Everyone has played Dots and Boxes at the back of a classroom. Draw a line, try not to give anything away, and grab squares when you can. Played like that, it feels like luck: at some point the safe lines run out, somebody has to open something, and the other player cleans up.
It is not luck. There is one idea that changes the whole game, and most people have never seen it.
A little history
The game was first described in print in 1889 by the French mathematician Édouard Lucas, who also gave us the Tower of Hanoi. It was already a schoolroom pastime. More than a century later, the American mathematician Elwyn Berlekamp wrote an entire book about its strategy. He had been studying it since childhood.
That a game played with a pencil could fill a book tells you something about what is under the surface.
The two phases
Every game has two parts.
In the opening, there are plenty of harmless lines. The only rule is: do not draw the third side of a square, because your opponent will draw the fourth and take it.
In the endgame, no harmless lines are left. The board has broken up into chains: corridors of squares where taking one lets you take the next, and the next. Whoever has to move must open a chain for the other player.
Our board has sixteen squares. Most games are decided by who gets the long chains.
Short chains and long chains
A chain of one or two squares is short. A chain of three or more is long. The difference matters enormously, and here is why.
The move nobody expects
Suppose your opponent has just opened a chain of five squares. The obvious thing is to take all five. Each one gives you another turn, so you take them one after another.
But after the fifth square, you still have to move. And if the only lines left open another chain, you have just handed your opponent the next one.
Now try this. Take three of the five. Then, instead of taking the fourth, draw the line at the far end of the chain. That leaves two squares sitting together as a pair, each with three sides drawn.
Your opponent takes both with a single line. They score two. But having taken them, they must move again, and now they are the one who has to open the next chain.
You gave away two squares. You kept the right to be the one who receives the next chain. If that chain has six squares, it was an excellent trade.
Staying in control
A player who does this is said to be in control. From that point on the pattern repeats:
- Opponent opens a long chain, because they have to.
- You take all but two.
- You hand back the last two.
- Opponent takes two and has to open the next chain.
On the very last chain, you take everything.
Each long chain costs you two squares and gives you the rest. With three chains of four, five and four squares, the player in control takes two, three and four, which is nine. The other player takes two and two, which is four. The player who "gave squares away" wins comfortably.
Why short chains are different
The trick needs at least three squares. With a chain of two, there is nothing left to take after you leave two behind.
So short chains do not help the player in control. In fact, the player who is not in control wants as many short chains as possible, because each one is a free gift with no price attached.
What to do when you are losing control
If you can see that you are going to be the one forced to open chains, your best hope is damage limitation.
- Open the shortest chain first. Always.
- Break up long chains early. During the opening, draw lines that cut a long corridor into two short ones.
- Open a chain of two in the middle. If you draw the line between the two squares, your opponent can take both but cannot use the hand-back trick. If you draw it at the end, they can decline. This is a small detail that wins games.
The fight in the opening
Once both players understand all this, the real game moves back to the opening. Who ends up in control depends on how many long chains form, and that can be steered, many moves in advance, by where the early lines go.
That is the part that fills books. There is a counting rule for it, which is beyond a short post. But you do not need it to beat almost everyone you know. You only need to refuse those two squares.
Try it
Play a game and deliberately let a few long chains form. When your opponent opens one, take all but two and draw the far line. Watch their face when they realise they have to move again.
Of all the turn-based HTML5 games on TwoSide, this is the one where a single idea makes the biggest difference. It turns a pastime into a contest.