Life In 19x19

Gérard TAILLE wrote: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.

Thanks for the correction.

Edit: To be clear, I did not show that playing in the environment first is best, I showed that playing in U dominates playing the sente, given the ko. The environment is Y + Z.

The main assumption is that Black is the ko winner, killing the corner in 3 net local plays, if she plays first. To keep things simple, we are ignoring the possibility of seki. Obviously, to be the ko winner Black needs ko threats which White answers. These threats are not shown. Any other threats in the ko fight are not shown, either.

Let's back up to where there is only the corner and the environment with temperature t, where the last player to play in it gains t/2 in the end, an ideal environment.

Black to play has three choices, to start the ko, to play in the environment, or to play the sente, allowing White to live in the corner, and then play in the environment.

2) Black plays in the environment, White saves the corner, and then Black plays in the environment again.
Result: -6 + 1½t

3) Black plays the sente first, White replies, and then Black plays in the environment.
Result: -5 + t/2

Black is indifferent between these plays when

-6 + 1½t = -5 + t/2

t = 1

(We knew that already. )

If t > 1 then Black prefers to play in the environment first.

What if Black starts the ko first? Black kills the corner in 3 net plays, while White plays three times in the environment in exchange.
Result: 20 - 2½t

If Black is indifferent between that and playing in the environment first, then

-6 + 1½t = 20 - 2½t

4t = 26

t = 6½

Black is komaster because she has not had to ignore any ko threats to win the ko. From this we derive the average value of a play in the ko when Black is komaster to be 6½ points.

So far, so good.

But it is unusual, except at low temperatures, to have several plays worth exactly the same. So we look at the corner plus U = {u|-u}, where u > t. And then we look at the corner plus U and V = {v|-v}, where u ≥ v > t.

Bill Spight wrote:

Gérard TAILLE wrote: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.

Thanks for the correction.

Edit: To be clear, I did not show that playing in the environment first is best, I showed that playing in U dominates playing the sente, given the ko. The environment is Y + Z.

The main assumption is that Black is the ko winner, killing the corner in 3 net local plays, if she plays first. To keep things simple, we are ignoring the possibility of seki. Obviously, to be the ko winner Black needs ko threats which White answers. These threats are not shown. Any other threats in the ko fight are not shown, either.

Let's back up to where there is only the corner and the environment with temperature t, where the last player to play in it gains t/2 in the end, an ideal environment.

Black to play has three choices, to start the ko, to play in the environment, or to play the sente, allowing White to live in the corner, and then play in the environment.

2) Black plays in the environment, White saves the corner, and then Black plays in the environment again.
Result: -6 + 1½t

3) Black plays the sente first, White replies, and then Black plays in the environment.
Result: -5 + t/2

Black is indifferent between these plays when

-6 + 1½t = -5 + t/2

t = 1

(We knew that already. )

If t > 1 then Black prefers to play in the environment first.

What if Black starts the ko first? Black kills the corner in 3 net plays, while White plays three times in the environment in exchange.
Result: 20 - 2½t

If Black is indifferent between that and playing in the environment first, then

-6 + 1½t = 20 - 2½t

4t = 26

t = 6½

Black is komaster because she has not had to ignore any ko threats to win the ko. From this we derive the average value of a play in the ko when Black is komaster to be 6½ points.

So far, so good.

But it is unusual, except at low temperatures, to have several plays worth exactly the same. So we look at the corner plus U = {u|-u}, where u > t. And then we look at the corner plus U and V = {v|-v}, where u ≥ v > t.

With this last post I understand that you made as assumption that Black as enough ko threats to be komaster. In this context I agree completly with you and in my post viewtopic.php?p=265279#p265279 I already write "I am wondering if the 6½ points you calculated assume implicitly that black is komaster (?)". Your last post confirm that point and I agree with you.

