To

If you use them in battle, but the victory does not come for a long time, their weapons are dulled, and their zeal weakens. If you besiege the city, their strength is depleted. If you subject the army to a long war, the state’s reserves are not enough.
...
Therefore, I heard about the success of fast military campaigns, and did not hear about the success of protracted. No state has benefited from a long war.

Sun Tzu "The Art of War "

In the previous article , I talked about how the seizure of go stones is performed, but did not mention one rule. The rule is so important that it has its own name - Ko .

Battle for eternity

Ko is a translation from the Sanskrit Hindu term kalpa or "Day of Brahma". Strictly speaking, there is a whole gradation of different types of kalpa, from short (average human life expectancy) to “great” (80 small kalps), but they all define very long periods of time. Most likely, this means “small kalpa”, that is, human life, during which players could continue the game (with an unlimited supply of stones), hitting the position depicted in the illustration at the beginning of this article. However, when playing bodhisattvas , the kalpa could well be the middle one.


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Players alternately take each other’s stones, repeating the same position over and over again. Since the number of stones taken by the parties is equal (with an accuracy of one stone, depending on the turn), the duration of such a game does not affect the final result. To break this vicious circle, the “Ko rule” was introduced. Position on the board should not be repeated!



After capturing the stone with black, White cannot immediately pick up the stone that was in the position of Atari , since this would lead to the repetition of the previous position (for the convenience of players, the forbidden position is marked with an empty square). White goes somewhere else in the board (or folds) and black can break the cycle by connecting his stones.

Often, especially in the final phase of the game ( yose ) of equal opponents, the result of the Ko play is important. In such cases, Ko-wrestling can start. In order to pick up a stone protected by the Ko rule, a player must make a move that changes position on the board. But this move must be such that the enemy is forced to answer it, because otherwise he will complete the Ko-fight, in his favor, by connecting the stones. Such a move is called Co-threat.

One should not think that Ko-wrestling can occur only in batches of strong players. As an example, I can cite a fragment of my party with dizews . This was my second game with a man (before that, I played with bots) and I hopelessly lost it at the very beginning. Actually, in this game, there would be nothing interesting if it were not for the complex, multi-stage Co-fight, practically at its very end.



Of the assets, at that time, I had two groups of “with eyes”, completely locked in white, and I really wanted to build a third one. In turn, the enemy tried his best to crush her. I needed Co-threats in order to win the pace and I found them. A move to S19, in other circumstances, would have been pure suicide, but it was a move that was necessary to react to, since otherwise I would have taken a large group of white stones, along with strengthening myself in the previously lost upper right corner.

Of course, White played R19, completely burying my attack. White did not take the stone on S19, although it was worth spending one move on it (in this case I wouldn’t have any threats left to complete the second “eye”). Then there was a small additional Ko-fight in the upper right corner, which gave me a couple more paces and an Atari for a group of two white stones, which allowed me to connect, completing the Ko-fight in my favor.

Unfortunately, this game also illustrates another, less pleasant fact for me. The subsequent analysis showed that this whole Ko-struggle was completely devoid of meaning. I could crush the whites at the bottom of the board, but my narrow-mindedness in one single scenario did not allow me to see all the possibilities.

Taking off your head - do not cry through hair

Probably many of us played in school in the " points ". This game does not require special equipment. In order to play it, it is enough to have a clean sheet of paper lined "in a cell" and two fountain pens of different colors. This game arose as a result of the incorrect interpretation of the rules of the game of go. Consider the rules of "points" in more detail:

• The game is played on a 39 × 32 pitch field.
• Players take turns (1 move - one point), you can not skip a move.
• When creating a continuous (vertical, horizontal, diagonal) closed line, an area is formed. If inside it there are points of the enemy, it is considered to be an area of ​​environment, which is forbidden to put a point to any of the players.
• When an opponent's point appears in a free area, the free area will be considered an area of ​​environment provided that the opponent's point was not the final one in his environment.
• Points that fall into the region of the environment, then do not participate in the formation of lines for the environment.
• Points placed at the edge of the field are not surrounded.
• The winner is the player who has surrounded a larger number of points of the opponent.

The main difference from go (besides the impossibility of playing “under stones”) is the way to capture groups. At the “points”, to capture, it is necessary to surround the enemy points with a continuous closed loop. At the same time, the presence of dam (including the “eye”), the presence of points of their own color in the captured territory, etc. are completely ignored. The previously surrounded area can be re-surrounded by a larger contour. Such an interpretation greatly simplifies the game, pushing beyond its limits all the problems of "life and death." In fact, the game turns into a race for the environment of larger and larger groups.

Points placed on the edge of the field can not be surrounded! This is the second most important difference from go. In Go, stones located along the edge of the field are easier to capture, since they initially have less dam. You can consider the areas located beyond the board, as a kind of, indestructible neutral wall, equally used by players. It is clear that the rule prohibiting the capture of items located on the edge of the board, radically changes the tactics of the game. In addition, the size of the playing field in the "points" is very large and not standardized. If most tactical methods are tied to the size of the field, then at “points” this parameter practically affects only the duration of the game, leaving tactics and strategy unchanged.

