This 'n' that

Bill Spight · Post by **Bill Spight** » Wed May 19, 2021 10:47 am

Gérard TAILLE wrote:
Bill Spight wrote:When using difference games for comparing moves in go when ko fights are not an issue, we do not need to rely upon CGT per se. The commonsense notion that a game is at least as good for one player versus the other if the other player plays first but cannot win is all we need.

Example:

When we set up the original 0 game, there will always be a even number of board points in play.
your conclusion is the following:
"if there is no ko fight, in the original position B is at least as good for Black as A."

That leads to another issue: assume a ko may arise in the environment. In that case we know that the result of the difference game is not reliable but here is my question : can you build an example of ko environment in which A appears strictly better than B ?

If not I will say B dominates A in a strong manner, otherwise I will only say B dominates A in a weak manner.

Well, I lost my reply somehow. I'll post it later.

In short, maybe so, maybe not.

Gérard TAILLE · Post by **Gérard TAILLE** » Wed May 19, 2021 10:55 am

https://senseis.xmp.net/?L2Group%2FDiscussion

Click Here To Show Diagram Code: [go]$$B $$ ----------------- $$ | . . . 1 . . . . $$ | a b O O X . . . $$ | . . O X X . . . $$ | . O O X . . . . $$ | . O X . X . . . $$ | . X X X . . . . $$ | . . . . . . . . $$ | . . . . . . . .[/go]

Bill, in the discussion referenced above you compared the white answers a and b to the hane

For a yose point of view white at "b" is not that good because:

Click Here To Show Diagram Code: [go]$$B $$ ----------------- $$ | . . . 1 . . . . $$ | . 2 O O X . . . $$ | 4 . O X X . . . $$ | . O O X . . . . $$ | 3 O X . X . . . $$ | . X X X . . . . $$ | . . . . . . . . $$ | . . . . . . . .[/go]

is needed to avoid a potential ko

In the other hand after white at "a" :

Click Here To Show Diagram Code: [go]$$B $$ ----------------- $$ | . . 4 1 . . . . $$ | 2 . O O X . . . $$ | . . O X X . . . $$ | . O O X . . . . $$ | 3 O X . X . . . $$ | . X X X . . . . $$ | . . . . . . . . $$ | . . . . . . . .[/go]

The possibility for white to answer by

shows that, for a yose point of view, white "a" in the first diagram is far better than white "b" in order to avoid leaving the sente move

Gérard TAILLE · Post by **Gérard TAILLE** » Sat May 22, 2021 6:57 am

https://senseis.xmp.net/?L2GroupWithDescent

The reference above deals with the following well known position:

Click Here To Show Diagram Code: [go]$$B $$ ----------------- $$ | . . . . B . . . $$ | . . O O X . . . $$ | . . O X . X . . $$ | . O O X . . . . $$ | . O X X . . . . $$ | . X . . . . . . $$ | . . X . . . . . $$ | . . . . . . . .[/go]

The best sequence, mentionned by Bill, seems the following:

Click Here To Show Diagram Code: [go]$$B $$ ----------------- $$ | 7 2 . 4 B . . . $$ | . 1 O O X . . . $$ | 3 . O X . X . . $$ | 6 O O X . . . . $$ | 5 O X X . . . . $$ | a X . . . . . . $$ | . . X . . . . . $$ | . . . . . . . .[/go]

The result is a two stage ko, white having a ko threat at a.

Let's take an example to understand this point:

Click Here To Show Diagram Code: [go]$$B $$ --------------------- $$ | . . . . X X X O . | $$ | . . O O X X O O O | $$ | . . O X X X O . . | $$ | . O O X X X X O O | $$ | . O X X X X X X X | $$ | . X . . X . . . . | $$ | . . X X . . . . . | $$ | X X X . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]

One of the best sequence is the following

Click Here To Show Diagram Code: [go]$$B $$ --------------------- $$ | 7 2 . 4 X X X O . | $$ | 8 1 O O X X O O O | $$ | 3 . O X X X O 9 . | $$ | 6 O O X X X X O O | $$ | 5 O X X X X X X X | $$ | . X . . X . . . . | $$ | . . X X . . . . . | $$ | X X X . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]

Click Here To Show Diagram Code: [go]$$Wcm10 $$ --------------------- $$ | 2 O 4 O X X X O . | $$ | O X O O X X O O O | $$ | X . O X X X O X 1 | $$ | O O O X X X X O O | $$ | X O X X X X X X X | $$ | 3 X . . X . . . . | $$ | . 5 X X . . . . . | $$ | X X X . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]

and white saves his group

As a consequence, due to this last white ko threat, black needs two ko threats in order to try and kill white.