Now we are facing another problem : what happens if neither black nor white has ko threats?
1) when white has to play in the corner ?
2) when black has to play in the corner ?
3) if black has to play in the corner in which case she has to provoque the ko and in which case she has to only play sente moves ?

I begin to have some interesting answers to these questions but I have now to analyse a little deeper.

Gérard TAILLE wrote:

Bill Spight wrote:

Gérard TAILLE wrote: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.

Thanks for the correction.

Edit: To be clear, I did not show that playing in the environment first is best, I showed that playing in U dominates playing the sente, given the ko. The environment is Y + Z.

The main assumption is that Black is the ko winner, killing the corner in 3 net local plays, if she plays first. To keep things simple, we are ignoring the possibility of seki. Obviously, to be the ko winner Black needs ko threats which White answers. These threats are not shown. Any other threats in the ko fight are not shown, either.

Let's back up to where there is only the corner and the environment with temperature t, where the last player to play in it gains t/2 in the end, an ideal environment.

Black to play has three choices, to start the ko, to play in the environment, or to play the sente, allowing White to live in the corner, and then play in the environment.

2) Black plays in the environment, White saves the corner, and then Black plays in the environment again.
Result: -6 + 1½t

3) Black plays the sente first, White replies, and then Black plays in the environment.
Result: -5 + t/2

Black is indifferent between these plays when

-6 + 1½t = -5 + t/2

t = 1

(We knew that already. )

If t > 1 then Black prefers to play in the environment first.

What if Black starts the ko first? Black kills the corner in 3 net plays, while White plays three times in the environment in exchange.
Result: 20 - 2½t

If Black is indifferent between that and playing in the environment first, then

-6 + 1½t = 20 - 2½t

4t = 26

t = 6½

Black is komaster because she has not had to ignore any ko threats to win the ko. From this we derive the average value of a play in the ko when Black is komaster to be 6½ points.

So far, so good.

But it is unusual, except at low temperatures, to have several plays worth exactly the same. So we look at the corner plus U = {u|-u}, where u > t. And then we look at the corner plus U and V = {v|-v}, where u ≥ v > t.

With this last post I understand that you made as assumption that Black as enough ko threats to be komaster. In this context I agree completly with you and in my post viewtopic.php?p=265279#p265279 I already write "I am wondering if the 6½ points you calculated assume implicitly that black is komaster (?)". Your last post confirm that point and I agree with you.

As I recall, that claim was about the local swing being 26 points instead of 25 if Black was the ko winner. It was possible a play in the ko gained on average 6¼ points instead of 6½ points. I take it that we are in agreement about the swing, even when Black is not komaster.

Gérard TAILLE wrote:Now we are facing another problem : what happens if neither black nor white has ko threats?
1) when white has to play in the corner ?
2) when black has to play in the corner ?
3) if black has to play in the corner in which case she has to provoque the ko and in which case she has to only play sente moves ?

I begin to have some interesting answers to these questions but I have now to analyse a little deeper.

If Black has no ko threats, how does Black win the ko?

There are cases where the komaster will allow the koloser to win a ko, what I have dubbed tunneling.

Bill Spight wrote: If Black has no ko threats, how does Black win the ko?

There are cases where the komaster will allow the koloser to win a ko, what I have dubbed tunneling.

Why do you want black win the ko ?
For me the black strategy is different : depending of the environment black may use the following strategy : she provoques the ko, then she loses the ko but gains in exchange some points in the enviroment.
Look at the following example:

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 . O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

the environment is the following:
Three gote points u, v, w with the values 4, 3½, 3 and I assume the remaining environment being an ideal environment at temperature t = 2½.

