Dots and Boxes Glossary

Ilan Vardi

• Chain

A chain is a string of 3 or more boxes. If you want to be perfectly precise, you say long chain.
If you move into a chain, then you allow your opponent to take all the boxes (zero doublecrosses) or else take all but the last two boxes (one doublecross).

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.

• Chain Fight

The chain fight is the battle to get the number of chains consistent with the chain rule. In 5x5 Dots, Yellow wins the chain fight with one or three chains, Green wins with zero, two, or four chains.

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.

• Chain Rule

The chain rule tells you how many chains you should make to force your opponent to open the first long chain or cycle:

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

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

For the usual board sizes, this gives:

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

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

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

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

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.

• Combinatorial Game Theory

This is the branch of mathematics which studies the theory of games such as Dots-and-Boxes. That is, games in which all the information is available to the players, there are no secret factors,such as cards to hide from your opponent, and no element of chance, such as dice. Other games analyzed by this theory are Go and Chess.

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.

• Control

Control means keeping the initivative in the final phase of Dots and Boxes. That is, when your opponent has opened up a chain or cycle and you continually force him to open up chains and cycles by leaving him the last two boxes in each chain, or the last four boxes in each cycle.

• Cycle

A cycle is a closed loop of four or more boxes.

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.

• Dabble

Dabble is a computer program written by J.P. Grossman. It plays at a very high level (about 2200 Yahoo), and can be easily downloaded from this website.

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.

• Doublecross

A doublecross is when you leave your opponent with two boxes, which he takes by making a single edge.

Leaving doublecrosses is the fundamental technique which allows you to implement the basic strategy.

• Doubletrap

A double trap is when you offer your opponent a 2-chain, giving him the possibility of either taking the two boxes or declining with a doublecross.

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.

• Final Phase

This is the part of the game in which there are only chains and cycles left.

• Joins

A join is the most common way of connecting two chains from a distance.

Diagram

A join cannot be disconnected without giving away one of the two chains.

• Life and Death

Life and death is my term for the problem of deciding if a chain can be forced in a region of the board.

It is a term borrowed from the game of Go.

• Loony Move

Loony move is a term from combinatorial game theory. In Dots, it is any move which allows a doublecross, that is, any move which enters a chain or cycle, or which makes a doublecross from a doubletrap.

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.

• Mathter

"To mathter" something means to master it using mathematical ideas and results. For example, one "mathters" a Dots opponent by beating him using mathematical techniques. This word is based on the word "Math-mastering" used by Aviezri S. Frankel to explain his lifelong interest in mathematical games, see the Introduction.

• Move

A move is drawing a single edge.

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.

• Normal game

A normal game is one in which both players take turns without moving into any chains or cycles (or declining a doubletrap with a doublecross) until all that is left is chains and cycles. At this point, a player will have to move into a chain or cycle. A normal game will continue with the other player responding by taking control and keeping it. A normal game ends when the player having control takes all the boxes in the last chain.

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.

• Neutral Phase

This is the part of the game in which all chains and their lengths have been determined. Players alternate placing edges without capturing boxes.

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

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.

• Nimdots Value

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:

• A losing position always has value 0.

• A position in which the possible moves enter positions with values A, B, C, will have value the smallest number not among A, B, C. For example, if from X, one can move only into positions with values 0, 1, 2, 4, then X has value 3.

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

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.

• Non-Chain Rule

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.

• The Prize

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.

A special name for a 4-cycle.

Understanding the quad is one of the keys for defending with Green in 5x5 (4x4 box) Dots.

• Short Chain

A short chain is a chain of one or two boxes.

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

• Short Chain Phase

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.

• 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

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.

• Turn

A turn is one complete set of consecutive moves by one player.

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.

• Zero Rule

This concerns the case in which there are no chains or cycles. Since zero is an even number, one could think that this case is advantageous for Green, who needs an even number of chains. However, this is exactly wrong.

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.