Go (board game)

(CC) Photo: S. Venga
Modern game board of Go the game.

Go is a board game played by two players. It is also referred to as Weiqi in Chinese (圍棋; 围棋), Baduk in Korea (바둑) and Igo or Go in Japanese (囲碁; 碁). Go is the world's oldest game that is played in its original form, with a documented history of over 2,500 years.

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Character

Go is played on a flat board with a grid of 19x19 intersections. Two sets of white and black stones are used. The game is played in turns and unlike Chess, black makes the first move in go. Each stone is placed on an intersection and the goal is to capture more territory than the opponent. In go, it often matters whether a given move is beautiful or produces good shape.

It is possible to play essentially the same game on a board of size different from the standard 19 by 19. Beginners, children and those in a hurry, often play on 13 by 13 board, or even on 9 by 9.

Geometric concepts of the game

The geometric-analytic representation of the go 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 called 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. And a set is called 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 the everyday (non-mathematical) language we say that 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 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, 1 point is white, and the score is 2.

Rules

There are a few different Rule Sets for playing Go. They are very similar to each other and for most games the different rules give similar results. The core rules are:

1. Each turn a player places a stone on one empty intersection. Afterwards it's the other player's turn.

2. A stone is a member of a group of stones if one stone is connected to another stone of the group with the same color via a line (there are no direct diagonal connections). If there is no stone that is connected via a line to an empty intersection in a group that group is dead and has to be removed from the board and the stones become prisoners.

3. You aren't allowed to make a move that directly reverses the move of your opponent (your move would remove the stone of your opponent and that stone closed the last intersection of one of your stones and you want to replay that stone).

4. If nobody wants to make a move and passes the game is over. All stones that could be captured through adding additional stones are dead and get removed from the board and become prisoners.

After Japanese rules the winner is the person whose sum of enemy prisoners together with the amount of empty intersections that surround his stones is greater than the other person's.

In Chinese rules you add the number of your stones to the number of your surrounded empty interactions and the player that has the most points wins.

Because making one prisoner reduces the amount of enemy stones exactly by one both methods usually get the same result, besides the fact that in half of the games there is a one point difference because white passed first and black played one stone more than white.

While the above rules are sufficient for most games, specific rules that exactly define the terms that are used in the rules can be a bit longer.

History

There is no exact date for the invention of Go. One legend dates the invention to the Emperor Yuo who taught the game to his eldest son Dan Zhu. Most modern writers think, that this legend (and a few similar legends), were written down in the Han period, to make the game appear older than it really is. They date the invention to the period of 1000-400 BC.

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 postulates 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. In order 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 deads 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).

Comparison to chess

Go is conceptually simpler than chess (especially when go rules are properly formulated):

• A go player has only one kind of pieces, called stones. A chess player has six kinds of pieces (king, queen, rook, bishop, knight, pawn, and one of her/his bishops runs on white squares, while the other one on the black--thus they are actually different too).

• A go player makes only one kind of moves, namely setting a stone on an intersection point (the effect may be different each time, causing sometimes a group of opponent stones to be removed). A chess pawn has four kind of moves: 1.going one step forward, 2.going two steps forward from its initial position, 3.capturing an opponent's piece one step askew from it and landing where the opponent's piece was, 4. capturing opponent's pawn en passant. On the top of it, the pawn, which reaches the last row gets promoted.

• Go has essentially only one (very natural) restriction on moves: a move which would lead to a repetition of position is illegal (not allowed). It has also another rule, which disallows a suicide, but it's only a cultural rule, not essential to the game. In chess too we have a cultural rule which disallows to put your own king in check. In addition, we have also several essential rules which contribute to the total chess rules complexity: castling, en passant, promotion and the rules about draw by repetition or by making 50 moves by both sides without any capturing and without any pawn move, or by stalemating your opponent.

Chess as a whole does not admit any natural, regular generalizations onto larger boards (but many chess endings do). In the case of go, one may play the game on the square boards of arbitrary n by n size, and also on rectangular m by n boards. More than that, one may play go on arbitrary finite simple graphs. Thus go is so mathematical that it provides a graph invariant: with arbitrary graph we may associate the result of the game played optimally by both players (mathematical theory of games states that such optimal strategies exist; it's a corollary to the respective Zermelo's theorem, 1913).

Go is one of the most complex games in the world, far outweighing games such as chess in the number of possible game positions. Indeed, computer chess programs, by year 2000, became as strong or even stronger that the strongest (human) chess players. On the other hand, computer go programs don't have a chance even against go prodigies so far.

Major Titles

There are 7 major go titles in Japan. The record for winning the most titles over the years is held by Japanese professional Cho Chikun, who has won 71 titles.

Tournament	Prize money	Current title holder
Judan		Cho Chikun
Tengen		Kono Rin
Oza		Yamashita Keigo
Meijin		Takao Shinji
Gosei		Cho U
Honinbo		Takao Shinji
Kisei		Yamashita Keigo

Cultural Dimensions

Go strategy is also studied as an metaphor for Asian strategy compared to western strategy.

Neuroscience Research

Neuroimaging research suggests players use the right hemisphere of their brain while playing Go, responsible for position and orientation, slightly more than the left side. The game requires a more global strategy than chess, and thus differs from the more left hemisphere activated activity of playing chess. A study of six subjects suggests successful Go playing requires practice and experience rather than sheer intellectual ability. Another study suggests the ability of professional Go players to assess visual information and nonmotor learning memory processes is mediated in part by the cerebellum and precuneus.