Can you find a better result than:

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------------------------
$$ | 5 2 7 8 X X X X X X O . . . . . . . . |
$$ | 4 1 O O X O O O O 6 O . . . . . . . . |
$$ | 3 . O X X X X X X X O . . . . . . . . |
$$ | . O O X X . O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Click Here To Show Diagram Code

[go]$$Bm9
$$ ---------------------------------------
$$ | X 2 X O X X X X X X O . . . . . . . . |
$$ | 4 X O O X O O O O O O . . . . . . . . |
$$ | X 6 O X X X X X X X O . . . . . . . . |
$$ | 8 O O X X . O O O 3 O . . . . . . . . |
$$ | 1 O X X X X X X X X O . . . . . . . . |
$$ | 9 X X . X X O O O 5 O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

takes t/2

[quote="Bill Spight"]
2) Black plays in the environment, White saves the corner, and then Black plays in the environment again.
Result: -6 + 1½t

3) Black plays the sente first, White replies, and then Black plays in the environment.
Result: -5 + t/2

Black is indifferent between these plays when

-6 + 1½t = -5 + t/2

t = 1

(We knew that already. )

It seems you consider here that the result in this scenario is 1 point sente for black. I realise now that this wording may not be correct for two reasons

Reason 1 : under t <= 1 a black move is not sente:

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------
$$ | . 4 3 1 X . . . . |
$$ | 2 . O O X . . . . |
$$ | . . O X X . . . . |
$$ | 6 O O X . . . . . |
$$ | 5 O X X . . . . . |
$$ | 7 X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

As you can see if t<=1 then the black moves 3 and 5 are correct and white will play first in the environment


Reason 2 : if a black move in the corner is evaluated to 1 point sente then we may expect that a black move in the corner must be strictly better than a ½ points gote. Look at the following example:

Click Here To Show Diagram Code

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

a black move at "a" gains only ½ points
and I can see the following sequence:

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------
$$ | . . 4 3 X . 1 O . |
$$ | 2 . O O X X X O . |
$$ | . . O X X O O O . |
$$ | 6 O O X . . . . . |
$$ | 5 O X X . . . . . |
$$ | 7 X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

and black manage to get the result -3.
How can you get a better result by playing directly in the corner ?

Gérard TAILLE wrote:

Bill Spight wrote: If Black has no ko threats, how does Black win the ko?

There are cases where the komaster will allow the koloser to win a ko, what I have dubbed tunneling.

Why do you want black win the ko ?
For me the black strategy is different : depending of the environment black may use the following strategy : she provoques the ko, then she loses the ko but gains in exchange some points in the enviroment.
Look at the following example:

$$B
$$ ---------------------------------------
$$ | . . . . X X X X X X O . . . . . . . . |
$$ | . . O O X O O O O u O . . . . . . . . |
$$ | . . O X X X X X X X O . . . . . . . . |
$$ | . O O X X . O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

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 . O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

the environment is the following:
Three gote points u, v, w with the values 4, 3½, 3 and I assume the remaining environment being an ideal environment at temperature t = 2½.

Can you find a better result than:

$$B
$$ ---------------------------------------
$$ | 5 2 7 8 X X X X X X O . . . . . . . . |
$$ | 4 1 O O X O O O O 6 O . . . . . . . . |
$$ | 3 . O X X X X X X X O . . . . . . . . |
$$ | . O O X X . O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------------------------
$$ | 5 2 7 8 X X X X X X O . . . . . . . . |
$$ | 4 1 O O X O O O O 6 O . . . . . . . . |
$$ | 3 . O X X X X X X X O . . . . . . . . |
$$ | . O O X X . O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

$$Bm9
$$ ---------------------------------------
$$ | X 2 X O X X X X X X O . . . . . . . . |
$$ | 4 X O O X O O O O O O . . . . . . . . |
$$ | X 6 O X X X X X X X O . . . . . . . . |
$$ | 8 O O X X . O O O 3 O . . . . . . . . |
$$ | 1 O X X X X X X X X O . . . . . . . . |
$$ | 9 X X . X X O O O 5 O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Bm9
$$ ---------------------------------------
$$ | X 2 X O X X X X X X O . . . . . . . . |
$$ | 4 X O O X O O O O O O . . . . . . . . |
$$ | X 6 O X X X X X X X O . . . . . . . . |
$$ | 8 O O X X . O O O 3 O . . . . . . . . |
$$ | 1 O X X X X X X X X O . . . . . . . . |
$$ | 9 X X . X X O O O 5 O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

takes t/2

Very good.