Let's then take this new position:

Click Here To Show Diagram Code: [go]$$B $$ --------------------- $$ | . . . . X X X O . | $$ | . . O O X X O O O | $$ | . . O X X X O . X | $$ | . O O X X X O . b | $$ | . O X X X X O a O | $$ | . X . . X X X O O | $$ | . . X X . . X X X | $$ | X X X . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]

Now black has two ko threats in "a" and "b" instead of only one in the previous diagram.
What is now the best sequence for both?
I discovered a quite unexpected sequence here ... but I am not quite sure. What is your analysis?

Gérard TAILLE · Post by **Gérard TAILLE** » Sat May 29, 2021 10:37 am

After one week it's time to give you the move I discovered

Click Here To Show Diagram Code: [go]$$B $$ --------------------- $$ | . . . . X X X O . | $$ | . . O O X X O O O | $$ | . . O X X X O . X | $$ | . O O X X X O . b | $$ | . O X X X X O a O | $$ | . X . . X X X O O | $$ | . . X X . . X X X | $$ | X X X . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]

wrong sequence:

Click Here To Show Diagram Code: [go]$$Bc Wrong sequence $$ --------------------- $$ | 7 2 . 4 X X X O . | $$ | 8 1 O O X X O O O | $$ | 3 . O X X X O . X | $$ | 6 O O X X X O . . | $$ | 5 O X X X X O 9 O | $$ | . X . . X X X O O | $$ | . . X X . . X X X | $$ | X X X . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]

and black manage to kill one of white groups.

In order to save his two groups white has to play:

Click Here To Show Diagram Code: [go]$$Bc correct sequence $$ --------------------- $$ | 5 2 . . X X X O . | $$ | 4 1 O O X X O O O | $$ | 3 . O X X X O . X | $$ | . O O X X X O . . | $$ | . O X X X X O 6 O | $$ | . X . . X X X O O | $$ | . . X X . . X X X | $$ | X X X . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]

The exchange

allows white to play tenuki BEFORE the beginning of the ko. That is the point; white is able to destroy black ko threats and saves his two groups.

With this new idea we have to review the previous moves in order to see if they are correct.

Let me propose a new position:

Click Here To Show Diagram Code: [go]$$Bc wrong sequence $$ --------------------- $$ | . 2 . 4 X a X O . | $$ | . 1 O O X X O O O | $$ | 3 . O X X ? ? O . | $$ | . O O X ? ? ? O O | $$ | . O X X X ? ? ? ? | $$ | . X . . X ? ? ? ? | $$ | . . X X X ? ? ? ? | $$ | X X X ? ? ? ? ? ? | $$ | ? ? ? ? ? ? ? ? ? | $$ ---------------------[/go]

this environment is a ko environment and this is the simpliest ko environment you can build.
Assume you use area counting.
Could you see that the three moves

,

and

are all bad moves ? Surprising isn't it ?

If that is true that means you cannot say that

,

are tesuji moves, unless you find another (simple?) environment in which these moves appears strictly better than other moves.

Click Here To Show Diagram Code: [go]$$B $$ --------------------- $$ | . . . b X . . . . | $$ | . a O O X . . . . | $$ | . . O X X . . . . | $$ | . O O X . . . . . | $$ | . O X X X . . . . | $$ | . X . . X . . . . | $$ | . . X X X . . . . | $$ | X X X . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]

My feeling is that it is not easy to find an environment for which black move "a" is strictly better than the obvious black move "b" (my diagram above shows an example where black move "b" is strictly better than black move "a").