The goal of the game is to capture the maximum number of enemy points, also leading to a strong simplification of the game. The goal of the Go game is to enclose the maximum territory by players (and, in the Japanese version of the rules, only blank points are taken into account when counting). The enemy’s taken stones, although they are counted, are not the main goal of the game. Dead groups located on the enemy’s territory are taken at the end of the game automatically, which makes it possible not to completely surround them, thereby reducing their own territory.

Summing up, it can be said that the way of playing “with a pencil and paper” can be considered as one of the radical ways of fighting Ko, but the version of the game that is common in European territory has very little in common with the original game. There is a version of the game that is closer to the rules of go, but, in our locality, it is much less well known.

Medicines worse than the disease

Some time ago, a user of VlK published an article devoted to the Ludi project, the purpose of which was to automate the process of developing new board games. I will not dwell on the project itself (this is a separate and very big topic), let me just say that with its help we really managed to develop several new, quite interesting games. One of the side results of the research was the publication of an article in which one of the authors of the project considers the problem of Co. on a hexagonal board.



Indeed, the classic “short Co.”, in such a situation, is impossible, but it is worth thinking about the reasons why this happens. Despite the fact that there are no diagonal connections in Go, diagonals are important! The diagonal allows you to "cut" the line by starting the environment of the enemy group. On a hexagonal board, the diagonal can not cut anything! There are no intersecting diagonals:



In addition, in contrast to the usual board, on the hexagonal, each stone (in the center of the board) has six neighbors instead of four. As a result of all this, the game loses its tactical diversity, becomes more static and predictable. In addition to all the above, Ko’s situation remains possible when playing more than two players:



Variants of the “Long Co.” are not considered in the article. In my opinion, the use of a hexagonal board is not the best method for combating Ko. Together with the "water" from the trough was thrown out the "child". By the way, this version of go is not fundamentally new. TriGo is a fairly well-known and well-studied game.

Of course, the use of hexagonal boards is not the only investigated possibility of combating Ko. Many variations of rules have been proposed that make the “Ko rule” redundant. So in One-Eyed Go, the “advantage of the attacker” rule was canceled. In this variant of go, it is forbidden to place a stone, depriving its group of the last dame (even if this move leads to the taking of a group of opponent's stones).

The option Capture in Atari suggests to immediately remove from the board the groups that have one dame left, citing the fact that, as a rule, such groups are still “dead”. In my opinion, somewhat dubious motivation. In No-ko Go it is proposed to prohibit not the repetition of the position, but the very beginning of the Ko-sequence. This sentence leads to funny incidents. For example, the white configuration shown on the left, according to these rules, is considered “live”. In the variant Geneva Ko Rule it is proposed to remove both stones that entered into the Ko-fight. In general, it can be said that such changes to the rules, “overriding” Ko's rule, change the very nature of the game so much that the final version seems to be very little.

From the depths of the ages

At present, it is difficult to determine in which country exactly and for how long Guo was born. According to one version, it happened in Tibet. The game called Gundru, pressed another game of the same name (something similar to Reversi with moving chips). Of course, the “Tibetan go” rules were significantly different from the modern ones.

• In Tibetan go board is used 17x17 lines (and not 19x19, as in modern)
• The game does not begin with an empty board, but with some initial arrangement of twelve white and black stones.
• White begins the game
• The moves are allowed no further than one point from the stone of its own color already on the board
• The player who first placed the stone in the center point of the board receives an additional 5 points when calculating
• If the opponent captures the corner of the board where the player’s stone is located, then the player’s score is 20 points deducted
• It is forbidden to go to the points that were removed from the previous move

In the context of our article, the last of the listed rules is especially interesting. Obviously, it excludes the possibility of a “short Co.”, making minimal changes to the game. The tactics of the game, of course, is changing, but not as radically as in Paper and Pencil Go or One-Eyed Go . There is a version of the game that further tightens this rule. In Stoical Go, it is forbidden to take stones if the opponent made a take on the previous move. Those who wish can try their hand at this game (however, I must warn you that the AI ​​of this implementation is extremely weak).

Ko defeated?

After such a long and detailed discussion, it may appear that the problem of Ko has been investigated along and across and there is nothing to add to what has been said. Of course, it is not. Most of the above approaches consider only the situation of a “short Co.” - a classic two-way cycle. But can the position be repeated after a longer chain of moves? Of course, yes:



When playing games, you can declare the Super Ko rule, but to put it into practice is technically rather difficult. It is not enough just to check the subsequent position for a match with the previous one. It is necessary to perform a comparison with all previous positions! When using computer implementations, Zobrist hashing can help .

But is it worth it? Can a “long Co.” arise in a real party? I will answer this question with the following quote:

On June 20, 1582, a party took place between the Honimbo Sansa and the Casio Reegan - the strongest players of the time - in the presence of Oda Nobunaga. The game was completed without result due to the triple co. The next day, June 21, Nobunaga's ally, the warlord Aketi Mitsuhide, rebelled. The ruler was blocked in the temple and made seppuku not to be captured. After these events, the triple K was considered a bad omen.