I think you are on to something. Let me get back to you on this.

Gérard TAILLE wrote:

Bill Spight wrote: 2) Black plays in the environment, White saves the corner, and then Black plays in the environment again.
Result: -6 + 1½t

3) Black plays the sente first, White replies, and then Black plays in the environment.
Result: -5 + t/2

Black is indifferent between these plays when

-6 + 1½t = -5 + t/2

t = 1

(We knew that already. )

It seems you consider here that the result in this scenario is 1 point sente for black. I realise now that this wording may not be correct for two reasons

Reason 1 : under t <= 1 a black move is not sente:

$$B
$$ ---------------------
$$ | . 4 3 1 X . . . . |
$$ | 2 . O O X . . . . |
$$ | . . O X X . . . . |
$$ | 6 O O X . . . . . |
$$ | 5 O X X . . . . . |
$$ | 7 X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------
$$ | . 4 3 1 X . . . . |
$$ | 2 . O O X . . . . |
$$ | . . O X X . . . . |
$$ | 6 O O X . . . . . |
$$ | 5 O X X . . . . . |
$$ | 7 X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

As you can see if t<=1 then the black moves 3 and 5 are correct and white will play first in the environment

is sente.

$$B Sente, var. 1
$$ ---------------------
$$ | . 4 . 1 X . . . . |
$$ | 5 3 O O X . . . . |
$$ | 7 . O X X . . . . |
$$ | . O O X . . . . . |
$$ | 6 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------

Click Here To Show Diagram Code

[go]$$B Sente, var. 1
$$ ---------------------
$$ | . 4 . 1 X . . . . |
$$ | 5 3 O O X . . . . |
$$ | 7 . O X X . . . . |
$$ | . O O X . . . . . |
$$ | 6 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

elsewhere

$$B Sente, var. 2
$$ ---------------------
$$ | . 5 6 1 X . . . . |
$$ | 4 3 O O X . . . . |
$$ | . 9 O X X . . . . |
$$ | 8 O O X . . . . . |
$$ | 7 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------

Click Here To Show Diagram Code

[go]$$B Sente, var. 2
$$ ---------------------
$$ | . 5 6 1 X . . . . |
$$ | 4 3 O O X . . . . |
$$ | . 9 O X X . . . . |
$$ | 8 O O X . . . . . |
$$ | 7 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

elsewhere

Gérard TAILLE wrote:Reason 2 : if a black move in the corner is evaluated to 1 point sente then we may expect that a black move in the corner must be strictly better than a ½ points gote. Look at the following example:

$$B
$$ ---------------------
$$ | . . . . X . a O . |
$$ | . . O O X X X O . |
$$ | . . O X X O O O . |
$$ | . O O X . . . . . |
$$ | . O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------

Click Here To Show Diagram Code

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

a black move at "a" gains only ½ points
and I can see the following sequence:

$$B
$$ ---------------------
$$ | . . 4 3 X . 1 O . |
$$ | 2 . O O X X X O . |
$$ | . . O X X O O O . |
$$ | 6 O O X . . . . . |
$$ | 5 O X X . . . . . |
$$ | 7 X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------
$$ | . . 4 3 X . 1 O . |
$$ | 2 . O O X X X O . |
$$ | . . O X X O O O . |
$$ | 6 O O X . . . . . |
$$ | 5 O X X . . . . . |
$$ | 7 X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

and black manage to get the result -3.
How can you get a better result by playing directly in the corner ?

Let’s set up an ideal environment for these two lines of play.

