Playing Dots

Chapter 1 of my Dots page

by

Ilan Vardi

(ilanpi on Yahoo Dots)

What every beginner needs to know

Every beginner who wants to improve his game should read the sections about basic strategy, that is, doublecrosses, control, and the chain rule.

Getting good

This basic knowledge should allow you to beat most people. The rest is mostly technical details that allow you to implement the basic strategy since Dots, like the board games chess and go, is mostly about technical details, once you understand the basic concepts. You will need to know how to "see," for example, how to recognize chains, as well as tactics like sacrifices. To defend with green, you will need to know all about the cycle (usually called "the box" by Yahoo players). Finally, you will need to know basic openings.

Sorry, but I won't be able to tell you how to win all your games. Computer analysis has shown that 5x5 Dots (4x4 boxes) is always a tie with best play by both players.

If you're already good

If you're a good Dots player (2500+ Yahoo), then you might want to give me some pointers about the game, and I will credit you on this website. Just send me mail.

You can record your games and send them to me, so you can show everyone some of your best games.

You should also try your hand at 6x6 Dots. You can do this on Yahoo by simply choosing the 10x10 board option, and only playing in the bottom left-hand 6x6 corner.

Finally, I suggest you try playing Go, which I find very similar to Dots, but which is a much more interesting and challenging game.

Basic notation

In this site, I will mostly talk about 5x5 (4x4 boxes) Dots, since this is the only interesting version of the game that is available on Yahoo, since the 10x10 is already too big and long, let alone 15x15 or 20x20 (are you listening, Yahoo?). Mathematicians like Berlekamp have mostly been playing 6x6 (5x5 boxes) Dots, but it seems that the games are very similar and my experience shows that my understanding of 5x5 Dots carries over quite well to 6x6. You can play 6x6 Dots on Yahoo by choosing the 10x10 board option, and only playing in the bottom left-hand 6x6 corner. You can also practice 6x6 Dots against Dabble.

Since many people reading this site will be used to Yahoo Dots, I will use their notation, for example, the first player will be Yellow and the second player will be Green. Mathematicians have traditionally called these Dodie and Evie, because Dodie plays the odd movies and Evie playes the even moves. In books, you will also see A for the first player and B for the second player.

I have made a list of technical terms in the glossary.




Table of Contents





An example game

The game of Dots is played on a rectangular array of dots. The players alternate placing edges connecting the dots. When a player makes a box, he puts his name in the box and moves again. At the end, the player with the most boxes wins. If the number of boxes is equal, then the game is a tie.

On Yahoo Dots, the first player is Yellow and makes yellow boxes and the second person is Green so makes green boxes.

Here is an example game on the 3x3 Dots board

The players start, with Yellow moving first

On his last move, Yellow allows Green to take a box.

Green takes the box, but he is forced to move again. Any move that Green makes allows Yellow to take all the other boxes.





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)

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)

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!



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:

For the usual board sizes, this gives:

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:

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 good practice for those aspiring to attain this level.

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, there will usually be a quad formed on the left, and the players will then form chains and 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.

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.



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.

The Doubletrap


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.




Becoming Expert

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 much more comprehensive and represents the essence of top level Dots play: Players must develop a catalogue of endgame positions and their values.

    This step represents 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 also need to adapt your play. 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.



The Preemptive Sacrifice

The preemptive sacrifice is the most effective technique for tying a game, despite having lost the chain rule fight. Learning to recognize it, use it, and prevent it, is an entrance requirement to the exclusive Dots expert club.

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.

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

In this section, I prove the fact that you must be at least three boxes ahead to win using the preemptive sacrifice on a 5x5 board, and two boxes ahead on a 6x6 board. You should look this over only if you have an abnormal mathematical curiosity.

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:

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

Y + C + X < N/2 and G + 2 + X > N/2

Subtracting Y + X + 2 from both these inequalities gives

G - Y > N/2 - Y - X - 2 > C - 2

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

G - Y > N/2 - Y - X > 1

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

G > Y + 2

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

G > Y + 1

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



Nibbling

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 Quad

The quad is a mystery to most beginners. One reason is that it affects the chain rule differently according to how it is taken and experienced players will sometimes fight for the wrong number of chains (according to the chain rule) when there is a quad around.

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.



It's not Easy Being Green

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".

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.




