## Getting better

Getting the right number of chains corresponding to your color will usually allow you to win your Dots game, so once you know the chain rule, most of Dots strategy is about trying to set up the right number of chains. This is not at all as simple as it might seem at first, and it is exactly where things get really complicated. This section will mostly be about how to get the right number of chains. But first, it might help to identify the different stages of a typical Dots game.

### Stages of a Game

A Dots game can be split up into four separate parts:

1. Opening: The first few moves of the game in which the players try to determine the nature of the middlegame. For good players, this consists of standard moves which they have played many times.
2. Middlegame: The part of the game following the opening in which the players are no longer playing standard moves are fighting for chain parity (getting the number of chains corresponding to their color), and, once the number of chains has been determined, to extend or limit chains.
3. Endgame: The part of the game when all chains, cycles, and their lengths have been determined. This part can be split into three phases:
1. Neutral phase in which the players alternate filling edges without affecting chain or cycle length or taking boxes, or the final score (with some exceptions).
2. Short chain phase in which players alternate capturing short chains.
3. Final phase in which only chains and cycles are left.

My definition of endgame is not quite standard. In particular, Go players will usually define the endgame as the part of the game when all territories have been determined. In Dots "a territory" means an area where at most one chain can live. However, 5x5 Dots is such a small game compared to Go, that experts in that game might consider the starting position of 5x5 Dots as an endgame. Mathematicians will also prefer to call the final phase the loony phase, since every move is a loony move or a response to a loony move, where a loony move is a mathematicial term describing the Nimdots value of a move which allows a doublecross. In other words, every move in the final phase allows a doublecross or can play a doublecross.

I felt it was necessary to define the endgame as I did because learning who wins in all such endgames positions appears to be one of the key steps required to become an expert Dots player.

### All About The Chain rule

There is a little bit more you need to know about the chain rule.

### The Chain Rule for Other Boards

The chain rule is also true for all size boards, but with very simple modifications:

### The General Chain Rule:

• If there is an odd total number of dots, then the first player (Yellow) should make an odd number of chains and the second player (Green) an even number of chains.

• If there is an even number of total dots, then the first player should make an even number of chains and the second player an odd number of chains.

For the usual board sizes, this gives:

• 3x3 dots (2x2 boxes): First player odd number of chains, second player even number of chains.

• 5x5 dots (4x4 boxes): First player odd number of chains, second player even number of chains.

• 6x6 dots (5x5 boxes): First player even number of chains, second player odd number of chains.

• 10x10 dots (9x9 boxes): First player even number of chains, second player odd number of chains.

To master the chain rule, you will also need to take cycles into account. In fact, a cycle counts as two chains, as far a control is concerned, so they don't affect the chain count, at least, at the basic level. However, cycles make things a lot more complicated so you should probably wait and check out the section on cycles and control a little bit later.

### The Non-Chain Rule

The chain rule is very simple to understand, but one of its direcct consequences (actually, a consequence of its proof), the non-chain rule, presents some problems for beginners (it certainly did for me) and is a source of mistakes. It states:

The Non-chain rule: Once the number of chains is determined, no choice of move will change which player will first have to move into a chain or cycle.

The exact statement of the non-chain rule does require a little more care. The rule only applies in a normal game, that is, the players will not move into a chain or cycle, or decline a doubletrap until the final phase of the game (when there are only chains and cycles left). For anyone who has read the proof of the chain rule, this simply says that there are no doublecrosses till the final phase.

### Good Greed

The non-chain rule immediately implies: You should always take a box that is offered to you, when it isn't part of a chain, cycle, or doubletrap.

This will come as a relief to people whose natural greed was thwarted by the kindler, gentler approach of the basic strategy which gives away two boxes per chain.

### The Short Chain Rule

The greedy non-chain rule immediately implies that during the short chain phase, the best strategy for the players is to exchange short chains starting from the smallest and ending with the largest. This gives the following handy principle for computing short chain scoring.

The Short Chain Rule: The player who has lost the chain fight (the first player forced to move into a chain) will get at least as many short chains as his opponent during the short chain phase of the game.

In other words, if you have been forced to first move into a long chain, then you will either tie or get more boxes during the short chain phase.