Click Here To Show Diagram Code

[go]$$B Sente
$$ ---------------------
$$ | . 4 3 1 X O 5 O . |
$$ | . 2 O O X X X O . |
$$ | . . O X X O 6 O . |
$$ | 8 O O X X X X O . |
$$ | 7 O X X X . 0 O . |
$$ | 9 X . . X X X O . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

Net result: 1 point for White

Click Here To Show Diagram Code

[go]$$B Environment
$$ ---------------------
$$ | . . 4 3 X O 1 O . |
$$ | 2 . O O X X X O . |
$$ | . . O X X O 8 O . |
$$ | 6 O O X X X X O . |
$$ | 5 O X X X . 9 O . |
$$ | 7 X . . X X X O . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

Net result: 1 point for White

Black is indifferent between these two plays. QED.

Bill Spight wrote: is sente.

$$B Sente, var. 1
$$ ---------------------
$$ | . 4 . 1 X . . . . |
$$ | 5 3 O O X . . . . |
$$ | 7 . O X X . . . . |
$$ | . O O X . . . . . |
$$ | 6 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------

Click Here To Show Diagram Code

[go]$$B Sente, var. 1
$$ ---------------------
$$ | . 4 . 1 X . . . . |
$$ | 5 3 O O X . . . . |
$$ | 7 . O X X . . . . |
$$ | . O O X . . . . . |
$$ | 6 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

elsewhere

$$B Sente, var. 2
$$ ---------------------
$$ | . 5 6 1 X . . . . |
$$ | 4 3 O O X . . . . |
$$ | . 9 O X X . . . . |
$$ | 8 O O X . . . . . |
$$ | 7 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------

Click Here To Show Diagram Code

[go]$$B Sente, var. 2
$$ ---------------------
$$ | . 5 6 1 X . . . . |
$$ | 4 3 O O X . . . . |
$$ | . 9 O X X . . . . |
$$ | 8 O O X . . . . . |
$$ | 7 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

elsewhere

OC white has to answer black 1. In that sense black is sente but I do not like this wording:

Click Here To Show Diagram Code

[go]$$B Sente, var. 1
$$ ---------------------
$$ | . . . a X . . . . |
$$ | b . O O X . . . . |
$$ | . . O X X . . . . |
$$ | . O O X . . . . . |
$$ | . O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

The point is the following : the exchange black "a" white "b" is not a gain for black but rather a gain for white because now black cannot provoque a ko.
Let's change the order: if white plays first "b" then white is happy if black answer with "a" because white "b" becomes sente.

In such situation I call the sequence black "a" white "b" a thank-you sequence and I call the sequence white "b" black "a" a sente sequence.

Sure you will understand why I am reluctant to say that black "a" is sente. In the other hand you may say that black "a" is a ko threat but that is quite different.

Gérard TAILLE wrote:

Bill Spight wrote: is sente.

$$B Sente, var. 1
$$ ---------------------
$$ | . 4 . 1 X . . . . |
$$ | 5 3 O O X . . . . |
$$ | 7 . O X X . . . . |
$$ | . O O X . . . . . |
$$ | 6 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------

Click Here To Show Diagram Code

[go]$$B Sente, var. 1
$$ ---------------------
$$ | . 4 . 1 X . . . . |
$$ | 5 3 O O X . . . . |
$$ | 7 . O X X . . . . |
$$ | . O O X . . . . . |
$$ | 6 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

elsewhere

$$B Sente, var. 2
$$ ---------------------
$$ | . 5 6 1 X . . . . |
$$ | 4 3 O O X . . . . |
$$ | . 9 O X X . . . . |
$$ | 8 O O X . . . . . |
$$ | 7 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------