Masters and Non-Mathematicians

The last part of this tutorial is for players who aspire to complete Dots mastery. I will describe my conception of optimal Dots from observation of the 5x5 Dots oracle. I will end by explaining Nimstring strategy in terms most accessible to Dots practitioners.

6x6 Dots

So you've finally reached the stage where you hardly ever lose on Yahoo Dots (2400+). You may even be starting to get bored with all those easy wins. Well, the next logical step is to play 6x6 Dots, which will present new and interesting challenges. In particular, it will avoid all the known openings, ties, and computer analysis of 5x5 Dots, but without the interminable and intractable games of 10x10 Dots. In fact, no one knows who the theoretical winner is in 6x6 Dots.

It appears that the biggest difference in 6x6 Dots is the much more important role of preemptive sacrifices. This must be due, in part, to the fact that in 6x6 Dots, you can use it to win a game with only a two box advantage, whereas a three box advantage is required in 5x5 Dots, see my proof of this result. The greater possibility of creating chains, due to the larger playing area increases the chance of sacrifices preventing a chain from forming, thereby giving a player a material lead which allows him to use the preemptive sacrifice effectively.

To play 6x6 Dots on Yahoo Games, just choose a 10x10 board and limit your moves to the bottom left-hand 6x6 corner.

In order to make this restriction clear, the players have cooperated to make an enclosure around the 6x6 board. Note that this does not change the sides: Yellow, who start the enclosure, will also play Yellow (first player) in the 6x6 Dots game.

When the 6x6 game is over, the loser must agree to resign the game, as in this diagram, since the Yahoo game is not actually over.

Therefore, playing 6x6 on Yahoo Dots requires a great deal of cooperation and trust between the players. It is therefore limited to expert players who are highly motivated to test their skill on this more challenging venue.

Handicap Dots

One way of challenging yourself against less able competition is to play Handicap Dots.

The only direct handicap possible on Yahoo Dots is to always play Green. This roughly corresponds to a 100 point rating difference. As you improve, you will note that playing Green is the only way to attract competition. Always playing Green is difficult, due to the fact than any error against a good opponent is fatal, while an error with Yellow can often be rescued to obtain a tie. Continually playing second is made more stressful due to the fact that Yellow dictates the opening, so can vary his game as he wishes, whereas Green must closely defend against the first player's choices and has little flexibility in his responses.

Strong players must come to terms with this handicap, especially since the Yahoo system doesn't automatically change colors. This Yahoo peculiarity renders Dots similar to Go in which the recognized stronger usually plays second. One can even follow Go tradition by having intermediate handicaps, for example, to let the weaker player go first in two out of three games.

In any case, many players refuse to go second, so it is important for both sides in such matches to be aware that refusal to play Green is acknowledgement of a sizeable handicap and recognition that the other player is stronger.

More severe handicaps are possible, such as giving your opponent some free boxes at the end. For example, giving a one box handicap simply means that the weaker player wins tied games, that is, "draw odds."

I recommend giving one box for every 200 Yahoo rating point differential, and always play Green when there is an extra 100 points difference.

This type of handicap is not directly feasible on Yahoo Dots, however, keeping box handicaps n mind will help you stay honest by always trying to play the optimal move, that is, the one which gives you the biggest possible winning margin.

These considerations stress the fact that Dots seems more similar to Go than to Chess, where players try to give themselves every advantage, usually setting up the White pieces for themselves at the beginning of a match. On the contrary, Go practice has included letting the weaker player move first, declaring a tie favorable to the second player, and having the first player give a number of stones (usually 5 and 1/2) to his opponent at the end of the game. However, the Go handicap system consisting of letting the weaker player start with more than one stone has no direct counterpart in Dots-and-Boxes, since Dots players do not place different types of edges, in the sense that a player can make boxes using edges placed by his opponent.



Studying Games

To achieve the highest level of play, you will need to start studying games very carefully. In particular, you will need to play over games in order to identify mistakes or find improvements. You can also record your opponents' games in order to find weaknesses in their strategy.

Using Dabble

The easiest way to record and play over games is to use Dabble. To record a game (that you are not playing with Dabble) you first, you turn off "Autoplay" in the "Game" menu. You then click on the sequence of moves. When you're done with that, you save to a Dabble file (*.dbl) using the "File" menu.

To play over the stored game, you first retrieve the Dabble file you previously saved using "Load" in the "File" menu of Dabble. You then click on the right arrow in the Dabble window to see each successive move.

The first method (Dabble with Autoplay off) is also the easiest way to review a game you recorded using a scoresheet.

