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.

**************************************************************************

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.

**************************************************************************

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."

Click Here To Show Diagram Code

[go]$$W upper left corner, 3-3 stone
$$ ------------------
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$W suicide defense
$$ ------------------
$$ | . . 8 . . . . . .
$$ | . . 4 3 . . . . .
$$ | 6 2 X . 7 . . . .
$$ | . 1 . . . . . . .
$$ | . . 5 . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

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.

Click Here To Show Diagram Code

[go]$$W reasonable attack and defense
$$ ------------------
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . 2 . . .
$$ | . . . . . . . . .
$$ | . . 1 . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

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).

Click Here To Show Diagram Code

[go]$$W continuation
$$ ------------------
$$ | . . . . . . . . .
$$ | 8 . . . . 6 5 . .
$$ | 7 4 X . . X . . .
$$ | . 3 . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

and so on, resulting in

Click Here To Show Diagram Code

[go]$$W
$$ ------------------
$$ | . . . . . X . . .
$$ | X X . . . X O . .
$$ | O X X X X X . O .
$$ | O O . O O . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

We have 8 points for Black. This is the current territory of the 3-3 stone.

**************************************************************************

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:

Click Here To Show Diagram Code

[go]$$W upper left corner, 3-3 stone
$$ ------------------
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$W reduction
$$ ------------------
$$ | . . 8 . . . . . .
$$ | . . 4 3 . . . . .
$$ | 6 2 X . 7 . . . .
$$ | . 1 . . . . . . .
$$ | . . 5 . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

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.

**************************************************************************

Attempted justification for 4 points II:

Here is attempt II to justify it:

Click Here To Show Diagram Code

[go]$$W upper left corner, 3-3 stone
$$ ------------------
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$W reduction
$$ ------------------
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . 2 . . .
$$ | . . . . . . . . .
$$ | . . 1 . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . 3 . . . . . .
$$ | . . . . . . . . .[/go]

Afterwards, territories are counted, starting with Black's:

Click Here To Show Diagram Code

[go]$$W reduction of Black's region
$$ ------------------
$$ | . . . . . . . . .
$$ | 6 . . . . 4 3 . .
$$ | 5 2 X . . X . . .
$$ | . 1 . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .[/go]

and so on, resulting in

Click Here To Show Diagram Code

[go]$$B Black's points
$$ ------------------
$$ | . . . . . X . . .
$$ | X X . . . X O . .
$$ | O X X X X X . O .
$$ | O O . O O . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .[/go]

Black has 8 points. Now, White's territory is determined:

Click Here To Show Diagram Code

[go]$$B reduction of White's region
$$ ------------------
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . X . . .
$$ | . 3 . . . . . . .
$$ | . 4 O . . . . . .
$$ | . . 6 5 . . . . .
$$ | . . 8 7 . . . . .
$$ | . 2 O . . . . . .
$$ | . 1 . . . . . . .
$$ | . . 9 . . . . . .
$$ | . . . . . . . . .[/go]

and so on. The white two-space extension group is considered alive. We get

Click Here To Show Diagram Code

[go]$$B White's points
$$ ------------------
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . X . . .
$$ | X X . . . . . . .
$$ | O O O . . . . . .
$$ | . . O X . . . . .
$$ | . . O X . . . . .
$$ | O O O . . . . . .
$$ | . X . . . . . . .
$$ | . . X . . . . . .
$$ | . . . . . . . . .[/go]

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:

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . X . . .
$$ | . . . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .[/go]

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:

Click Here To Show Diagram Code

[go]$$B
$$ ------------------
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

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.

It would help in some of your reviews to not write about yourself in the third person. It's a little odd.

Are you arguing against a justification for something that you have not read?

Boidhre, as you can infer from my reasoning, I am convinced that a good justification for 4 points (without extra alpha parameter) cannot be found (especially since a good one for 8 points is available). If Yi has offered some explanation for 4 in his book, I like to hear about it, but I expect to be able to refute it with reasons similar to those shown.

I have this book in Korean so let me copy some text from this book.

The hide tag shows the original Korean text.

Diagram 1 (the value of komoku)
In the case of the most solid and territory oriented komoku, how much should it be calculated in points for one move.