Click Here To Show Diagram Code

[go]$$B Sente, var. 2
$$ ---------------------
$$ | . 5 6 1 X . . . . |
$$ | 4 3 O O X . . . . |
$$ | . 9 O X X . . . . |
$$ | 8 O O X . . . . . |
$$ | 7 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

elsewhere

OC white has to answer black 1. In that sense black is sente but I do not like this wording:

$$B Sente, var. 1
$$ ---------------------
$$ | . . . a X . . . . |
$$ | b . O O X . . . . |
$$ | . . O X X . . . . |
$$ | . O O X . . . . . |
$$ | . O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------

Click Here To Show Diagram Code

[go]$$B Sente, var. 1
$$ ---------------------
$$ | . . . a X . . . . |
$$ | b . O O X . . . . |
$$ | . . O X X . . . . |
$$ | . O O X . . . . . |
$$ | . O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

The point is the following : the exchange black "a" white "b" is not a gain for black but rather a gain for white because now black cannot provoque a ko.

I never said that Black a gains anything. In fact, sente gains nothing on average.

Gérard TAILLE wrote:Let's change the order: if white plays first "b" then white is happy if black answer with "a" because white "b" becomes sente.

sets up a delayed sente for Black, which is sometimes advantageous for White.

Gérard TAILLE wrote:

Bill Spight wrote: If Black has no ko threats, how does Black win the ko?

There are cases where the komaster will allow the koloser to win a ko, what I have dubbed tunneling.

Why do you want black win the ko ?
For me the black strategy is different : depending of the environment black may use the following strategy : she provoques the ko, then she loses the ko but gains in exchange some points in the enviroment.
Look at the following example:

$$B
$$ ---------------------------------------
$$ | . . . . X X X X X X O . . . . . . . . |
$$ | . . O O X O O O O u O . . . . . . . . |
$$ | . . O X X X X X X X O . . . . . . . . |
$$ | . O O X X . O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

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 . O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

the environment is the following:
Three gote points u, v, w with the values 4, 3½, 3 and I assume the remaining environment being an ideal environment at temperature t = 2½.

Can you find a better result than:

$$B
$$ ---------------------------------------
$$ | 5 2 7 8 X X X X X X O . . . . . . . . |
$$ | 4 1 O O X O O O O 6 O . . . . . . . . |
$$ | 3 . O X X X X X X X O . . . . . . . . |
$$ | . O O X X . O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------------------------
$$ | 5 2 7 8 X X X X X X O . . . . . . . . |
$$ | 4 1 O O X O O O O 6 O . . . . . . . . |
$$ | 3 . O X X X X X X X O . . . . . . . . |
$$ | . O O X X . O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

$$Bm9
$$ ---------------------------------------
$$ | X 2 X O X X X X X X O . . . . . . . . |
$$ | 4 X O O X O O O O O O . . . . . . . . |
$$ | X 6 O X X X X X X X O . . . . . . . . |
$$ | 8 O O X X . O O O 3 O . . . . . . . . |
$$ | 1 O X X X X X X X X O . . . . . . . . |
$$ | 9 X X . X X O O O 5 O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Bm9
$$ ---------------------------------------
$$ | X 2 X O X X X X X X O . . . . . . . . |
$$ | 4 X O O X O O O O O O . . . . . . . . |
$$ | X 6 O X X X X X X X O . . . . . . . . |
$$ | 8 O O X X . O O O 3 O . . . . . . . . |
$$ | 1 O X X X X X X X X O . . . . . . . . |
$$ | 9 X X . X X O O O 5 O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

takes t/2

Many thanks, Gérard. You are always an inspiration.

I have studied this position a bit more, and I believe that I understand it better. As always, I follow Berlekamp's dictum that to understand a position, start with its thermograph. The above position has the corner plus three simple gote with local temperatures greater than the temperatures of the ideal environment.

To understand the corner better, I made the three simple sente identical, with a temperature equal to that of the ideal environment, t = 4. IOW, we have the corner plus an ideal environment with 6½ ≥ t > 1.

In the sgf file below I compared these two lines of play with Black playing first.