Gérard TAILLE · Post by **Gérard TAILLE** » Sat May 29, 2021 2:47 pm

Click Here To Show Diagram Code: [go]$$B $$ --------------------- $$ | . . . b X . . . . | $$ | . a O O X . . . . | $$ | . . O X X . . . . | $$ | . O O X . . . . . | $$ | . O X X X . . . . | $$ | . X . . X . . . . | $$ | . . X X X . . . . | $$ | X X X . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]

In addition to my previous post you can look at the following difference game

Click Here To Show Diagram Code: [go]$$B $$ --------------------------------------- $$ | . O . . X . . . . . . . . . O O . . . | $$ | . X O O X . . . . . . . . . O X X . . | $$ | X . O X X . . . . . . . . . O O X . . | $$ | . O O X . . . . . , . . . . . O X X . | $$ | . O X X . . . . . . . . . . . O O X . | $$ | . X X . . . . . . . . . . . . . O O . | $$ | . X . . . . . . . . . . . . . . . O . | $$ | X X . . . . . . . . . . . . . . . O O | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Black to play : draw
White to play : white wins

That proves that black "b" looks better than black "a" (in a non ko environment).
As a consequence you may play black "a" only if you are more or less komaster. Do you agree with that?

Bill Spight · Post by **Bill Spight** » Sun May 30, 2021 1:42 pm

If Black to play kills the top left corner with ko, it takes 3 net local play to get 20 points. White to play lives with 1 net local play to get 6 points, That's a swing of 26 points by territory counting with 4 net plays, an average gain of 6½ points.

OTOH, a Black play mirroring the White push in the top right corner is a 1 point sente, holding White to 5 points.

It is hard to believe that there is no environment where Black in the top left kills the corner with ko in exchange for fewer than 25 points or so elsewhere in 3 net plays.

Note: The 25 points is not exact, because the value of sente after the exchange may differ between the different lines of play.

Edit: The 1 point sente raises the local temperature to 13, which gives Black plenty of leeway of when to start the ko.

Gérard TAILLE · Post by **Gérard TAILLE** » Mon May 31, 2021 6:57 am

Bill Spight wrote:If Black to play kills the top left corner with ko, it takes 3 net local play to get 20 points. White to play lives with 1 net local play to get 6 points, That's a swing of 26 points by territory counting with 4 net plays, an average gain of 6½ points.

OTOH, a Black play mirroring the White push in the top right corner is a 1 point sente, holding White to 5 points.

It is hard to believe that there is no environment where Black in the top left kills the corner with ko in exchange for fewer than 25 points or so elsewhere in 3 net plays.

Note: The 25 points is not exact, because the value of sente after the exchange may differ between the different lines of play.

Edit: The 1 point sente raises the local temperature to 13, which gives Black plenty of leeway of when to start the ko.

You seem to confirm what I said previously : a black move on the point 2-2 in order to start a ko is a good move if black is komaster.
But what happen if neither side has ko threat in the environment? Depending of the temperture which player will play first in the local area?

With a direct ko the calculation is easy : the tally is equal to 3 and a local move can be evaluated to the swing value divided by 3. That'is fine.
Here the ko is far more complex (you can see a yose ko and a two stage ko). If there are no threat available in the environment I am not sure the value of a local move is as high as 6½ points but it is difficult for me to calculate the correct evaluation.
I am wondering if the 6½ points you calculated assume implicitly that black is komaster (?).

Bill Spight · Post by **Bill Spight** » Mon May 31, 2021 8:15 am

Gérard TAILLE wrote:I am wondering if the 6½ points you calculated assume implicitly that black is komaster (?).

With a local swing of 26 points and a difference of 4 local moves, the average gain per move is 6½ points. If the ko winner can kill the corner without ignoring any of the opponent's ko threats, then she is the komaster, by definition.

In 3 net plays the ko winner gains 19½ points locally. If the koloser also gains 19½ points elsewhere in 3 net plays, the exchange is equitable.

Komaster analysis does not give a hard limit. It is an idealization which allows a mast value and temperature to be found. In practice, the environment is not typically ideal, and in this case seki is a possibility which needs to be considered.

Bill Spight · Post by **Bill Spight** » Mon May 31, 2021 8:58 am

To keep things simple, let's ignore the seki possibility and assume an environment with temperature t with several plays that gain t and that getting the last play in the environment gains t/2. The ko winner plays first.

If there are no other plays larger than those in the environment then the ko winner gets 20 points locally and the koloser gets 2½ t elsewhere. (Only t/2 for the third play in the environment.) Result: 20 - 2½ t.

For comparison let the ko winner start in the environment. Then let the opponent save the corner for -6 points. The result is 1½ t - 6.

The ko winner is indifferent between these two results if

20 - 2½ t = -6 + 1½ t

I.e, if t = 6½

Now suppose that there is a simple gote, {u | -u}, on the board with u > t.

1) The ko winner starts the ko and kills the corner. Result: 20 - u - 1½ t.