The short chain rule shows that it is easiest to count points in short chains backwards from the largest to the smallest.

### The Zero Rule

Mathematicians and other incredibly picky people will have noted that zero is also an even number, so that the chain rule should also say that the player fighting for an even number of chains should be happy getting no chains at all. In fact, this can be a mixed blessing, for the following reason.

The short chain rule also says that in a game without any chains, it is the loser of the chain fight, Yellow in 5x5 Dots, who will be favored in the short chain phase. But since there aren't any chains to snap up, he will also be favored to win the game. For 5x5 Dots, this gives

DIAGRAM

The Zero Rule: If there are no chains or cycles (that is, quads), Yellow will have the advantage. With even material, Yellow will always win, unless there is an even number of 1-chains and also an even number of 2-chains.

For other board sizes, this rule also applies, but with Yellow replaced by the player who needed an odd number of chains who will have the advantage.

What happens when there are quads around? Since they don't affect the chain count, and Yellow has lost the chain fight, he will be forced to move into a quad first. This means that with exactly one quad, Green will get all the points in that quad and probably win the game.

With two quads, the players will trade quads at the end, so Yellow will be favored, just as in the zero rule. However, the quad trade means that three quads will again favor Green.

### Care and Feeding of Chains

The chain rule works best if there are very long chains, so if you think you can get the right number of chains, you should try to make them as long as possible. The next sections will show you how to do this.

### Life and death

In Dots, it is very important to be able to determine whether one is able to force the creation of a chain in a region of the board. I will call this type of problem "Life and Death" in analogy with the crucial question of of Life and Death in Go.

DIAGRAM

A chain has life. If you don't stop it, it will keep on growing and take over a whole sector of the board. It is from this point of view, a chain is similar to a live group of stones in the game of Go.

You will need to understand life and death for a few key positions.

### The 4-Corner

The most useful position to understand is the following

DIAGRAM

I have called this the 4-Corner, but in Berlekamp's book, this Dots position is called the 2x2 Icelandic game.

### Openings

Note: This section will be entirely devoted to openings for 5x5 Dots (4x4 boxes), since the final outcome of the opening moves in this game has been evaluated rigorously by David Wilson using computer analysis. However, the general principles are the same for 6x6 Dots, with the goals of the first and second player reversed, of course.

The opening is the initial part of the game in which players make their basic strategic choices. For experienced players the opening will consist of standard moves which they will have memorized. If you do not know basic openings, you can find yourself in a losing position after your first move! This is especially true for Green, who usually has a hard time defending.

### Basic concepts

The basic strategy says that Yellow is trying to get one or three chains while Green is trying to get two chains. This determines the opening strategy as follows:

• Yellow tries to

• make a chain run through the center (easily suppressing the space on the edges so no other chains are possible),
• split the board into two parts, one of which is large (a chain can be created) and one of which is small (a chain cannot be created),
• split the board into three parts and create one chain in each part.

• Green tries to split the board into two equal parts, so he can make one chain in each.

### Fight for the Center

One sees that the first issue of contention in a Dots game is a fight for the center. If Yellow manages to make a chain run through the center, then he will most likely win and if Green manages to split the board into two equal parts, then he will probably tie and have good winning chances.

### Standard Openings

OK, you may have understood some of the basic issues, but you still need to know where to play! Luckily, David Wilson has already done a thorough analysis of all opening moves and the outcome of the game with best play. His results are available on his web site. He also named some of the basic openings, and I will follow his terminology.

The most important point is that the following opening move by Yellow is best because it threatens to win the game right away!

Of course, this opening move is not unique, because it makes no difference if you use one of the seven other identical versions which are just reflections or rotations of this move.

OK, Yellow is threatening to make a second move preventing Green from splitting the board into two and David Wilson's analysis shows that if Green allows Yellow to make this move, then Green will lose the game, with best play by both sides.

So Green must prevent Yellow's threat, and he has only two responses which don't lose the game. The names "Yahoo Opening" and "Wilson Opening" are due to David Wilson.

All other Yellow opening moves are also acceptable, but they they do not fight for the center, so do not pose an immediate threat to Green. I will therefore concentrate on these openings, which are the most frequently used by good Dots players. For an analysis of the other openings, see David Wilson's Dots site.