Diagram 2 (6 points)
It is actually difficult to draw a borderline with a single stone. But to draw an imaginary line going across from the komoku to each 1st lines below, a counclusion comes out that it is about 6 points.

Diagram 3 (the weakness of komoku)
Komoku is a little difficult to be solid territory in one move. For instance, white's approach at A or B will devide the occupation of corner territory. Therefore, to make komoku as clear territory, Black needs an enclosure at A-C.
.
.
.
Diagram 18 (the value of sansan)
Let's now discover the territory value of sansan.
How much points is this most territory oriented move.

Diagram 19 (clear 4 points in one move)
The corner black territory is about 4 points with an imaginary boundary; little smaller than the one of komoku's and harder to develop for it's lowness. But this sansan has an advantage that a single move can perfectly enclose the corner.
.
.

_________________
Wait, please.

My point was, what he meant by "worth 4 points" and what you think he means by "worth 4 points" could be quite different, this is the issue with arguing against something you haven't read.

Thank you for the citation and translation! From this text, let me guess Dia. 19, which seems to look like this to represent "an imaginary boundary" around 4 intersections in the corner:

Click Here To Show Diagram Code

[go]$$W
$$ ------------------
$$ | . . C . . . . . .
$$ | . . C . . . . . .
$$ | C C X . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

If so, there are two possible explanations why Yi suggests the value 4 and this imaginary boundary, but does not suggest an alpha:

1) He draws the imaginary boundary purely visually, like a computer program in the 80s drawing a trivial influence region, without any justification.

2) According to Fairbairn's review, Yi uses sente / forcing endgame-like reduction sequences to derive boundaries. In this case, he must have imagined a sequence like this:

Click Here To Show Diagram Code

[go]$$W suicide defense
$$ ------------------
$$ | . . 8 . . . . . .
$$ | . . 4 3 . . . . .
$$ | 6 2 X . 7 . . . .
$$ | . 1 . . . . . . .
$$ | . . 5 . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

If so, he would have been making two mistakes:
- forgetting that (maintained) life is a requirement for territory,
- forgetting his own words that 3-3 is the "most territory oriented move" (according to your citation above).

Either (1) or (2) would be poor. His text "to draw an imaginary line going across from the komoku to each 1st lines below" for the 3-4 suggests that it is more likely that he uses (1), i.e., just draws lines to the edges without any justification why such determine the corner stone's safe territory value.

(Boidhre, thanks to MJK, it is pretty much clarified. That Yi appears to use (1) makes the almost missing justification by Yi actually worse than I feared. It seems that I was too generous in assuming there would be some solid justification about the 3-3 territory value at all in the book. The four corner intersections are specific in that White cannot play and live there, but this could also be said about a white stone on the 3-1 point under the black 3-3 stone. So even with this kind interpretation of Yi's proclaimed safe territory points, the value 4 is too small. Yi's "explanation" really boils down to just lazi drawing of visual lines to the edges, doesn't it?)

His remarks (according to Fairbairn's review) about corner enclosures make more sense. For them he uses proper reduction sequences and speaks of development potential and an alpha.

Click Here To Show Diagram Code

[go]$$Wc This is diagram 19 in the book.
$$ ------------------
$$ | . . x . . . . . .
$$ | . . x . . . . . .
$$ | x x X . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

What I understand from this book is, Lee Changho isn't saying the value of 3-4 and 3-3 is precisely 6 and 4 points each; but that those points are counted by "the imaginary boundary purely visually, like a computer program in the 80s drawing a trivial influence region, without any justification", also mentioning "an alpha" by phrases such as "the weakness of komoku" "a little difficult to be solid territory in one move" "clear 4 points in one move" "perfectly enclose".

_________________
Wait, please.

Would you say that Yi is methodically inconsistent in his book with respect to methods of determining territory values or do you perceive his method (applied before the endgame) in general as "precise value V (for this region) plus not determined parameter Alpha", where V is sometimes determined by justified sequences and sometimes by visual guessing?

I would say Lee (this is the official romanized family name rather than "Yi") does not think it is important to count the precise value of a single corner stone. He would rather want to say that "komoku is roughly six points, but you should be prepared to share corner occupation after an approach move. Or you can enclose the corner with an extra stone" "sansan is roughly four points but it is very solid and clear in corner occupation in just one move, although it is not easy to develop" therefore, "It doesn't matter much and it's just your choice."