2) The ko winner takes u and then the koloser saves the corner. Result: -6 + u + t/2.

If the ko winner is indifferent between these two results then

20 - u - 1½ t = -6 + u + t/2

26 = 2u + 2t

t = 13 - u

And so on.

Bill Spight · Post by **Bill Spight** » Mon May 31, 2021 9:14 am

Oh, let's look at an environment with temperature t with two simple gote on the board, {u|-u} and {v|-v}, u ≥ v > t.

1) The ko winner kills the corner in ko. Result: 20 - u - v - t/2.

2) The ko winner takes u, the koloser saves the corner, and then the ko winner takes v. Result: -6 + u + v - t/2.

The ko winner is indifferent between these two results when

u + v = 13

The environment is irrelevant.

Gérard TAILLE · Post by **Gérard TAILLE** » Mon May 31, 2021 2:45 pm

Bill Spight wrote:Oh, let's look at an environment with temperature t with two simple gote on the board, {u|-u} and {v|-v}, u ≥ v > t.

1) The ko winner kills the corner in ko. Result: 20 - u - v - t/2.

2) The ko winner takes u, the koloser saves the corner, and then the ko winner takes v. Result: -6 + u + v - t/2.

The ko winner is indifferent between these two results when

u + v = 13

The environment is irrelevant.

That is not that clear Bill. If I understand correctly you assume there are no ko threat in the environment and you really use the value u and v instead of a non-existant ko threat. If it is the case then, because white can always save her corner, the problem is not to compare variants in which the corner is killed, but to know :
1) when it is interesting for black to play in the corner and
2) is it interesting for black to play the ko rather than sente moves.

Bill Spight · Post by **Bill Spight** » Mon May 31, 2021 4:32 pm

Gérard TAILLE wrote:
Bill Spight wrote:Oh, let's look at an environment with temperature t with two simple gote on the board, {u|-u} and {v|-v}, u ≥ v > t.

1) The ko winner kills the corner in ko. Result: 20 - u - v - t/2.

2) The ko winner takes u, the koloser saves the corner, and then the ko winner takes v. Result: -6 + u + v - t/2.

The ko winner is indifferent between these two results when

u + v = 13

The environment is irrelevant.
That is not that clear Bill. If I understand correctly you assume there are no ko threat in the environment and you really use the value u and v instead of a non-existant ko threat.

No. The game is the corner plus U = {u|-u} plus V = {v|-v}. That's all. U and V are not ko threats, they are simple gote.

Gérard TAILLE wrote:If it is the case then, because white can always save her corner,

No. Black (the ko winner, who plays first) gets to decide whether to kill the corner or not.

Gérard TAILLE wrote:the problem is not to compare variants in which the corner is killed, but to know :
1) when it is interesting for black to play in the corner and
2) is it interesting for black to play the ko rather than sente moves.

That is why we find the conditions under which the choice is indifferent.

Suppose that Black takes U and then White takes V. Now Black kills the corner. Since u + v = 13 and u ≥ v, the most that v can be is 6½ , and v > t. So White should not take V.

Edit: OK, let's add the sente against the corner.

Since the sente raises the temperature to 13, it may be played before U.

3) Black plays sente against the corner, which White saves, and then Black takes U, White takes V, and Black gets t/2
Result: -5 + u - v + t/2

Let's compare that with Black killing the corner.

20 - u - v - t/2 >?< -5 + u - v + t/2

t >?< 25 - 2u

Now let's compare it with Black taking U.

-6 + u + v - t/2 >?< -5 + u - v + t/2

t >?< 2v - 1 > 2t - 1

t >?< 1

So for the sente to be better for Black than taking U, t ≤ 1.

Gérard TAILLE · Post by **Gérard TAILLE** » Tue Jun 01, 2021 6:55 am

Bill Spight wrote:
Gérard TAILLE wrote:
Bill Spight wrote:Oh, let's look at an environment with temperature t with two simple gote on the board, {u|-u} and {v|-v}, u ≥ v > t.

1) The ko winner kills the corner in ko. Result: 20 - u - v - t/2.

2) The ko winner takes u, the koloser saves the corner, and then the ko winner takes v. Result: -6 + u + v - t/2.

The ko winner is indifferent between these two results when

u + v = 13

The environment is irrelevant.
That is not that clear Bill. If I understand correctly you assume there are no ko threat in the environment and you really use the value u and v instead of a non-existant ko threat.
No. The game is the corner plus U = {u|-u} plus V = {v|-v}. That's all. U and V are not ko threats, they are simple gote.