### The Yahoo Opening

The Yahoo opening will typically continue as follows.

DIAGRAM

Here Yellow threatens to take control of the center, winning the game. Once again, Green only has one reponse that can tie the game.

DIAGRAM

After this move, Green has successfully split the board into two parts. A natural move by Yellow is to try to split the top part in half, in order to obtain three separate regions.

DIAGRAM

This is the typical position arising from the Yahoo opening. Experience seems to indicate that Yellow can force a single chain in the bottom half of the board. It also appears that Yellow can split the top into two regions and force Green to commit himself first in one of the regions, and therefore win the chain fight. A good example of this strategy is given in this section.

However, Green can defend against this strategy without too much trouble by making a quad and reaching a standard tie.

The Yahoo opening therefore appears to lead to quick ties, once Green has understood how to use the quad, that is, knows how to defend correctly even he has lost the chain fight. This ability is what characterizes expert play, so this opening is recommended for players who have not yet mastered the game.

### The Wilson Opening

Once again, Yellow makes a move threatening to control the center and win the game.

DIAGRAM

Green has only one response leading to the following position.

DIAGRAM

In this position, one quad or chain is possible on the left, and similarly one or two quads on the right. The outcome will usually be decided by the number of short chains, so the players must be extremely careful in their choices. In fact, general strategical principles are not sufficient to understanding this opening, and players must essentially know all the possiblities.

Therefore this opening is much more difficult for both players, but also gives more winning chances to both players than the Yahoo opening. This is confirmed by David Wilson's perfect game, in which each side plays a move giving his opponent the least amount of good replies. For this reason, the Yahoo opening is a better choice for players who have not yet reached a very high level of play.

### Common Mistakes

Since 5x5 Dots (4x4 boxes) is known to be tie with best play, any win must be due to a mistake by one of the players. So, from a purely formal standpoint, getting good at Dots means the elimination of all mistakes. But, for the rest of us humans ( Dabble too), we have a lot more basic mistakes to take care of before we can even think about attaining perfection.

### Forgetting who you are

The most common error, by far, is forgetting which side you are. That is, you will play 5x5 Dots (4x4 boxes) and will make two chains as Yellow or one chain as Green, that is, the exact opposite of what the chain rule says, and so a guaranteed loss.

Interestingly, even the top players will make this mistake, so don't feel too bad if it happens to you.

### Errors in Winning Ways

These errors were found in the problems given on pages 544-545 of the 1983 second corrected printing of Volume 2 of Winning Ways. The errors relate to the answers given on page 536 of Winning Ways.

• Diagram (c): It is stated in the solutions that this is a 9-7 win for the first player (Yellow). However, the position is a 8-8 tie with best play.

In this position, Green is threatening to sacrifice a two chain on the left, joining two chains to make a very long 9 chain. Since there already is another chain, this will win the game for Green.

Therefore, as stated in Winning Ways, Yellow must sacrifice two boxes, creating 3 long chains consistent with the chain rule. However, the second player (Green) can tie by taking the boxes and immediately making a preemptive sacrifice in the top right hand corner chain. This prevents that chain from getting any longer, and allows Green to tie the game.

• Diagram (g): It is correctly stated in the solutions that this is a win for the first player. However, the suggested solution uses a nibbling strategy, losing the chain fight, but winning the game 8-7. However, the first player can force a chain by this move, winning the game 9-6.

• Diagram (h): It is stated that this is a 12-13 win for the second player. However, the position is a 13-12 win for the first player.

It is correctly stated that the second player (Green) will try to make a chain in the right hand region. However, it is incorrectly stated that the first player (Yellow) can prevent this by making repeated sacrifices. In fact, the second player can force a chain with this move, following the initial sacrifice by the first player.

Since the second player can make a chain, the first player will not immediately sacrifice in that region, and will instead limit the growth of that chain to length 4. This, followed by the suggestion in the solution of a preemptive sacrifice on the left will ensure a win for the first player.

A preemptive sacrifice is natural in this situation, because the first player is already ahead by two boxes and I showed in the section on the preemptive sacrifice that he now has the possibility of winning by using this sacrifice, since it is a 6x6 board (the 5x5 board requires a three box lead).