawn each. The bottom row is already a long chain. The safe line you draw decides how the boxes above it split into chains, so count before you draw: you want an odd number of long chains.
Your move
Draw the safe line that leaves an odd number of long chains.
Change the count
Back to the board from the quiz, two turns later. You gave away the 2-box chain at the top of the right column, drawing the line between its boxes, and Red took both. Red has just opened the other 2-box chain at its end, with the marked line on the right edge. Two long chains are left, so if you take these two boxes, you have to open a long chain and Red keeps control.

That was Red's mistake: a 2-box chain opened at its end can be handed back. A double-cross changes the count. Hand the two boxes back: Red takes them with one line, then has to open a long chain, and control is yours.
Your move
Hand both boxes back to Red.

What you learned

Whoever doesn't have to open the first long chain has control, and keeps it with the double-cross.
On Playboard's board, Blue gets control with an odd number of long chains and Red with an even number.
A double-cross played before the long chains flips that count.
Short chains only pass the turn, as long as each is given away so it can't be handed back.