A Dots game can be split up into four separate parts:
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.
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: 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.
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 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 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.
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 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.
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.
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:
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.
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 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.
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.
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.
Interestingly, even the top players will make this mistake, so
don't feel too bad if it happens to you.
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.
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.
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).