This book is about practical positional judgement during the game, explained with some basic fundamentals and various examples from Lee's own games. It wants you to have the right feeling of the game for who is winning; not to know how much exactly a single corner stone is.

_________________
Wait, please.

Robert, you are the one making mistakes, not Yi. In your diagram labelled Suicide Defence there is no way Black would answer 5 at 6 and 7 at 8. You will see from the pdf file that Yi (like other authors) specifically uses the sagari-sagari method for determining edge boundaries in such cases (and also that he makes appropriate deductions when one side genuinely has to make extra moves).

Rather than worshipping numbers, he is (as MJK also points out) simply making a conceptual point - even one stone can have a prospective territory.

You need to understand the purpose of this diagram and similar diagrams, and of the nature of sequences constructed for the and only the purpose of positional judgement. These sequences let the attacker reduce and the defender depend passively the boundaries of his currently attacked region where it is currently being attacked. White 5 attacks from the left side, so Black defends his territory boundary on the left side. Likewise 7 reduces the upper side and 8 defends the upper side boundary of the black territory region. (Except that, here, the black group dies; so this default construction must be modified.) Cho, Lee and others also construct such sequences (for other positions), where they are not about maximal efficiency in the global context, but only about local territory determination of the defender's region.

So, WRT constructing the sequence as a positional judgement sequence, I do not make mistakes. As you can see in my first post, I also explain why White 1 and Black 2 would be unreasonable. In this respect, the sequence has these mistakes, but the sequence is there for showing what happens if these initial mistakes are being made and then the sequence continues as if it were a normal reduction and peaceful answer sequence constructed for positional judgement.

Since you claim that Yi does not make mistakes, you need to explain your opinion. In particular, why may one draw perpendicular lines without having any justification for them in the first place?



I see that they use this method, and I notice that they use it without first having any justification for it. E.g, one might also draw lines 4-1 via 3-3 to 1-4. An explanation is missing why perpendicular lines would be any more meaningful.



Which are Lee's appropriate deductions for the 3-3?

I see, but... if the game is close, it matters what exactly every value is. From what I understand about Lee's book so far, it implies at least three kinds of parameters (symbols by me; the book seems to call everything Alpha): X for influence / outside development potential, Y for insecure parts of territorial frameworks, Z for territory of early corner stones outside the visual line drawing. Since there can be several regions (with index i) on the board, there can also be several Xi, several Yi and a few Zi. How does one derive the "right feeling" from such a heap of parameters? Are all the parameters then summarised (by feeling?) as one Alpha parameter, which favours either Black, White or neither player? Or maybe sometimes as "inconclusive"?

EDIT:

E.g., is it like "Black leads by T points and has the favourable Alpha", so Black leads? Or "Black leads by T points, White leads by Alpha", so White must use Alpha to catch up on points"? Or "Black leads by T points, Alpha is inconclusive, so the game is either Black's lead or close"? Is this how Lee thinks when using positional judgement to make strategic decisions?

Sorry, but I can't really understand what you are saying about. Perhaps it's because you have a very different approach with me on this game. Strong go players especially pros do have a accurate feeling on the game on who is winning. They know where the natural boudaries should be by a huge mass of experience. This should be the reason why I can rely on Lee Changho's book. However, I'm not saying the boundaries given for a single 3-3 stone in Lee's book is natural, but that the moments when positional judgement is imporant is after the middlegame starts where there are no such single corner stones left; in my opinion, this book has more justifiable information in positional judgement during the middlegame for various positions, which I found more practical and useful than trying to figure out the value of a corner stone alone.

_________________
Wait, please.

I use positional judgement from move 1 of the game on and find precise values very useful for decision making. From your description, I understand that there is the alternative of accumulating a huge mental "database" for feeling-driven judgements. I prefer to avoid such; with my methods I can avoid such to a pretty good extent.

Since I prefer precision like "the territory count favours Black by 4.5 points, the center's influence stone count favours White by 5 stones, it is Black's turn, other judgement aspects can be neglected in this position, so the game is close", I would not be happy with Lee's method, which determines territory precisely for an on average too small part of the board, leaving too much of it in an Alpha parameter.