Scoresheet for 5x5 Dots

Click on this picture and print it to make scoresheets for 5x5 Dots (4x4 boxes). Using this scoresheet, you can record the games you play on Yahoo Dots. You can also bind these scoresheets together to make a book of your best games.

Scoresheet for 6x6 Dots

David Wilson has written a scoresheet for 6x6 Dots (5x5 boxes). It is best suited for recording face to face games with two playes facing each other at a table, for example, in a real (non-internet) Dots tournament.



Consulting the Oracle

If you aspire to perfection, then I recommend you consult the 5x5 Dots Oracle. This is a program written by David Wilson which gives all the optimal theoretical outcomes for the opening moves. That is, for about the first 10 turns of the game, it gives, at each step, the final outcome of the game, if both sides play perfectly.

You can use this site to analyze your openings, and see whether you or your opponent made a mistake in the opening. You can also test openings, and search for variations which limit the number of good options for your future opponents.

The oracle also provides a challenge to players who have reached the top of the Yahoo hierarchy. Indeed, lack of Yahoo competition becomes an issue fairly quickly, so it appears that one way to improve to the next level is to try to understand the often obscure winning moves decreed by the oracle.



The Nimstring Method

The game of Nimstring, which was invented by Elwin Berlekamp, is a variant of Dots which has a very nice mathematical theory, but which is also useful for playing some Dots positions. The name "Nimstring" comes from an equivalent form of Dots called String and Coins by Berlekamp. In fact, I will talk about the more directly relevant game of Nimdots here.

Beyond Life and Death

The Nimstring method will help you out if you tend to get nervous when positions like this come up.

DIAGRAM

That is, you don't feel comfortable with all those unresolved life and death issues going on simultaneously and you always hurry up and resolve the problem. Well, if you are consistently losing those games, then it is time for you to get with the Nimstring Program.

The Mirror Strategy

Most good Dots players are men, and like any self-respecting male, a good Dots player is afraid of commitment. Consider the following position.

Whoever wins the battle on the top part of the board will be able to take all 8 boxes on the bottom and win the game. Note that chain issue has not been decided in the two regions at top, and that the first person to resolve the chain issue in one of the regions will lose, since the other player responds by resolving the chain issue in his favor in the other region.

The simple way to achieve this is to simply use a mirror strategy, copying the exact move made in one region into the other until the chain issue is resolved in one region.

Interestingly, the chain rule doesn't come into consideration, because the player using the mirror strategy will be always be able to resolve the chain parity in his favor.

When are Two Regions Equivalent?

Interestingly, the mirror strategy method works even for two regions which are not exactly identical. For example, in this positon

This is again a win for Yellow, but what is interesting about it is that there is again a mirror strategy. That is, for each move in one region there is a corresponding response in the other region (possibily preceded by capturing a box) which turns the chain parity question in the players favor.

In this sense, the two regions at the top are equivalent to each other. In general, two regions will be equivalent if for each delaying move (or move resolving the chain issue) in one region there is a corresponding move in the other region which gives the same delaying options (or move deciding the chain issue either way).

Beyond Parity

Before talking about Nimstring strategy in detail, it is useful to understand how the delay strategy works in a position where the chain issue has already been resolved.

So, how many move can you delay in a region where the chain issue is resolved? The answer is that you can delay you can delay zero moves if you must move into a chain or take the last box (that is you can't delay at all!) in that region, and you can delay exactly one move in the region if there is one move left on the board which doesn't move into a chain or force you to move again outside the region (capturing the last box), and

When there are more moves available, the answer is given by 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.

This means that no matter what the choice of move, the player who will first have to move into a chain in the region will always be determined by the parity of the number of turns left until someone has to move into a chain (or take the last box) in the region.

It therefore makes sense to say that a region is equivalent to 0 if the player to move will be the first have to move into a chain, and equivalent to 1 if the player can move into a position where his opponent will first have to move into a chain, that is, he can move into a 0 position.

To repeat myself once more, the non-chain rule says that every region in which the chain number is resolved is equivalent to 0 or 1.

Note that moving into a chain always loses the delaying battle. This is because the opponent can then decide which side he wants to continue the game as (strategy stealing). This is a special case of the more general

Zero game: A Nimdots position is a win for the second player if and only if its value is 0.

