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The Mystery of Go
I J Good
21 January, 1965
GO, the Japanese national pastime, was recently described by Ralph Fox, a Princeton professor of mathematics, as the most interesting game in the world. At any rate many expert chess players, including Emanuel Lasker, who was World Chess Champion for 28 years, have held that Go is more interesting than chess, and it is not easy to think of any third game that is a serious rival. Another, unrelated chess master, Edward Lasker, believes that Go will replace chess as the leading intellectual game of the Occident just as it has reigned supreme in the Orient for some four thousand years.
In Japan the game is known as I-go or Go, in China as Wei-k'i or Wei-Chi, in Korea as Patok. It is played by a high proportion of educated people in Japan, including many Geisha girls, and ability at Go is relevant to promotion in many firms.
There is a very large literature of Go in the Orient, but I have been able to trace references to only twelve books in English that deal primarily with Go. Some of these are listed at the end of this article.
The basic rules are simple, although there are some rare situations where there is some difficulty in deciding who has won. Owing to the possibility of such situations, there are some minor variations in the rules from place to place and from time to time, but the basic rules are clear-cut.
The fascination of Go depends on the simplicity of the basic rules, on the great variety of situations to which these simple rules lead, on the mystery of the strategic principles, on the gradual unfolding of this mystery, and to some extent on an analogy with land warfare. (This analogy could be improved by playing on boards resembling maps.)
Another game with simple rules and practical complexity is the child's game of Boxes, but it has complexity without qualitative variety, and is too difficult for humans owing to the paucity of general principles for good play. Nominally a game of pure skill, in practice Boxes is almost a game of chance. But in Go, one's understanding and play can improve without limit. Whereas in Boxes there are only about three grades of players, in Go there are some thirty grades, such that a player could give, say, 2 to 1 odds that he would beat a player in the next lower grade. The number of distinguishable grades is a reasonable measure of the extent to which a game is really a game of skill. Fortunately, players of very unequal strengths can have an interesting game of Go by making use of a convenient, standard method of handicapping.
The rules are basically so simple that perhaps a game very much like Go is played in many extra-terrestrial places, even within our own galaxy. All the pieces are of the same nature and there are no moves.
There is a board consisting of 18 x 18 squares, giving 19 x 19 vertices (see diagram above).
There are two players, Black and White, who take it in turn to place black and white stones (counters) on these vertices. By convention, Black plays first. Once a stone is placed it can never be moved, but it can be captured and removed from the board. The object is to control more territory than your opponent at the end of the game. Mathematicians may see the game as a kind of combinatorial topoology; biologists, as competition between organisms.
The simplest form of capture of a single stone is by surrounding it:
All other captures are generalisations of this, but the word surround is misleading in the general situation. As is implicit in the diagram above, the rules of Go do not refer to diagonals: only horizontal and vertical connections are relevant. Here, a small army of two Black stones has just been captured:
Capture an army
If one player is known to be appreciably stronger than the other, he handicaps himself by allowing the weaker player to take black and to place extra Black stones on the board before the start of play. These handicap stones occupy the positions marked with heavy dots on the Go board, and are generally recognised to have the greatest strategic value.
An army is a connected group of men, or stones, all of the same colour, such that one could travel from any one man of the army to any other, by single, non-diagonal steps, coming to a man of the army at every step. A single stone, not adjacent to another stone of the same colour, is itself an army. A typical army is:
In order to capture such an army, White must occupy all the vertices that are adjacent to the army, both inside and outside it, thus:
Capture an Army
These vertices, when unoccupied, are called breathing spaces or liberties. White must occupy the eye at d4 last, in capturing the Black army; otherwise this placement would be suicidal, and an apparently suicidal placement is permitted only if it results immediately in a capture.
In the following further examples of captures, it will be noticed that armies having stones on the edge of the board have fewer liberties than similar armies in the middle of the board:
In this last example, White must occupy the eye at a2 last.
In the next situation, White captures two Black armies by playing at e3, and, of course, no White stones are removed from the board: they never are immediately after a White move. Playing at e3 would be suicidal and illegal if vertex f4 had been unoccupied:
Capturing Two Black Armies
We have already noted the significance of an eye, or empty territory, inside an army. It is clear that an army with two or more eyes, as in these two examples, is safe for the rest of the game provided that the player does not fill in his own eyes:
The opponent cannot legally fill one eye while the other one is open, because suicide is against the rules.
The ladder. - Consider now the following position:
Black may wish to try to save his army, but he cannot do so, if White carries out a ladder attack, thus:
With an additional stone at b4, if White again attempts a ladder attack, Black can turn the tables on him. White is forced to play at c5 and then at f3, to protect his stones at c4 and e3:
Stopping the Ladder Attack
The outcome exemplifies one of the principles of Go, that an attack should not usually be launched unless it is going to be successful. Note that in this example the single Black stone, remote from the start of the potential ladder, is exerting an influence on the game out of all proportion to its apparent strength. The influence that a stone can have on play in another part of the board is of great importance in Go strategy.