1) Black plays first in the environment and then White plays in the corner, and then Black plays in the environment again.

2) Black starts the ko, which is sente. In the fight, with neither side having a ko threat, White plays once in the environment and Black plays twice, and then White plays in the corner to win the ko, and then Black plays in the environment again.

These two lines of play should produce the same net result. As the sgf file indicates, they do.



The three simple gote give leeway by adjusting their miai values for starting the ko to be the best play. What is intriguing is that values that create an environment that is very close to ideal does that.

Also, if it is White to play before the global temperature drops to 1, White will almost surely play in the environment. So Black will have many chances to gain from starting the ko at her turn. Well spotted, Gérard!

Bill Spight wrote: Many thanks, Gérard. You are always an inspiration.

I have studied this position a bit more, and I believe that I understand it better. As always, I follow Berlekamp's dictum that to understand a position, start with its thermograph. The above position has the corner plus three simple gote with local temperatures greater than the temperatures of the ideal environment.

To understand the corner better, I made the three simple sente identical, with a temperature equal to that of the ideal environment, t = 4. IOW, we have the corner plus an ideal environment with 6½ ≥ t > 1.

In the sgf file below I compared these two lines of play with Black playing first.

1) Black plays first in the environment and then White plays in the corner, and then Black plays in the environment again.

2) Black starts the ko, which is sente. In the fight, with neither side having a ko threat, White plays once in the environment and Black plays twice, and then White plays in the corner to win the ko, and then Black plays in the environment again.

These two lines of play should produce the same net result. As the sgf file indicates, they do.



The three simple gote give leeway by adjusting their miai values for starting the ko to be the best play. What is intriguing is that values that create an environment that is very close to ideal does that.

Also, if it is White to play before the global temperature drops to 1, White will almost surely play in the environment. So Black will have many chances to gain from starting the ko at her turn. Well spotted, Gérard!

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------------------------
$$ | . . . . X X X X X X O . . . . . . . . |
$$ | a . O O X O O O O 1 O . . . . . . . . |
$$ | . . O X X X X X X X O . . . . . . . . |
$$ | . O O X X O O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X O O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
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$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Oops I do not understand your variation 2. After in the environment white cannot play in the corner when the temperature of the environment is as high as t = 4. White must answer in the environment waiting for a drop of this temperature. Basically white's line of play is not quite difficult. Black's problem is more difficult. She has to decide:
1) when to play in the corner
2) which move to play in the corner

Gérard TAILLE wrote:

Bill Spight wrote: Many thanks, Gérard. You are always an inspiration.

I have studied this position a bit more, and I believe that I understand it better. As always, I follow Berlekamp's dictum that to understand a position, start with its thermograph. The above position has the corner plus three simple gote with local temperatures greater than the temperatures of the ideal environment.

To understand the corner better, I made the three simple sente identical, with a temperature equal to that of the ideal environment, t = 4. IOW, we have the corner plus an ideal environment with 6½ ≥ t > 1.

In the sgf file below I compared these two lines of play with Black playing first.

1) Black plays first in the environment and then White plays in the corner, and then Black plays in the environment again.

2) Black starts the ko, which is sente. In the fight, with neither side having a ko threat, White plays once in the environment and Black plays twice, and then White plays in the corner to win the ko, and then Black plays in the environment again.

These two lines of play should produce the same net result. As the sgf file indicates, they do.



The three simple gote give leeway by adjusting their miai values for starting the ko to be the best play. What is intriguing is that values that create an environment that is very close to ideal does that.

