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Bent four in the corner is dead
Path: SecondCourseOnKo · Prev: NoResult · Next: EternalLife
Keywords: Life & Death, Ko, Rules
This page discusses the fact that under Japanese rules, a group which is surrounded by a live enemy group and whose eyeshape can be reduced to bent four in the corner is dead.
Let's talk over this position from bent four in the corner. 1. Japanese rulesBy Japanese rules the white group in the above diagram is considered dead and is removed without playing any ko. The reason for this is as follows. Since White can never play a nor b, Black could fill all outside liberties after play has stopped at the end of the game, and then play a to create the ko explained in bent four in the corner and shown in the following diagram:
The rule is now that in the determination of life and death at the end of the game, after play has stopped, the only ko-threat for a ko is a pass for that ko. If White makes such a pass, Black wins the ko, anyway.
The first written Japanese rules, in 1949, also ruled that "bent four in the corner is dead". This ruling has been discussed ever since, because the fact that some (even many)[1] games contain unremovable ko threats makes this rule unfair towards the defending side in this position. Karl Knechtel: How about a double bent-four situation like this?
It seems to me that when one side starts a sequence (by extending to either a, forcing a play at the corresponding b, and replacing the circled stone), that the other corner can provide ko threats for the first fight. If the response to a is the other a (applying mutual damage), then the first player has the choice of capturing four stones and starting a sequence, or capturing a large group and being captured in return (in which case there is nothing more to dispute). What asymmetries can be introduced here? The Japanese rules suggest to me that both "sandwiched" groups die leaving an even result, but is there a way that the first player can do better than that? (Obviously there is no way the second player can do better, because of passing - the same reason that the proper value of komi must be nonnegative.) Suppose one of the "sandwiched" groups contains some extra stones; if the player with the large group is to move, can both groups be beneficially saved? Andre Engels: At least under rules when bent four is dead, there is absolutely no possibility of gaining from playing in this situation. Playing in the left does not gain Black anything: if White does not react, Black has simply lost a point. Playing on the right does not gain anything either - White just captures and has nothing to worry about. This type of position seems to be only interesting in rulesets where dead stones have to be captured. Graham: So what is the ruling for this configuration ?
I don't think either player will be in a rush to fill the mutual inside liberties, since once these are filled on a group the other player would start the bent four ko and kill the outside group and hence break the whole-board deadlock. So is this seki even under Japanese rules ? 2. Other rulesetsUnder Chinese rules or under rulesets where life and death have to be proven rather than simply decided, the ko should be played, because unremovable ko threats can make a difference. 3. CautionWhen the surrounding group of a group which is supposed to be dead by this rule is not alive with two eyes itself, the rule doesn't apply: in that case the aforementioned ko will be played out. [2]
Juha Nieminen: I came up with this position to demonstrate how it's possible to construct a situation where the "bent four in the corner is dead" rule is unfair. Assume no komi and no prisoners. According to the Japanese rules the black group at the upper left is dead and White does not have to demonstrate this. By this rule White wins by 14 points. However, there's absolutely no way White can win this game by actually playing it through. There are basicly three things White can try, which I have demonstrated below:
If White plays the bent four by playing at a in the diagram above, Wlack captures with b and this sequence follows. Now White has two options: Either ignore black's ko-threat or answer to it. If White answers the threat by playing at c, Black will capture the ko by playing at 2. Now White has no ko-threats, so must pass, after which Black captures by playing at d. The end result is that Black wins by 2 points. If White does not answer the threat but captures by playing at d, Black will capture at e and win the game by 5 points.
White could also try to remove the ko-threat with 1, but Black naturally captures with 2. Even though now the black group at the upper left is truly dead, Black still wins by 5 points.
If we considered the situation at the upper left corner a seki, Black would win by 2 points. Thus there's absolutely no way White could win this by playing. The Japanese rule may be anomalous but it's not unfair. It applies to both players and they know about it before they start the game. Isn't the standard way to show it is anomalous just to posit a double ko elsewhere on the board? Authors: Andre Engels, HolIgor, Juha Nieminen, Karl Knechtel, and Dieter (who also did a WikiMasterEdit). [1] Many games in number, but a tiny fraction of all games. [2] Well, it can be more complicated than that too, but I think I'll leave that for some other Robert. Path: SecondCourseOnKo · Prev: NoResult · Next: EternalLife This is a copy of the living page "Bent four in the corner is dead" at Sensei's Library. ![]() |