Beyond the chain rule

You have understood the fundamentals of the game and can usually set up the correct number of chains. But are you are still losing consistently to Dabble and the better Yahoo players? Well, you have probably reached the big barrier separating good players and expert players, somewhere between 2000 and 2200 in Yahoo ratings.

The Dots strategy that I have described so far was simple: you try to set up the correct number of chains corresponding to your color. However, this strategy has limitations, and players may find that they are winning the chain fight, but losing the game.

There appear to be two important steps required to reach the expert level.

1. The first step is to nderstand the limitations of the chain rule, in particular, the role of the preemptive sacrifice, nibbling, and other ways to win or tie a game, despite losing the chain fight.

2. The second step is to understand the quad. It appears that for 5x5 Dots there is no simple strategy like the chain rule which tells you how to play with quads: You have learn all the endgames with quads and who wins them.

   This step is the key to top level Dots play: Players must develop a catalogue of endgame positions and their values. In fact, this is also an important conceptual advance: Instead of just thinking ahead to achieve immediate goals, players must also learn to first visualize the endgame positions they want to obtain, and adapt their immediate move to reach that goal. That is, the strategical emphasis changes from simply making general moves delineating territory, into long range moves seeking to achieve specific endgames.

I will try to explain how to learn these two steps in this section. But, in order to assimilate it, you will need to work hard. First, you must learn all the endgame positions with quads. The only effective way to do this is to figure it out yourself by making a list of the endgames and going through them on your own. I will try to help you do this by outlining the basic results.

This may sound trite, but to get better, you will need to start playing with better players. You may have avoided doing this before, because it was depressing to lose all the time, but if your rating is getting around 2000, then you should be able to do better against Dabble and players rated about 2200, so playing them won't simply be a source for discouragement.

Playing Dabble is now more important than ever. Dabble will consistently be able to win with Yellow and tie with Green, despite having lost the chain parity fight. You must study Dabble's techniques and learn to prevent it from using them, as well as applying them in your games.

It is also time to start challenging players rated about 2200 or higher. You should be able to tie and even win a few games. However, players at the 2400 level can still probably beat you every game, so don't insist too much with them.

For example, in the above position, Yellow has managed to make three chain, and appears to have the game locked up, since one of the chains is already quite long. However, Green can save the farm by moving in the unfinished chain (he will have to be the first to move into a chain, sooner or later). This effectively reduces the length of that chain, and leads to a tie. This position is typical of the preemptive sacrifice, because you must already be ahead to use it effectively. The following rules apply, with best play by both sides.

• If you are not ahead in boxes, the preemptive sacrifice will always lose.

• It is possible to tie the game using the preemptive sacrifice only if you are at least one box ahead.

• It is possible to win the game using the preemptive sacrifice only if you are at least three boxes ahead.

The preemptive sacrifice can easily be overlooked, and its unexpected use can be a cause of some anxiety. These rules give some peace of mind, since they say that you can't lose to it if you aren't already 3 boxes down.

These rules also seem to indicate that a preemptive sacrifice is most effective if it is played immediately after your opponent has sacrificed some boxes in order to win the chain fight.

In 6x6 Dots and other games with an odd total number of boxes, it is possible to win using the preemptive sacrifice if you are only two boxes ahead. For example, in this position, from the book Winning Ways, Yellow is two boxes ahead, so can use the preemptive sacrifice to win the game. DIAGRAM

It is easy to check these rules by noting that accepting or declining the preemptive sacrifice simply inverses the role of the players. I give a formal proof below for people who are interested in seeing it.

Aftermath

When your opponent makes a preemptive sacrfice, you are faced with the choice of accepting all the boxes, or leaving the last two boxes making a doublecross.

In the first case, you simply continue with the basic strategy, making the number of chains corresponding to your color. In the second case, the early doublecross means that you have made one chain and used it like you would in the final phase. With one chain done with, you need to make one less than you would normally. In other words, for 5x5 Dots:

An early doublecross means a change of sides: Yellow must now get make zero or two chains, Green must make one or three chain. This count does not include the sacrificed chain.

Skip This Proof

Using basic algebra, let N be the total number of boxes (16 in 5x5 Dots and 25 in 6x6 Dots), Y and G be the number of Yellow and Green boxes before the sacrifice, C the number of boxes in the sacrificed chain. Let X be the maximum number of boxes that can be captured by the side taking the last box in the chain. Since X is the maximum number of boxes the first player can attain in a Dots-and-Boxes game played on the same board, but with the Y Yellow Boxes, G Green Boxes, and C chain boxes removed, its value is independent of what Yellow and Green did beforehand.

Without loss of generality, assume that Green makes the preemptive sacrifice to win the game. One has two possibilities:

• Yellow accepts all the boxes in the chain and his final number of boxes is
  Y + C + X

• Yellow declines the last two boxes in the chain. Green therefore gets the following number of boxes
  G + 2 + X

In order for Green to win in both cases, the first Yellow value must be less than N/2 and the second Green value must be greater than N/2, that is

Subtracting Y + X + 2 from both these inequalities gives

Now, C must be at least 3, since chains have at least three boxes, so one gets

Now, if N is even like in 5x5 Dots, then the term in the middle is a whole number so that the right and left sides must be at least two apart, that is

But if N is odd, like in 6x6 Dots, then the middle term is half an integer and one can only conclude

This proves the conditions for a win using the preemptive sacrifice. The proof for ties is exactly similar.

To nibble means "to eat in small bites" and this technique can be used effectively against an opponent who wants to get the correct chain count at all costs.

For example, in this position Yellow seems to have attained all his goals: With only two boxes sacrifice, he has formed a very long chain and has prevented Green from making a second chain.

