Go, the rules of

Geometric concepts of Go

Geometric-analytic representation of the board

The Go cross-section points can be represented, as in analytic geometry, by ordered pairs of integers, $\ (x,y)$ , where the two coordinates vary between $\ 0$ and $\ 18.$ For instance, the row of points near the player of the white stones consists of:

(0,18) (1,18) . . . (18,18)

while the row of points near the player of the black stones consists of:

(0,0) (1,0) . . . (18,0)

Two points, $\ (x,y)$ and $\ (x',y'),$ are called adjacent, or the nearest neighbors (as in the theory of lattice systems of statistical mechanics) if they are next to each other in a row or in a column; formally, if:

|x-x'|+|y-y'|\ =\ 1

For instance, points (2,5) and (2,6) are adjacent; also (2,5) and (1,5); while points (2,2) and (3,3) are not.

A sequence of Go points is called a path when its each pair of consecutive points is adjacent.

A point is adjacent to a set of points if it is adjacent to at least one point of that set. We also say that a set is adjacent to another set if the two are not apart, i.e. if there exists an adjacent pair of points which has one point in each of the two sets.

Connected sets

A set $\ A$ of Go points is disconnected if it splits into a union of two disjoint non-empty sets, $\ A=B\cup C$ , such that no point of $\ B$ is adjacent to any point of $\ C$ ; such two disjoint sets $\ B$ and $\ C$ are said to be apart. A set is connected when it is not disconnected. It turns out that a set is connected if and only if for every two of its different points there exists a path which starts at one of these points and ends in the other one.

Every set of Go points is uniquely a union of its maximal connected subsets, called its connected components. Each two components are disjoint and even apart one from another, meaning that points from two different components are never adjacent.

Remark: The empty set, and each 1-point set, is connected.

2-point sets: A 2-point set is connected if and only if its points are adjacent.

Board configuration and groups of stones

Each time you have black and white stones on some of the go points (cross-sections) you get a (board) configuration. Formally, a board configuration is an arbitrary function

\ f\ :\ \{0,\dots ,18\}\times \{0,\dots ,18\}\ \rightarrow \ \{-1,0,1\}

Equality $\ f(x,y)=1$ is interpreted as: black stone occupies point $\ (x,y);$
equality $\ f(x,y)=-1$ is interpreted as: white stone occupies point $\ (x,y);$
equality $\ f(x,y)=0$ is interpreted as: point $\ (x,y)$ is vacant.

In everyday (non-mathematical) language, a configuration is any distribution of black and white stones on (the cross-points of) the board. Then we call a collection of the black (respectively white) stones a group if the points which these stones occupy form a connected component of the set of all points occupied by the black (resp. white) stones.

In the section devoted to a version of precise rules of Go, the black color will be associated with 1, and the white color with -1. For instance a phrase like color $\ (-1)^{k}$ will mean color black for even values of $\ k,$ and it will mean color white for odd values of $\ k.$

Removal of stones

Given a configuration $\ f,$ and a set $\ A$ of stones (each of of either color), the removal of stones of $\ A$ means formally the replacement of the given configuration $\ f$ by configuration $\ g,$ such that $\ g(x,y):=0$ for every point $\ (x,y)$ of $\ A$ , and $\ g(x,y):=f(x,y)$ for every other point $\ (x,y)$ of the board.

Liberties and eyes

A vacant point adjacent to a (point occupied by one of the stones of a given) group is called a liberty of that group.

A group of stones which has no liberties is called dead.

If a vacant point is adjacent to black (resp. white) stones only then it is called a black (resp. white) eye or 1-point eye.

More generally, given a configuration $\ f,$ a component $\ A$ of the set of all vacant points is called a black (resp. white) eye if there does not exist a configuration $\ g\leq f$ (resp. $\ g\geq f$ ) such that a point of $\ A$ is a white (resp. black) 1-point eye with respect to configuration $\ g.$

In the ordinary language of Go, a black eye is not a result of a fist landing on someone's face, but it is a connected set of vacant points, surrounded by black stones, and such that it is not possible to create a white 1-point eye by filling all but one of these vacant points with white stones so that the remaining single vacant becomes a 1-point white eye (whether or not we also set white stones on the remaining vacant points, outside of the given connected group of vacant points is irrelevant because it will not affect the status of the points of the given vacant component).