Also, if it is White to play before the global temperature drops to 1, White will almost surely play in the environment. So Black will have many chances to gain from starting the ko at her turn. Well spotted, Gérard!

$$B
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$$ | . . . . X X X X X X O . . . . . . . . |
$$ | a . O O X O O O O 1 O . . . . . . . . |
$$ | . . O X X X X X X X O . . . . . . . . |
$$ | . O O X X O O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X O O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
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$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------------------------
$$ | . . . . X X X X X X O . . . . . . . . |
$$ | a . O O X O O O O 1 O . . . . . . . . |
$$ | . . O X X X X X X X O . . . . . . . . |
$$ | . O O X X O O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X O O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
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$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Oops I do not understand your variation 2. After in the environment white cannot play in the corner when the temperature of the environment is as high as t = 4. White must answer in the environment waiting for a drop of this temperature.

Berlekamp's komaster theory stipulates that for ko positions there are enough plays at the temperature of the environment that the ko may be played out before the temperature drops. In that case White's reply in the environment is simply treading water; the temperature is never going to drop. Remember, CGT does not take into account whose turn it is. I omitted White's play for the corner when White plays first because we know that it is the jump to A-18.

By contrast with the case that the three simple gote are different, then we don't know that it is and von Neumann game theory applies. That's how we determined that White's play in the environment is better when the global temperature is greater than 1.

Bill Spight wrote: Berlekamp's komaster theory stipulates that for ko positions there are enough plays at the temperature of the environment that the ko may be played out before the temperature drops. In that case White's reply in the environment is simply treading water; the temperature is never going to drop. Remember, CGT does not take into account whose turn it is. I omitted White's play for the corner when White plays first because we know that it is the jump to A-18.

By contrast with the case that the three simple gote are different, then we don't know that it is and von Neumann game theory applies. That's how we determined that White's play in the environment is better when the global temperature is greater than 1.

For any white go player, playing in the corner when t = 4 is a very bad move (it looks like a 1 point reverse sente move). You mentionned "Berlekamp's komaster theory" and I cannot believe that such theory stipulates that such white play in the corner might be good. Obviously there is a misunderstanding somewhere. Maybe here is two important points:
1) Neither black nor white have ko threats => "Berlekamp's komaster theory" does not apply (?)
2) For white point of view the ko does not exist. The ko can exist only if black plays first and only if black choose the ko variant. If black provoques really this ko then white OC will fight in the corner without waiting temperature drop.

Gérard TAILLE wrote:

Bill Spight wrote: Berlekamp's komaster theory stipulates that for ko positions there are enough plays at the temperature of the environment that the ko may be played out before the temperature drops. In that case White's reply in the environment is simply treading water; the temperature is never going to drop. Remember, CGT does not take into account whose turn it is. I omitted White's play for the corner when White plays first because we know that it is the jump to A-18.

By contrast with the case that the three simple gote are different, then we don't know that it is and von Neumann game theory applies. That's how we determined that White's play in the environment is better when the global temperature is greater than 1.

For any white go player, playing in the corner when t = 4 is a very bad move (it looks like a 1 point reverse sente move). You mentionned "Berlekamp's komaster theory" and I cannot believe that such theory stipulates that such white play in the corner might be good.

The problem is me and Berlekamp. Berlekamp developed his komaster theory before I came along. In 1994 I attended his talk in which he presented the theory, but he apparently had developed it before 1990. He defined it in terms of tax, like the rest of thermography at that time. However, komaster theory obviously did not apply to multiple kos and superkos. In 1998 I redefined and expanded thermography to cover multiple kos and superkos by basing it on play in an ideal environment. When you define it that way you accommodate komaster theory by requiring all ko plays to be made before the temperature of the ideal environment drops. That is why made all three plays the same size and said that if Black starts by taking one of the three plays it does no good for White to take one, as well. There will always be enough plays of the same size for White to take one back. In Berlekamp's original formulation, there are no plays outside the corner, and you just apply the tax. Thermography is usually an approximation of the whole board game. If you can solve the whole board you do not need thermography.

Bill Spight wrote:In 1998 I redefined and expanded thermography to cover multiple kos and superkos by basing it on play in an ideal environment.

Do you mean an arbitrarily dense "rich environment"?