Nimdots: The Theory of Delaying Moves

To make the theory of delaying moves clearer, one invents a new game called Nimdots in which the goal is to force the other player to take the last box (or move into a chain). In other words, Nimdots is played exactly like Dots: You place edges on a rectangular array of Dots and whenever you take a box, you have to move again, except that the total number of boxes captured is not taken into account, the loser is the last person to take a box.

In mathematical works on Dots, this game is always called Nimstring, but the two games are completely equivalent.

It is fairly clear that in the two examples of delaying strategy, the winner of the game is the player who wins the Nimdots positions at the top half of the board.


Nimdots Values

The game of Nimdots has a much simpler mathematical theory than ordinary Dots because every Nimdots position can be characterized by a single number, its value.

As already mentioned, as far as delaying moves are concerned, every Dots position in which the chain count is determined has value 0 or value 1, that is:

Every Nimdots position in which the number of chains is determined has value 0 or 1.

Recall that this simply says that deciding which player will first have to move into a chain only depends on the parity of the number of turns left to play in that region.

For positions in which chain parity was not determined yet, there was the concept of equivalent positions, that is, in which the same delaying options exist in each region. Positions which are equivalent in this sense will be said to be equivalent Nimdots positions and will have the same Nimdots value.

Two Nimdots regions are equivalent if you can move into the same equivalent Nimdots positions. So, one can see that the Nimdots value can be computed from knowledge of the Nimdots values of all the positions you can play into. This method of computing Nimdots values should work, since final positions are simpler, since they have fewer options. It may seem hard to believe that this always works, but this way of building up values for positions is guaranteed to work for all combinatorial games.

For example, one can try to computer the value of the region

From this position, all moves resolve the chain question, so all moves will result in a value of 0 or 1. If you make a chain, then the result is 0, as decided above. If you decide to prevent a chain by sacrificing two boxes, then you leave your opponent the last delaying move (another two box sacrifice). This means that the second choice moves into a position with value 1.

Therefore, the options from the diagram are to move into 0 or into 1. By definition, this position will be said to have value 2.

More generally, if you have a position in which the options are to move into any of 0,1,2, then the Nimdots value will be 3.

However, the rule is slightly more interesting if there is a missing value:

MEX Rule: The Nimdots value of a position in which you can move into the values A, B, C,... is the minimal excluded value among A, B, C,....

For example, if, in a Nimdots position X, you can move exactly into positions with values 0,1,2,4,5, then the position X has value 3.

The reason for this rule can be understood from the mirror strategy. Consider the Nimdots position X from which you can only move into any of the values 0, 1, 2, 4, 5. If you play that position against any other position Y which has value smaller than 3, then the first player can win on his first move: If the Y value is 2, for example, then you move into a position of value 2 from X and continue with the mirror strategy. The same if the Y position has value 0 or 1.

However, if the Y position has value 3, then you will always lose if it is your move. The reason is that if you move into 0, 1, or 2 from X, then your opponent will also move into 0, 1, or 2 from Y, then use the mirror strategy. If you move into 4 or 5 in X, then he will move into 3 from those positions leaving you with a 3 and 3 and a loss due to the mirror strategy.

Finally, you can see that if the Y position has a value bigger than 3, then you can win by moving into 3 from that position, leaving a 3 and 3.

So, this argument shows that position X will win against every value not equal to 3 but lose to the value 3, so X must have value 3. This same argument always works in general.

The Value of the 4-Corner

As an example of the Nimdots computation, I will indicate how one computes the Nimdots value of the 4-Corner.

To compute this value, one must build up a catalogue of Nimdots values, starting with the simplest ending positions, which all have value 0 (the player must move into a chain, or have to move on the board). The final answer is that the value of the 4-corner is equal 2. Since there many positions to cover, I will only follow one branch in the large tree of possible continuations.

The justification for the value 2 is that the possible positions one can move into have values 0, 1, 3. The value 2 follows from the MEX rule, since 2 is the smallest number not appearing.

DIAGRAMS

The first position has value 3, because the possible positions one can move into have values 0, 1, 2.

DIAGRAMS

The first position has Nimdots value 2, because the possible positions one can move into have values 0, 1.

DIAGRAM

The first position has Nimdots value 0, because it is a loss for the player to move, he has to move into a chain.

Using Nimdots Values

The basic difficulty with the Nimstring Strategy is that Nimdots values are difficult to compute. To make things harder, the knowledge of the Nimdots value of a region is not very helpful if you don't know the Nimdots values of all the positions you can move into. Therefore, to use the Nimstring Strategy, it seems that (human) players should stick to memorizing the values of certain positions, as well as all the values of all the positions that can be reached from that basic position. A good start is to figure this out for the 4-Corner.