Gérard TAILLE wrote:If it is the case then, because white can always save her corner,
No. Black (the ko winner, who plays first) gets to decide whether to kill the corner or not.

Gérard TAILLE wrote:the problem is not to compare variants in which the corner is killed, but to know :
1) when it is interesting for black to play in the corner and
2) is it interesting for black to play the ko rather than sente moves.
That is why we find the conditions under which the choice is indifferent.

Suppose that Black takes U and then White takes V. Now Black kills the corner. Since u + v = 13 and u ≥ v, the most that v can be is 6½ , and v > t. So White should not take V.

Edit: OK, let's add the sente against the corner. Since the sente raises the temperature to 13, it may be played before U.

3) Black plays sente against the corner, which White saves, and then Black takes U, White takes V, and Black gets t/2
Result: -5 + u - v + t/2

Let's compare that with Black killing the corner.

20 - u - v - t/2 >?< -5 + u - v + t/2

t >?< 25 - 2u

Now let's compare it with Black taking U.

-6 + u + v - t/2 >?< -5 + u - v + t/2

t >?< 2v - 1 > 2t - 1

t >?< 1

So for the sente to be better for Black than taking U, t ≤ 1.

Let's take an example in order to find where is the misunderstanding:

Click Here To Show Diagram Code: [go]$$B $$ --------------------------------------- $$ | . . . . X X X X X X O . . . . . . . . | $$ | . . O O X O O O O u O . . . . . . . . | $$ | . . O X X X X X X X O . . . . . . . . | $$ | . O O X X X O O O v O . . . . . . . . | $$ | . O X X X X X X X X O . . . . . . . . | $$ | . X X . X X X O O w O . . . . . . . . | $$ | . X . . X X X X X X O . . . . . . . . | $$ | X X . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

I assume it remains on the board only the corner, the two gote points u and v, and a third gote point w
If I understand correctly your posts the best strategy for black is to provoque immediatly a ko in the corner.

How do you manage to get a better result than the following trivial sequence, with white playing first in the corner!

Click Here To Show Diagram Code: [go]$$B $$ --------------------------------------- $$ | . . 6 5 X X X X X X O . . . . . . . . | $$ | 4 . O O X O O O O 1 O . . . . . . . . | $$ | . . O X X X X X X X O . . . . . . . . | $$ | 8 O O X X X O O O 2 O . . . . . . . . | $$ | 7 O X X X X X X X X O . . . . . . . . | $$ | 9 X X . X X X O O 3 O . . . . . . . . | $$ | . X . . X X X X X X O . . . . . . . . | $$ | X X . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Bill Spight · Post by **Bill Spight** » Tue Jun 01, 2021 8:32 am

Bill Spight wrote:
Gérard TAILLE wrote:
Bill Spight wrote:Oh, let's look at an environment with temperature t with two simple gote on the board, {u|-u} and {v|-v}, u ≥ v > t.

1) The ko winner kills the corner in ko. Result: 20 - u - v - t/2.

2) The ko winner takes u, the koloser saves the corner, and then the ko winner takes v. Result: -6 + u + v - t/2.

The ko winner is indifferent between these two results when

u + v = 13

The environment is irrelevant.
That is not that clear Bill. If I understand correctly you assume there are no ko threat in the environment and you really use the value u and v instead of a non-existant ko threat.
No. The game is the corner plus U = {u|-u} plus V = {v|-v}. That's all. U and V are not ko threats, they are simple gote.

Gérard TAILLE wrote:If it is the case then, because white can always save her corner,
No. Black (the ko winner, who plays first) gets to decide whether to kill the corner or not.

Gérard TAILLE wrote:the problem is not to compare variants in which the corner is killed, but to know :
1) when it is interesting for black to play in the corner and
2) is it interesting for black to play the ko rather than sente moves.
That is why we find the conditions under which the choice is indifferent.

Suppose that Black takes U and then White takes V. Now Black kills the corner. Since u + v = 13 and u ≥ v, the most that v can be is 6½ , and v > t. So White should not take V.

Edit: OK, let's add the sente against the corner. Since the sente raises the temperature to 13, it may be played before U.

3) Black plays sente against the corner, which White saves, and then Black takes U, White takes V, and Black gets t/2
Result: -5 + u - v + t/2

Let's compare that with Black killing the corner.

