Territory Value of the 3-3 Stone
Posted: Sat Jul 20, 2013 11:45 pm
There are, in particular, these opinions on the territory value (excluding the influence value or its territory equivalent) of the 3-3 stone in an empty corner:
Jasiek: the 3-3 stone is 8 points.
Yi Ch'ang-ho: the 3-3 stone is 4 points. [1]
Since both statements cannot be correct, a discussion follows.
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Value of early corner stones:
Currently, the predominating opinion says that the miai value of the first board play and so the value of each early stone is (about) 14 points. [2] The miai value represents both a stone's territory value and the territory equivalent of its influence value.
Another predominating opinion says that the 3-3 stone is more territory-orientated than influence-orientated. Therefore, it is reasonable to imply that the 3-3 stone's territory value is worth more than 50% of the 14 points miai value. In other words, the territory value must be greater than 50% * 14 = 7 points.
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Justification for 8 points:
Using Jasiek's methods of positional judgement [3], current territory expresses the territory value and is defined via the opponent's expected endgame reductions in sente and the player's peaceful answers with the following, relevant exception: "The defender switches direction only if a) this is necessary for maintaining life [...]" Other principles are relevant for White's first reduction move: "Construct reasonable peaceful reductions from the outside [...]" and "If the attacker has only remote support by friendly stones, his early reduction plays are accordingly reasonable."
For the sake of making territorial positional judgement, White makes endgame reductions in sente. However, Black's peaceful defense would be wrong, because his group dies. One does not defend territory by dying; instead, one would be giving the opponent very much territory by allowing him to make territory by killing.
White 1 is another mistake, which violates the two "reasonable" principles.
White 1 (or its symmetrically equivalent approach from the upper side) approaches at a reasonable distance. Black 2 applies the exception about maintaining life by switching the direction from the left side (where White reduces) to the upper side (where Black defends).
and so on, resulting in
We have 8 points for Black. This is the current territory of the 3-3 stone.
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Attempted justification for 4 points I:
How might Yi Ch'ang-ho justify his assessment of the value 4 points? Here is attempt I to justify it:
For the sake of making territorial positional judgement, White makes endgame reductions in sente. Black defends peacefully. Result: Black seems to enclose 4 points of territory. This would be Yi Ch'ang-ho's mistake, because he would be overlooking that the black group dies and that therefore this attempt of a justification is a failure.
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Attempted justification for 4 points II:
Here is attempt II to justify it:
Afterwards, territories are counted, starting with Black's:
and so on, resulting in
Black has 8 points. Now, White's territory is determined:
and so on. The white two-space extension group is considered alive. We get
White has 4 points.
Next, Yi Ch'ang-ho might calculate the difference of Black's and White's points as 8 - 4 = 4 points. However, he would have made a methodical mistake. These 4 points are not the territorial value of the 3-3 stone in an empty corner, but they are the territory count of this position:
Note that, here, the difference of played black and white stones is 0. For comparison, in the following position, the difference of played black and white stones is 1:
The latter cannot result in the former position by means of a privilege (sente) sequence. Hence, the calculated 4 points do not determine the territorial value of the 3-3 stone in an empty corner. Also attempt II is a failure.
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Attempted justification for 4 points III:
It is improbable, but maybe John Fairbairn made a mistake in his review and, in his book, Yi Ch'ang-ho wrote "4 points plus alpha" for the territory value of the 3-3 stone? That would be a very rough estimate, but at least correct for alpha := 4, so that 4 + alpha = 8.
Jasiek: the 3-3 stone is 8 points.
Yi Ch'ang-ho: the 3-3 stone is 4 points. [1]
Since both statements cannot be correct, a discussion follows.
**************************************************************************
Value of early corner stones:
Currently, the predominating opinion says that the miai value of the first board play and so the value of each early stone is (about) 14 points. [2] The miai value represents both a stone's territory value and the territory equivalent of its influence value.
Another predominating opinion says that the 3-3 stone is more territory-orientated than influence-orientated. Therefore, it is reasonable to imply that the 3-3 stone's territory value is worth more than 50% of the 14 points miai value. In other words, the territory value must be greater than 50% * 14 = 7 points.
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Justification for 8 points:
Using Jasiek's methods of positional judgement [3], current territory expresses the territory value and is defined via the opponent's expected endgame reductions in sente and the player's peaceful answers with the following, relevant exception: "The defender switches direction only if a) this is necessary for maintaining life [...]" Other principles are relevant for White's first reduction move: "Construct reasonable peaceful reductions from the outside [...]" and "If the attacker has only remote support by friendly stones, his early reduction plays are accordingly reasonable."
For the sake of making territorial positional judgement, White makes endgame reductions in sente. However, Black's peaceful defense would be wrong, because his group dies. One does not defend territory by dying; instead, one would be giving the opponent very much territory by allowing him to make territory by killing.
White 1 is another mistake, which violates the two "reasonable" principles.
White 1 (or its symmetrically equivalent approach from the upper side) approaches at a reasonable distance. Black 2 applies the exception about maintaining life by switching the direction from the left side (where White reduces) to the upper side (where Black defends).
and so on, resulting in
We have 8 points for Black. This is the current territory of the 3-3 stone.
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Attempted justification for 4 points I:
How might Yi Ch'ang-ho justify his assessment of the value 4 points? Here is attempt I to justify it:
For the sake of making territorial positional judgement, White makes endgame reductions in sente. Black defends peacefully. Result: Black seems to enclose 4 points of territory. This would be Yi Ch'ang-ho's mistake, because he would be overlooking that the black group dies and that therefore this attempt of a justification is a failure.
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Attempted justification for 4 points II:
Here is attempt II to justify it:
Afterwards, territories are counted, starting with Black's:
and so on, resulting in
Black has 8 points. Now, White's territory is determined:
and so on. The white two-space extension group is considered alive. We get
White has 4 points.
Next, Yi Ch'ang-ho might calculate the difference of Black's and White's points as 8 - 4 = 4 points. However, he would have made a methodical mistake. These 4 points are not the territorial value of the 3-3 stone in an empty corner, but they are the territory count of this position:
Note that, here, the difference of played black and white stones is 0. For comparison, in the following position, the difference of played black and white stones is 1:
The latter cannot result in the former position by means of a privilege (sente) sequence. Hence, the calculated 4 points do not determine the territorial value of the 3-3 stone in an empty corner. Also attempt II is a failure.
**************************************************************************
Attempted justification for 4 points III:
It is improbable, but maybe John Fairbairn made a mistake in his review and, in his book, Yi Ch'ang-ho wrote "4 points plus alpha" for the territory value of the 3-3 stone? That would be a very rough estimate, but at least correct for alpha := 4, so that 4 + alpha = 8.