Safe groups and families of groups of stones

Let's look at the simple case before stating the general full definition of a safe family of groups.

A group of black (resp. white) stones is safe if it surrounds two eyes by itself, meaning that even if we remove from the board all other stones of the same color (outside of the given group), there would be at least two different eyes (with respect to the modified configuration) of the given color.

In general, a family of groups of stones of the same color is safe if after removing all stones of the same color, which do not belong to any of the groups of the family, each group of the family will be adjacent to at least two different eyes (w.r. to the modified configuration) of the given color.

If a family consists of just one group then we get back the simple case (the two definitions above are equivalent for a group and a single-group family).

Configuration score

The conceptual notion of the configuration score is virtually necessary in order to define the (practical) notion of the score of a game—to be defined in the section on rules. But these two related notions should not be confused.

Let $\ f$ be an arbitrary fixed configuration (fixed means that we consider just the same one configuration throughout this whole section).

Definition 1 Let $\ (x,y)$ and $\ (x',y')$ be arbitrary go points. We say that it is possible to reach the latter point from the earlier one if there exists a path from $\ (x,y)$ to $\ (x',y')$ such that all intermediate points of that path are vacant.

Definition 2 We say that point $\ (x,y)$ is black (resp. white) if it is occupied by a black (resp. white) stone or if it is possible to reach a black (resp. white) stone, but not white (resp. black), from $\ (x,y).$ Otherwise, when a point is neither black nor white, we say that such a point is neutral.

Definition 3 The configuration score is the number of black points minus the number of the white points.

Remark The configuration score may sound to a Go player at the same time familiar and strange (even silly). This is because a go player almost always thinks about the future configuration and never literally in the terms of the present configuration. Even when the two players agree to end the game, they, as a rule, do not consider the final configuration on the board but one of the equivalent future configurations which would occur if the players cared to make certain obvious moves. Thus they consider the score of one of those potential future final configurations, and not of the final configuration which actually occurred in the game. But we need the simple notion of the configuration score, as defined above, in order to precisely define the actual game score.

Examples

When there are no stones on the board (i.e. the configuration function is identically equal to zero) then all points are neutral, hence the configuration score is zero.

When there is only one black stone on the 19x19 board then the configuration score is equal to 19x19 = 361.

When there is only one black and one white stone on the board then all vacant points are neutral, and the score is zero.

When $\ f(1,0)=f(0,1)=1$ (two black stones) and $\ f(9,9)=-1$ (a single white stone in the center) and all other points are vacant, then 3 points are black (2 stones and point (0,0)), 1 point is white (a stone), and the score is 3 - 1 = 2.

A version of precise, complete rules of Go

A Go record is a finite sequence $\ f_{0},\dots ,f_{n+2}$ of configurations (where $\ n$ is a non-negative integer) such that the following six conditions hold:

$f_{0}\$ is identically $\ 0$ (the board is empty)
$f_{n}\ =\ f_{n+1}\ =\ f_{n+2}$
if $\ k<m\leq n$ and $\ f_{k}=f_{m},$ then $\ k+1=m<n$
configuration $\ f_{m}$ does not have any dead group of any color
for every $k=0,\dots ,n$ there is at most one point $\ (x,y)$ , called the click $\ k$ -point, such that $\ f_{k}(x,y)=0$ and $\ f_{k+1}(x,y)=(-1)^{k}$ (the click value)
if $\ (x,y)$ is a click $\ k$ -point then configuration $\ f_{k+1}$ is obtained from configuration $\ f'_{k}$ by removing the dead groups of color $\ (-1)^{k+1}$ , where configuration $\ f'_{k}$ differs from configuration $\ f_{k}$ only at the click point $\ (x,y)$ by assuming the click value $\ f'_{k}(x,y)=(-1)^{k}$ .

for every $k,m=0,\dots ,n.$

A Go game is the process of making Go moves by two players, of the black and of the white stones, where the player of black stones selects the odd numbered configurations $\ f_{1},f_{3},\dots ,$ and the player of white stones selects the even numbered configurations $\ f_{2},f_{4},\dots ,$ in such a way that they produce a finite sequence of configurations, which satisfies the above listed five assumptions. Each player selects the consecutive configuration based on the full information of the previous configurations, obtained by observing each previous generation from the moment it was selected to the moment the next configuration was selected.