20 - u - v - t/2 >?< -5 + u - v + t/2

t >?< 25 - 2u

Now let's compare it with Black taking U.

-6 + u + v - t/2 >?< -5 + u - v + t/2

t >?< 2v - 1 > 2t - 1

t >?< 1

So for the sente to be better for Black than taking U, t ≤ 1.

Gérard TAILLE wrote:Let's take an example in order to find where is the misunderstanding:

$$B
$$ ---------------------------------------
$$ | . . . . X X X X X X O . . . . . . . . |
$$ | . . O O X O O O O u O . . . . . . . . |
$$ | . . O X X X X X X X O . . . . . . . . |
$$ | . O O X X X O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X X O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
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Click Here To Show Diagram Code
[go]$$B $$ --------------------------------------- $$ | . . . . X X X X X X O . . . . . . . . | $$ | . . O O X O O O O u O . . . . . . . . | $$ | . . O X X X X X X X O . . . . . . . . | $$ | . O O X X X O O O v O . . . . . . . . | $$ | . O X X X X X X X X O . . . . . . . . | $$ | . X X . X X X O O w O . . . . . . . . | $$ | . X . . X X X X X X O . . . . . . . . | $$ | X X . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]
I assume it remains on the board only the corner, the two gote points u and v, and a third gote point w
If I understand correctly your posts the best strategy for black is to provoque immediatly a ko in the corner.

No, W does not qualify as an environment such that getting the last play in the environment gains w/2 in the end.

Click Here To Show Diagram Code: [go]$$B $$ --------------------------------------- $$ | . . . . X X X X X X O . . . . . . . . | $$ | . . O O X O O O O u O . . . . . . . . | $$ | . . O X X X X X X X O . . . . . . . . | $$ | . O O X X X O O O v O . . . . . . . . | $$ | . O X X X X X X X X O . . . . . . . . | $$ | . X X . X X X O O y O . . . . . . . . | $$ | . X . . X X X X X X O . . . . . . . . | $$ | X X . . . . . X O z O . . . . . . . . | $$ | . . . . . . . X X X O . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Now Y and Z do form such an environment.

Let's try playing the sente first.

Click Here To Show Diagram Code: [go]$$B $$ --------------------------------------- $$ | . 6 5 1 X X X X X X O . . . . . . . . | $$ | . 2 O O X O O O O 3 O . . . . . . . . | $$ | . . O X X X X X X X O . . . . . . . . | $$ | . O O X X X O O O 4 O . . . . . . . . | $$ | . O X X X X X X X X O . . . . . . . . | $$ | 8 X X . X X X O O 7 O . . . . . . . . | $$ | . X . . X X X X X X O . . . . . . . . | $$ | X X . . . . . X O 9 O . . . . . . . . | $$ | . . . . . . . X X X O . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Result: -5 + 8 + 4 + 2 = 9

Now let's try playing in U first.

Click Here To Show Diagram Code: [go]$$B $$ --------------------------------------- $$ | . . 6 5 X X X X X X O . . . . . . . . | $$ | 2 . O O X O O O O 1 O . . . . . . . . | $$ | . . O X X X X X X X O . . . . . . . . | $$ | . O O X X X O O O 3 O . . . . . . . . | $$ | . O X X X X X X X X O . . . . . . . . | $$ | 8 X X . X X X O O 4 O . . . . . . . . | $$ | . X . . X X X X X X O . . . . . . . . | $$ | X X . . . . . X O 7 O . . . . . . . . | $$ | . . . . . . . X X X O . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

Result: -6 + 8 + 6 + 2 = 10

Playing in U dominates taking the sente first.

Edit: If there were no ko, taking the sente first would dominate. Kos introduce weirdness.

Edit2: Thanks to Gerard, I corrected the results. But still, playing in U dominates taking the sente first, given the corner ko.

Gérard TAILLE · Post by **Gérard TAILLE** » Tue Jun 01, 2021 10:00 am

Oops I am now a little lost.
The results you calculated are wrong (instead of 6 + 8 + 6 + 2 = 22 you must read -6 + 8 + 6 + 2 = 10) but I corrected them easily.
In your example with u,v,y,z = 4,3,2,1 you showed that playing in the environment is best.
Could you clarify when you have to provoque a ko in the corner?
Note that in your example you have both u+v < 13 and u < 6½ and black prefers playing in the environment.

Life In 19x19

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