## The Basic Strategy

### Chains

A chain is a connected string of boxes where any move one player makes allows the other player to take all the boxes in the chain. Understanding chains is the key to the game of Dots.

Since you want to avoid moving into a chain, both sides will play quiet moves creating long chains until one player is forced to move inside a chain. In this game, there were two chains, and Yellow was the first to be forced to move into a chain. He decided to play If you are Green, what would you play? Yellow has just allowed you to take the six boxes up top. If you took all the the boxes in the chain, then you had to move into the bottom part. This gives the position No matter where you play now, you will have to make a move into the bottom part. This will allow Yellow to take all the remaining boxes, and win the game. Well, if you play Dots like this, then you are like most beginners: You play by making edges until there are a lot of chains around. You then alternate moving into a chain and taking all the boxes in a chain.

You might be surprised that there is a much better way to play Dots. Learning it will allow you to consistently beat all your friends, at least, those who don't know about it.

### Doublecrosses

Let's go back to the position. Now, watch closely, as I show you how to improve your game.

You start off the same, by taking the first three boxes up top. Then take the fourth box, as before. But this time, you leave the last two boxes! Instead of taking the last two boxes, Green leaves them for Yellow to take. OK, why did Green make this bizarre move? It's because Yellow is now forced to take the last two boxes. Then, Yellow is the one faced with the second chain. No matter where he moves, he will let you take all the remaining boxes, and you win the game. You will notice that the new ingredient was to leave the last two boxes in your chain to your opponent: When you left the last two boxes, you made a doublecross. Doublecrosses are the the key to all Dots playing strategy. You will need to use them well to become a good Dots player.

### Control and how to keep it

The doublecross is the basis of Dots strategy. Using it consistently is called control. For example, take this position where you are playing Green and you managed to force Yellow to play into a chain. Should you take all four boxes in the chain, or use the doublecrossing strategy of the previous section? The answer is that you win by continually doublecrossing your opponent, that is, always leaving him the last two boxes and forcing him to open the next chain. Green won the game 10-6 using this strategy. But what would have happened if he had taken the first four boxes that Yellow offered him? The answer is that Yellow could have tied the game by using the same strategy, that is, keeping control and winning all the boxes in the last chain.

### The Chain rule

Now you know how to keep control and win the game once your opponent has moved into a chain. Since there is a good chance your opponent knows how to do that too, in a good game of Dots, the loser will usually be the one who first had to move into a chain.

That means that to win a game of Dots, you need to know how to force him to be the first to move into a chain. This seems to be a hopelessly complicated problem, but Amazingly, there is a very simple way to tell which player will have to open up the first chain.

The Chain Rule: On the 5x5 board (4x4 boxes)

• If the number of chains 1 or 3, then no matter what the choice of moves, Green will be forced to open up the first chain.

• If there are 2 or 4 chains, then no matter what the choice of moves, Yellow will have to open up the first chain.

Note: When I say "no matter what the choice of moves", I mean all moves which do not enter a chain if there is another possibility open.

Since opening up the first chain usually loses, this gives the following simple strategy for playing Dots.

The Chain Fight: On the 5x5 board (4x4 boxes)

• Yellow must try to make an odd number of chains.

• Green must try to make an even number of chains.

Important Remark: When I use the word "chain", it is always about a chain of length 3 or more and I don't count chains of length 2. You should be able to see why: It is because the chain should always be long enough to allow you to leave the last two boxes as a doublecross. This is not possible with a chain of length 2 since your opponent can give up a chain of length 2 without allowing you a doublecross. Chains of length 1 or 2 are called "short chains".

Mathematical Remark: The chain rule is a mathematical result, but it has nothing to do with the chain rule that you may be forced to study in Calculus class. First of all, the Dots chain rule will actually help you do something useful, win Dots games. Secondly, you won't need to know anything about its proof to use it and win games.

Go forth and win: So now you know the most important facts about Dots strategy. You should now be able to go back and amaze your friends!