However, the game is a tie: Yellow will have to sacrifice two more boxes to prevent a second chain, then will have to give up four more boxes in the short chain phase. This example pretty well sums up the nibbling strategy:

1. Continually threaten to make one more chain than your opponent would like, forcing him to continually sacrifice boxes.

2. Short chain your opponent, that is, win more boxes than your opponent during the final short chain exchange.

Experienced players will note that the first method almost always wins for Yellow when Green tries to prevent one chain at all costs (instead of trying to make a second chain or cycle). This is why Yellow usually wins when there is very little space to make chains, for example, after sacrifices filling up the center.

The second method appeals to the Short Chain Rule which guarantees the short chain advantage to the player losing the (long) chain fight. You must use this advantage to save the Dots war, once you've lost the chain battle.

The theory is simple: If you take the quad (from a one edge offer) then you give up control. To keep control, you must decline the quad. This means:

1. Taking a quad from a single edge offer means an effective change of sides, with respect to the chain rule (one doublecross). Yellow must now have zero or two chains and Green one or three chains (in 5x5 Dots).

2. Declining the quad (by placing a second edge) preserves sides with respect to the chain rule (two doublecrosses). Yellow and Green must have the usual number of chains.

Well, that is the theory, now for the practice. Understanding the quad is not too hard, because the first person to move into a chain or quad will usually move into the quad first. It then comes down to analyzing what happens after the quad is taken. Note that you only have to the count from that point on, since one side or the other will have to take the quad, the rest of the moves will be the same no matter who took the quad.

It seems to me that the quad is has a levelling effect in most situations. The reason is that the player taking the quad will earn 4 points, and the player giving up the quad will usually take the last chain, so usually 4 points, so there is a fairly balanced split.

This levelling effect implies that sacrifices must be used very sparingly when there is a quad around. In particular, the preemptive sacrifice cannot be used when there is a quad, because acceptance gives at least 3 points, yielding 7 points when added to the guaranteed 4 points in the final phase. This almost certainly means a win for the person accepting the preemptive sacrifice.

One Quad

The basic situation with one quad can be completely characterized completely for 5x5 Dots (4x4 boxes). One has the following rules, which assume that all the capturing of short chains, chains, and quads is done at the very end (no sacrifices). I will also assume that chains do not terminate in quads (this usually gives the advantage to the winner of the chain fight):

1. One quad and one chain is a win for Yellow, unless there are at least three short chains of length two.

2. One quad and two chains is a tie, except when if there is one short chain of length two and two short chains of length one, in which case Yellow wins.

3. One quad and three chains is a tie, except if there is more than one short chain, in which case Yellow wins.

Two Quads

Two quads is also fairly easy to understand and the situation can be characterized completely for 5x5 Dots (4x4 boxes). One has the following rules, which assume that all the capturing of short chains, chains, and quads is done at the very end (no sacrifices). I will also assume that chains do not terminate in quads (this usually gives the advantage to the winner of the chain fight):

1. Two quads and two chains is always a win for Green

2. Two quads, one chain, and one short chain is a tie

3. Two quads, one chain, and two or more short chains is a win for Yellow

Knowledge of these rules should make it a lot easier to solve the above Dots problem. In Problem Q1, Yellow wins somewhat unexpectedly by immediately making a chain of length 4, which disconnects this chain from the bottom row. No matter where Green moves next, Yellow will be able to separate the bottom row into two disconnected parts, each of which is a short chain, and so a win for Yellow, by the third rule. Any other Yellow move will allow Green to keep the number of short chains down to one.

In particular, Yellow can move into the position of Problem Q2. Green now ties by moving into the bottom edge, second on the left. This threatens to make a join with the chain. If Yellow stops this by disconnecting the chain, then Green makes a second chain and wins.

Kermit's lament has never touched so many hearts as in 5x5 Dots game rooms. Indeed, the second player has a very difficult time holding his own. Computer analysis shows that Green only has two correct responses to Yellow's strongest first move, whereas Yellow has absolutely no theoretically losing move until his third turn.

A perhaps more compelling reason for Green's sorrow is that he will usually make two chains, sacrificing two boxes in one of them, therefore giving Yellow a two box handicap. For example, Yellow is more than ready to sacrifice two boxes to ensure one chain, usually winning the game if he can achieve his chain goal. On the other hand, if Green sacrifices two boxes to ensure two chains, then he will be already be at a four box disadvantage, due to the doublecross he will give away at the end, so he will have difficulty winning if his chains are not very long.

Another Green disadvantage is that the preemptive sacrifice will always fail against a unique chain, since accepting all the boxes leaves zero chains, which always favors Yellow, by the zero rule.

However, the strategies outlined in this section, the preemptive sacrifice, nibbling, and the quad are exactly the ticket to help Green.

Standard Ties

It seems that the best way for Green to save the day is to know a number of "saving positions" and try to reach these. The methods of this section give a few such positions which I call "standard ties".

• Nibbling tie
• Two quads, one chain tie
• One quad, two chain tie

Standard Losses

The standard ties have very similar counterparts which are losses for Green, and he must be aware of these to defend correctly. Yellow, on the other hand, will try to reach these positions.

• Nibbling loss Here Yellow has managed to limit short chains, so that Green doesn't win the short chain exchange, and so loses the game.
• Two quads, one chain loss Bumping up the number of short chains to two allows Yellow to get a foothold in the bottom half
• One quad, two chain loss Since Green has won the chain fight (two chains), he loses the short chain fight, since there are three short chains. But the presence of the quad means that points are split in the final phase, and Green loses.