The score of the Go game is the configuration score of the last configuration. The player of black stones strives at maximizing the score, while the player of the white stones strives at minimizing.

Who is the winner?

When, according with the standard rules the game starts with an empty board then experience and common sense show that the player who makes the first move, which is the player of black stones, should get a positive score, when playing against an opponent of equal strength. The score hovers mostly around the values 5 to 8 when two equal, strong players play. Thus tournament directors or some go organization set the so-called komi at 5.5 or 6.5 or 7.5 level, which means that the player of the black stones is considered to be the winner if the score of the game is greater than komi; otherwise, when the score is smaller than the komi value then the player of the white stones is considered to be the winner.

Games between players of unequal strength

Players of clearly unequal strength may start, for the sake of greater enjoyment of the game, from a configuration $f_{0}\$ different from the identically 0-configuration. Depending on the difference in their strength, the initial configuration may be selected in such a way as to make the chances of winning more equal for the two players.

Recording a game

A game can be recorded and stored for instance in an article, book, or in a computer file. To store a game, it is enough to store the click points and word pass or a special symbol when there is no click point. Then the game can be replayed (the consecutive board configurations of the game can be recovered). The consecutive click points can be stored in more than one way. In the books on go, in general and especially for the beginners, intervals of consecutive click points can be shown on one board diagram, when no captures were involved (when no dead groups were removed). After a capture it is preferable to provide a new board diagram with the next interval of click points represented. Each click k-point is represented on the diagram by a stone of color (-1)^k+1, with the numeral k printed on it.

Another way is to write down the sequence of click points (and passes), like this:

1 (2 3) 2 (15 15) 3 (4 2) etc.

Then the reader may drop the stones on the respective points (or imagine them)—first a black stone should be set on point (2,3), next a white stone on (15,15), next a black on (4,2), etc.

Comments about, and explanations of, the rules

In practice, the last part of the game is not really played. Instead, the two players predict what the score would be if they continued by making obvious, reasonable moves.
Rule three says that a configuration cannot be repeated except for two consecutive configurations. (It follows that no configuration may appear three times).
Let $\ f$ be an arbitrary board configuration which has no dead groups. Let $\ (a,b)$ be a vacancy, i.e. let $\ f(a,b)=0$ . Let configuration $\ f'$ be identical with $\ f$ except for $\ f'(a,b)=1$ (a black stone was set on $\ (a,b)$ . Assume that now there is at least one dead group of white stones with respect to $\ f'$ . Then the configuration $\ g$ , obtained from $\ f'$ by removing all white dead groups of stones obviously does not have any white dead groups. A momentary reflection will show that configuration $\ g$ does not have any black dead group either, i.e. $\ g$ simply is free of any dead groups of either color.

This observation is essential in the context of rule six above—if a click point causes removal of a dead group of the opponent stones then afterwards all our groups remain alive, and the move is legal. In short: capturing prevents suicide.

Theoretically, it is possible that a player cannot put a stone legally on any vacancy. Then it is necessary to play pass (in a real game, on a 19x19 board, such a situation is unthinkable, while it is possible when a game is played on a very small board).

Go on very small boards

We can illustrate some of the rules of go easily on small boards.

On the 1x1 board the only game is: 1 pass 2 pass. The score of this game is 0.

Indeed, it is not legal to set a stone on the only point of the board because such a stone would have no liberties. Thus it would be dead. But suicide is not allowed. Thus pass is the only first move, and the only second move.

On the 2x1 board {0,1}x{0} = {(0 0), (1 0)} the best move for the first player is pass:

Indeed, if the first move is for instance 1 (0 0), then the second player may play 2 (1 0) (it would be silly to say pass), thus capturing the black stone on (0 0). Now it's illegal for black to play 3 (0 0), because it would result in the configuration after move 1. Thus 3 pass is the only move by black at this stage of the game. Now white says 4 pass, the game is over, and the score is -2.

Thus black indeed should start with 1 pass. It follows that now 2 pass is the best that white can do.

Conclusion Under the best play of both sides the result of the game is 0.