The problems involved in computing Nimdots values can be worth the trouble, because there are some general rules which make them easy to use.

Zero game: A Nimdots position is a win for the second player if and only if its value is 0.

This says that in a Nimdots position, you have a theoretical win if and only if its value is not 0. In that case, you win by moving into a position which has value 0.

In terms of Nimdots values, the mirror strategy simply says.

Mirror rule: Combining two Nimdots positions with equal values gives a value of 0.

This is the simplest case of a completely general rule for combining Nimdots values.

Nimdots addition: To compute the value of the Nimdots position X composed of the independent Nimdots positions A, B, C,..., having values a, b, c,... you write a, b, c, in base 2 and add without taking carries.

DIAGRAM

For example, in this position, the board is split into three separate regions. One first computes the Nimdots value of each region. The analysis of the 4-corner shows that the Nimdots value of the top left region is 2 and the value of the top right region is 3. In the bottom region, the first player can force a unique chain, so the non-chain rule states that the Nimdots value is either 0 or 1. Since there are an odd number of total Dots in the bottom, the chain rule implies that this game is a win for the first player. Since there are an even number of edges placed in the bottom region, it is the first player's move, so the Nimdots value must be 1.

The three regions have values 1, 2, 3, and to compute the Nimdots value of the whole board, one uses Nimdots addition. Converting into base 2, one has 1 = 01, 2 = 10, 3 = 11. Adding without carries gives the result 00 = 0. One concludes that the position is a Nimdots loss for the player to move.


Nimdots and 5x5 Dots

In this position, Yellow has managed to attain all the basic objectives of the Nimstring strategy: He has set up a zero value Nimdots position at the top, a long chain at the bottom as well as winning the Nimdots position there as well. Therefore, Green will be the first to resolve the chain issue in one of the top regions, allowing Yellow to win the chain fight. In this continuation, Yellow uses the mirror strategy to his advantage and easily wins the game. However, Green did not make full use of the resources available to him, and has a number of tying moves. Perhaps the simplest one is the following. This moves threatens to make a quad, leading to a standard tie. It is seen that Yellow cannot prevent Green from making that quad. Indeed, Green's clever move has the advantage that Yellow can prevent the quad only by resolving the chain issue in that region, in which case Green wins by making the corresponding move in the other region.

This seems to indicate that the quad is the key to Green's defence when Nimstring considerations are not his favor. Since Nimstring will always favor Yellow when the board is split into three in this way, it makes sense that Yellow should try to prevent a quad as early as possible. In particular, Yellow can try to do this in the opening. I therefore adopted the following "anti-quad" opening to mathter unsuspecting opponents.

This forces Green to adopt different equalizing strategies, since this opening effectively prevents the standard quad ties.


Nimstring Strategy in Action

In the game ilanpi (Yellow) versus x0x_iceman2_x0x (Green) of February 10, 2003, the play from the above diagram continued.

Yellow's Turn 9 also threatens to connect the two left regions making a single chain and winning the game since Yellow will be able to prevent a chain on the right. It should be noted that Yellow a slightly stronger pivot move on his ninth turn, that is, placing the edge perpendicular and to the top left of the actual ninth move. This carries the same threats, but leaves yellow a stronger position on the bottom after Green has made the same response as in the game. As it is, Green has very few saving moves.


David Wilson's 5x5 Dots oracle shows that there are exactly five Green moves preserving a tie, that is, all other Green moves will lose the game, with best play by both sides.

It is not an understatement that understanding why these moves, and only these moves tie, remains somewhat obscure to the human mind.


The Game continued with Yellow's Turn 11 which split the board into three separate regions.

The Nimdots value of this position is 0, as was previously computed. This means that the position is a Nimdots loss for the player whose turn it is to move. This player will have to declare himself first in the top half of the board.

Green's 18th move must lose because it is a preemptive sacrifice which always loses if the number of boxes is equal, as I have proved. A similar proof shows that entering a chain always loses the corresponding Nimdots game, no matter what the number of boxes. One can therefore say that Green's 18th move was the culmination of Yellow's Nimstring strategy.

Indeed, if Yellow obstinately continued to try to force Green to commit himself first in the top two regions, then the game would have been a tie. For example, the game could have continued as follows:

1