The control of territory. - At the end of a game there are two methods of scoring. They are known as Chinese scoring and Japanese scoring and are almost equivalent. In Chinese scoring, every vertex occupied by White, or adjacent to a White stone but not to a Black stone, is counted to White, with a similar rule for Black. To apply this method of scoring it is in principle necessary to play the game to the very end.
Partly for this reason, Japanese scoring is nearly always used in practice, although Chinese scoring shows more clearly that the object of the game is to control more territory than the opponent. In Japanese scoring, occupied vertices are not counted: instead the number of unoccupied vertices surrounded by one's armies is added to the number of the opponent's stones that one has captured (prisoners). In practice, the prisoners are used to fill the opponent's territory when the game has ended, in order to simplify the counting process. Note therefore that, for Japanese scoring, it is necessary to retain prisoners, although they can, of course, be exchanged, one-for-one, with the opponent's prisoners. A full set of stones consists of 180 of each colour, so exchange of prisoners is never a practical necessity - unless stones have been lost through cracks in the floor-boards.
The end of the game. - At any turn it is legal for a player to pass, that is, to refuse to place a stone. (In Japanese scoring, be should pay by handing over a prisoner.) If the two players both pass in succession - White having the last turn - then the game is over provided that there is agreement about which armies still on the board would eventually be rounded up, and which can therefore be removed as prisoners. If there is disagreement, which is unusual, then it is necessary to play on.
Ko. - Rules are required, as in chess, in order to prevent the game from running into a cycle. The main rule is the Ko rule, and it applies only to immediate repetitions. It is illustrated here:
If White plays at d3, capturing one Black stone, Black cannot immediately play back at e3, capturing one stone and leading to the original position with White still to play. Instead, if Black is anxious to win the Ko, he must make a threat elsewhere, large enough to force White to answer, and only then can he recapture at e3. Then White might make a threat elsewhere and afterwards capture in the Ko, and so on. Eventually Black might win the Ko by filling at d3; or White, by filling at e3. These Ko fights are sometimes very exciting indeed and the outcome of the game often depends on them.
If a non-immediate repetition of position occurs, with the same person to play, then the game is drawn. This outcome is very rare, but it can happen if there are three or more Kos on the board. The reader might like to verify that it does not happen if there are only two Kos.
Seki. - An army is sometimes safe even without eyes, but it dares not be aggressive. In this example:
neither player dares play at d1. The situation is known as a Seki or impasse. The unoccupied vertex d1 does not count to either player as it is adjacent to armies of both colours, neither army being lost.
Avoiding book openings. - A weakness in chess is that there is too much knowledge about the openings, and it is constantly increasing. It is not good for a game if its mastery requires much rote learning. In Go, although play in the corners of the board is somewhat stereotyped among masters, one can become a competent Go player by occidental standards with very little knowledge of these so-called Joseki.
In chess, the parrots can be defeated by playing Randomised Chess, wherein the pieces on the back lines are permuted at random. Similarly, Randomised Go could be defined in terms of a random deletion of some of the vertices near the corners of the board. The corners could even be abolished by playing on a cylinder or an anchor ring: this could be done, without using a magnetic set, by identification of opposite sides of the ordinary board, either one pair of sides or both pairs. Another form of Go is that with more than two players, all against all, with one colour for each player. Since it is possible for players to form coalitions, this form of Go bears some resemblance to power politics and is liable to create an emotional scene.
But ordinary Go is fascinating enough for people with an IQ between 110 and 190 - in fact, too fascinating. Sometimes the game becomes an addiction, like smoking, food, drink, television, chess and women. It seems to appeal especially to scientists and mathematicians, because of the emergence of the Gestalt - a unity with a significant pattern - out of a collection of discrete entities and axioms.
Go on a computer? - In order to programme a computer to play a reasonable game of Go - rather than merely a legal game - it is necessary to formalise the principles of good strategy, or to design a learning programme. The prlnciples are more qualitative and mysterious than in chess, and depend more on judgment. So I think it will be even more difficult to programme a computer to play a reasonable game of Go than of chess.
The experienced player will often be unable to explain convincingly to a beginner why one move is better than another. A move might be regarded as good because it looks influential, or combines attack and defence, or preserves the initiative, or because if we had not played at that vertex the opponent would have done so; or it might be regarded as bad because it was too bold or too timid, or too close to the enemy or too far away. If these and other qualitative judgments. could be expressed in precise quantitative terms, then good strategy could be programmed for a computer; but hardly any progress has been made in this direction.
Some books on Go published in English.-
Edward Lasker, Go and Go-Moku (Dover Publications, New York, 1960).
Kensaku Segoe, Go Proverbs Illustrated (Japanese Go Association, Tokyo, 1960).
Kaku Takagawa, How to Play Go (Japanese Go Association, 1958).
Kaku Taikagawa, The Vital Points of Go (Japanese Go Association, 1958).
A. Smith, The Game of Go (Charles Tuttle, Rutland, Vermont, 1956).
Count Daniele Pecorini and Shu Tong; The Game of Wel-chi (Longgmans, Green and Co., 1929).
In this account I have generally followed the rules as specified in an article in Go Monthly Review (Tokyo), Vol. 4, No.7 (1964),pp 51-52, 55. Go Monthly Review is available from the British Go Association, 12 Third Avenue, Wembley, Middlesex.