From judgement, I know who is winning. I do not use "natural boundaries", but I determine the boundaries by determining the related sequences of territorial positional judgement well. For this purpose, there are principles specifying how to construct the sequences.

Imaginary judgement lines I draw almost only for a single, huge moyo, when a parameter (player must make at least X excess points, when ignoring territory beyond the lines) is more convenient than accurate calculation of half territory in the moyo.

Plainly the traditional territory value for positional judgement is very conservative, something that might be called solid territory. But how much local territory should we estimate for the 3-3 stone?

One way to answer that question might be statistically, looking at many pro games. Here I will take a different approach.

Click Here To Show Diagram Code

[go]$$W 3-3 stone, estimating territory
$$ ----------------------
$$ | C C C C C . . . . . .
$$ | C C C C C . . . . . .
$$ | . . X . . B . . . . .
$$ | . . . , . . . . . , .
$$ | . . . . . . . . . . .
$$ | . . W . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . , . . . . . , .
$$ | . . . . . . . . . . .[/go]

Here we have added the and stones. Plainly without the 3-3 stone the position has a value of 0, so we can estimate how much territory the 3-3 stone adds. Using the traditional way we get an estimate of 10 pts. (the points). But we should subtract the territory value of the stone. Since it has no base, we cannot properly estimate its territory value.

Click Here To Show Diagram Code

[go]$$W 3-3 stone, estimating territory (II)
$$ ----------------------
$$ | C C C C C C C C . . .
$$ | C C C C C C C C . . .
$$ | . . X . . B . . X . .
$$ | . . . , . . . . . , .
$$ | . . . . . . . . . . .
$$ | . . W . . . . . . . .
$$ | C C . . . . . . . . .
$$ | C C . . . . . . . . .
$$ | . . O . . . . . . . .
$$ | . . . , . . . . . , .
$$ | . . . . . . . . . . .[/go]

Here we have given the stone a base, adding a counterbalancing Black stone on the top. Now, still using the traditional estimate, we get 16 - 4 = 12 pts. The extra two points come from the two points under the stone. That seems to be an artifact. The extension added 4 pts. to the White position, but 6 pts. to the Black position.

We can adjust for that by adding 1 pt. underneath a stone on the 3d line. Then an extension adds 6 pts. to each position. When we do that we get the following diagram.

Click Here To Show Diagram Code

[go]$$W 3-3 stone, estimating territory (III)
$$ ----------------------
$$ | C C C C C C C C . . .
$$ | C C C C C C C C C . .
$$ | . C X . . B . . X . .
$$ | . . . , . . . . . , .
$$ | . . . . . . . . . . .
$$ | . C W . . . . . . . .
$$ | C C . . . . . . . . .
$$ | C C . . . . . . . . .
$$ | . C O . . . . . . . .
$$ | . . . , . . . . . , .
$$ | . . . . . . . . . . .[/go]

That estimate yields 18 - 6 = 12 pts.

Now, we expect from the value of komi that a stone on the 3-3 is worth around 14 pts. (Maybe less, as it has gone out of style.) If 12 pts. of that comes from local territory, that leaves 2 pts. for center influence. That seems reasonable.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Is it possible that attempts to find precise values on a near-empty board like this oversimplify the reality of the situation?

_________________
be immersed

Precise values? When you get different estimates of 4, 8, and 12, where is the precision?

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Hey, they're all within an order of magnitude. If that's not precise, what is?

Maybe there are some games where you'll get 4 points in the corner, some where you'll get 8, and some where you'll get 12. Maybe it doesn't have to be a single number, because there can be a variety of different board positions in practice.

It's in this mindset that concepts and ideas are more powerful than a formula that tells you F(StoneA, StoneB, StoneC) = 8 points. Simplifying the function to a single number loses details. Instead of reducing StoneA, StoneB, and StoneC to a single value, knowing certain properties of the stones may be more useful and widely applicable to whatever position you may find yourself in.

When there are more stones on the board and the position becomes calculable, at that point, you can apply some function to the stones to come up with a precise number.

But my feeling is that in an empty board, it's too much of a simplification to reduce the complexity of the position to a single, simple number.

_________________
be immersed