Short and Long Chains in Dots and Boxes

Near the end of a game, every line you draw gives your opponent a box. The boxes that are left fall into chains, and the player who counts them wins.

Step 1 of 3

Count the chains
No safe line is left: every line now gives a box its third side. The boxes that remain form chains, runs of boxes where taking one box hands you the next. The arrows trace them: the left column (4 boxes), the top three boxes of the right column (3), an arch through the two middle columns (6), and the two boxes at the top of the middle (2). The box in the bottom-right corner is a chain of its own (1).

A chain of one or two boxes is short, and three or more is long. The difference is the trick in the next lesson: whoever takes a long chain can hand its last two boxes back and keep control of the game. From a long chain that still leaves them a box or more. From a short chain it leaves them nothing, or can't be done at all.
How many are long?
No safe line is left on this board either. Trace each chain from one edge of the board to the other, and count only the long ones.
Question
Give away the short chain
In another game, only two chains are left: the left column, four boxes long, and the single box in the corner. Every line gives Red something, so give Red as little as you can. Red takes it, and then has to open the column for you.
Your move
Give Red as few boxes as you can, then take the whole column.

What you learned

When every line gives something away, the boxes left form chains: taking one box hands you the next.
A chain of one or two boxes is short. Three boxes or more make a long chain.
If you have to give something away, give away the shortest chain.

Unit 1The rules

Unit 2Chains

Take it to the table.

Put what you just read into practice. No sign-up needed.