On the 3x1 board {0,1,2}x{0} = {(0 0), (1 0), (2 0)} the player of black stones can get score +3 (absolutely the best possible) by playing to the middle: 1 (1 0). White has only one legal reply, namely 2 pass (setting a white stone on any of the two vacant points would amount to a suicide). Now black say 3 pass, and the score is +3.

Non-negativity of the score under the best black play

By playing in the best possible way, the first player (i.e. of the black stones) should be able to achieve a non-negative score against the best (or any) play of the second player:

Indeed, if the second player didn't have a strategy which would assure a non-positive score then the above claim is true (according to the respective Zeromelo theorem, black would have a strategy which would assure a positive score). And if the second player had a strategy which would assure a non-positive score then the first player may start with a pass. If the second player replies with a pass too, then the score is 0, and the claim holds. Otherwise, the second player plays a stone. Then the first player may pretend that white is black, black is white, and that s/he is a white player, while the opponent is the black player. Thus s/he will use the white strategy of achieving a non-positive score (under the guise of pretense), thus in reality achieving a non-negative score. Thus in this case the theoretical value of the go game would be 0.

Remark The non-negativity of the score claim is a purely theoretical result because nobody knows what is the best white way of playing (thus black does not know how to pretend to be white).

Major differences between rule sets

The method of counting the result of the game

There are area counting, territory counting and Ing counting. Territory counting has the problem that you lose points when make move in your own territory at the end of the game to capture a potential dead enemy group, if the player don't agree on the status of the group. Territory counting also needs an non-integer number as komi to prevent a draw where

Ing counting has the practical problem of requiring a exact number of 180 stones for each player, which means you might have to count the stones in the box before you begin to play. One advantage of Ing counting is that you can teach it to young kids that don't know who to count to play with the rules.

Some consider Area counting as having the disadvantage of taken more time to count.

Points in Sekis

After Territory style rules points in Sekis that are completely surrounded by one player usually don't count (the white group might have a one point eye and the black one a two point eye in addition to the shared eye of both groups). Area counting usually counts points in those surrounded eyes but not in the shared eye. Ing Rules count in addition to the surrounded eyes also points in shared eyes where the intersection is nearer (in [Manhattan distance]) to the stones of one player.

Ko Rules:

Classical Ko rule

A player can't reverse the last move of his opponent.

Super Ko

No board position may appear multiple times in a game. (the precise mathematical rules above use the Super Ko rule)

Special Positions

Bent four in the Corner

Moonshine life

Triple Ko

Specific Rules by Country

Korean Rules

The Korean Rules are similar to the Japanese Rules.

American Go Association Rules

French Rules

The French rules are similar to the American Go Association Rules.

United Kingdom Rules

On default the rules of the American Go Association are used, but deliberate illegal play leads to forfeit. On the other hand the Japanese rules are also often used in British tournaments with a komi of 6. In addition games that get voided because of a triple ko or a similar position count as jigo (draw). ^[4]

German Rules

The Japanese Rules are usually used in Germany. Komi depends on the given tournament.

New Zealand Rules

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Timesystems

Byo-Yomi

A game played with classical Byo-Yomi give a player after his main time is over x Byo-Yomi periods. A player has y seconds to play a move. If the player takes longer than y seconds he loses one of his Byo-Yomi periods and it he gets x new seconds to make a move. If he doesn't use all his x seconds he gets again x new seconds at his next turn If he loses all his Byo-Yomi Periods

Kanadian Byo-Yomi

A player has a given x seconds to play y moves after he used his regular time. If he plays all his y moves he his clock gets again reset to x seconds and the process repeat. Should the player use all of his x seconds he loses.

Progressive Byo-Yomi

Like the Kanadian Byo-Yomi but y increases every time it is used up. 10 stones, 20 stones, 30 stones, 40 stones, etc would be an example for an usual progression of Progressive Byo-Yomi.

Notes

[1] ttp://www.cs.cmu.edu/~wjh/go/rules/Japanese.html

[2] ttp://www.cs.cmu.edu/~wjh/go/rules/Chinese.html

[3] ttp://www.cs.cmu.edu/~wjh/go/rules/SST.html

[4] ttp://www.britgo.org/rules/compare.html

[5] ttp://homepages.ihug.co.nz/~barryp/rules.htm

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