A short chain of one or two boxes is not usually called a chain.
A cycle is not a chain.
Counting chains is the basis of Dots strategy due to the chain rule.
Winning the chain fight does not mean you have won the Dots game.
Learning to lose the chain fight but win the game is one of the keys to becoming a Dots expert.
Winning the chain fight does mean that you have won the Nimdots game.
The chain rule tells you how many chains you should make to force your opponent to open the first long chain or cycle:
For the usual board sizes, this gives:
Using the chain rule to make the correct number of chains will usually win the game for you.
There is a mathematical proof of the chain rule.
The word "combinatorial" is used to differentiate it from "Game Theory", which is the study of strategy in games which the opponents may be allowed to hide their information, and in which chance can play a role. Poker is a good example of such games, but game theory has mostly been used in economics and politics.
Basically, Combinatorial Game Theory is about playing games, whereas Game Theory is about making money.
If you move into a cycle, then you allow your opponent to take all the boxes (one doublecross) or else leave you with the last four (two doublecrosses).
The most usual case in 5x5 dots is the 4-cycle. Understanding this cycle is a little difficult, but it is the key to defending with Green.
A cycle is not a chain.
In terms of control and the chain rule, a cycle is equivalent to two chains (keeping control means making two doublecrosse). Therefore, cycles do not affect the basic chain count.
Practicing with Dabble is a good way to improve your game. You can also specify other board sizes, so Dabble can be used to practice 6x6 Dots, a more interesting version of Dots not yet available on Internet Dots servers.
You can use Dabble to record and replay Dots games. Just turn off "autoplay" in the "game" menu, then play the moves of the game you wan to store. You then save the game using the "save" entry in Dabble's "file" menu. To review the game, load the file and use the arrows in the Dabble menu to replay the moves.
Leaving doublecrosses is the fundamental technique which allows you to implement the basic strategy.
DIAGRAM
Making this move is very dangerous and is a common mistake. This is because playing a doubletrap allows your opponent to decline with a doublecross, changing the parity of the chain count, or take both boxes, preserving chain count parity.
The player presented with a doubletrap can decide once and for all the parity of the chain count, and so, in most situations, the outcome of the game, due to the chain rule.
For this reason, you should only play a doubletrap when you are absolutely sure that you are losing, in the hopes of confusing your opponent.
The idea of the doubletrap is the hope that your opponent makes the wrong choice, usually declining with a doublecross changing the chain parity in your favor.
Diagram
A join cannot be disconnected without giving away one of the two chains.
It is a term borrowed from the game of Go.
A loony move allows a doublecross, so effectively gives the opposing player the choice of side he wants to continue the game as (strategy stealing). Because of this, a loony move always means the loss of the corresponding game of Nimdots.
A player can make many moves in one turn.
If there are more than one moves in one turn, they are denoted by letters. For example, the three moves on turn 15 will be called 15a, 15b, 15c.
Dabble uses the term "move" to denote a turn.
This is the ususal way in which a Dots game is played by moderately good players. A good player can often beat a player in a normal game, even if the number of chains does not correspond to his color.
Understanding who plays last in a normal game is the first step in the proof of the chain rule.
The neutral phase happens just before the short chain phase.
Moves in the correct phase are always straightforward and do not affect the result, except for one exception.
You should never give away a box during the neutral phase.
Sacrificing boxes during the neutral phase is a common mistake by beginners.
Nimdots is the game played exactly like Dots, except that the number of boxes taken isn't counted, and the loser is the person completing the last box.
Nimdots is useful when there is more than one region of the board where life and death has not been resolved. It gives a systematic method for determining which player will be the first to resolve this issue in one of the regions. This should generally allow the other player to resolve the other region in his favor, and win the game.
The mathematical interest of Nimdots is that each Nimdots position can be completely characterized by a single number, its Nimdots value, which is a number greater or equal to zero.
Computation of Nimdots values many regions is simplified by the following rule: 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.
From the strategical point of view, the Nimdots game can be completely understood by giving each position a value, which is a number greater or equal to zero. This value can be computed recursively from evaluation of positions one can move into, according to the following rules:
For example, the 4-corner has Nimdots value equal to 2.
Therefore, To win a Nimdots position, one simply moves into a position with value 0. If this is not possible, then the position itself has value 0, and the game is a theoretical loss for the person to play.
The computation of Nimdots values is considerably simplified though the use of Nimdots addition.
The theory of Nimdots values is a special case of a general theory due to Sprague and Grundy, which states that any game like Nimdots, in which the players have exactly the same options in any given position (an impartial game) will be characterized by such values.
Nimstring is a game which is equivalent to Nimdots. It was invented by Elwin Berlekamp to help analyze Dots-and-Boxes positions mathematically.
Nimstring is the game played exactly like Strings-and-Coins, except that the number of coins taken isn't counted, and the loser is the person freeing the last coin.
Nimstring is characterized by Nimstring values, which are the exact equivalent of Nimdots values.
Nimstring is dual to Nimdots, just as Strings-and-Coins is dual to Dots-and-Boxes. Therefore, it is slightly more general than Nimdots, since it can be played on any graph.
This rule says: In a normal game, once the number of chains is determined, no possible move will change which player will have to first enter a chain or cycle.
This is the number of boxes in the longest chain. It is what you get if you win the fight for the number of chains, and keep control in the final phase.
Understanding the quad is one of the keys for defending with Green in 5x5 (4x4 box) Dots.
A short chain not usually called a chain, because you cannot use it to leave a doublecross.
You can count short chains using the short chain rule
This is the part of the game in which all that is left is short chains, long chains, and cycles, and in which players alternate giving away short chains.
Without exception, players should always give away the shortest short chains first.
The short chain phase happens right after the neutral phase and right before final phase of the game.
The scoring can be done more efficiently using the short chain rule.
This helps you how to count who gets the most short chains duing the short chain phase of the game. It says:
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 also says that you should count points in short chains backwards from the largest to the smallest.
The short chain rule also implies the zero rule dealing with the case when there are no chains or cycles.
The short chain rule is a simple consequence of the chain rule
Strings-and-Coins is a game invented by Elwin Berlekamp to generalize Dots-and-Boxes to general graphs.
The game is played on a graph, where the edges are strings and the vertices are coins.
The players alternate cutting strings. When a player cuts all the edges surrounding a coin, he takes the coin and moves again. The player with the most coins wins the game.
Strings-and-Coins is played on the dual graph of Dots-and-Boxes, which means the following: For any Dots-and-Boxes position, the corresponding Strings-and-Coins game is constructed by considering the boxes as coins, and the edges of the Dots-and-Boxes game as strings. It is seen that placing an edge in the Dots-and-Boxes position separates two boxes, so this has exactly the effect of cutting a string in the corresponding Strings-and-Coins position. From this point of view, the two games are equivalent.
Strings-and-Coins has the advantage that it can be played on any graph. It can also simplify certain Dots-and-Boxes observations. For example, the fact that in any Dots-and-Boxes positions, there is no distinction between the two side edges of a corner box is easy to see in Strings-and-Coins, since it says when there are two two strings with each having one connection to the same coin, then the two strings cannot be distinguished.
On each turn, a player makes one move for each box that he takes, plus one extra move (except on the last turn).
Turns are denoted by numbers. Yellow takes turn 1, Green turn 2, etc. The first player (Yellow) always makes the odd turns, and the second player (Green) always makes the even turns.
The definition of turn is important in the proof of the chain rule.
The Zero Rule: If there are no chains or cycles, 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.