Life In 19x19

Posted: **Tue Jun 08, 2021 1:38 pm**

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

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

BTW, if you do not know the white move at "a" and you play white "b" then black is komaster and is able to kill the top left white corner.
Otherwise if white plays "a" he will be able to save his group by playing first at "a" and black has no more ko threat. But note that by playing at "a" white do not play in the environment in order to gain t points and that means that the black ko threat allows black to add t + 1 to her score (t for a play in the enviroment and +1 for the white move in white territory).
I do not know what is the correct wording here. Certainly black in not "komaster" but her ko threat cannot be ignore and will assure some good points.

Posted: **Tue Jun 08, 2021 5:55 pm**

Gérard TAILLE wrote:
Gérard TAILLE wrote:
Bill Spight wrote:Oh, yes. Generally speaking the point of Black starting on the 2-2 is to make the ko, not to generate ko threats.

Up until now we have focused on the play that you discovered where Black makes the ko, despite neither player having a ko threat, now or later.

Evaluation of kos, except for placid kos, depends upon what assumptions we make. Berlekamp's brilliant komaster analysis depends upon assumptions that are usually not met exactly on the go board, but which provide useful approximate limits. When neither side is komaster, he came up with the idea of a neutral threat environment (NTE), where each player has the exact opposite of the ko threats of the other, where the ko threats are sufficiently large. How good an approximation NTE produces in practice is another question, but let's give it a shot.

$$W White first
$$ ---------------------
$$ | C 5 4 2 X . . . . |
$$ | 1 C O O X . . . . |
$$ | C C O X X . . . . |
$$ | C O O X . . . . . |
$$ | . O X X . . . . . |
$$ | 3 X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$W White first $$ --------------------- $$ | C 5 4 2 X . . . . | $$ | 1 C O O X . . . . | $$ | C C O X X . . . . | $$ | C O O X . . . . . | $$ | . O X X . . . . . | $$ | 3 X . . . . . . . | $$ | . X . . . . . . . | $$ | X X . . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]
gets a local score of -5 in one net play.

$$B Black first, no ko
$$ ---------------------
$$ | . 4 3 1 X . . . . |
$$ | . 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]$$B Black first, no ko $$ --------------------- $$ | . 4 3 1 X . . . . | $$ | . 2 O O X . . . . | $$ | . . O X X . . . . | $$ | 6 O O X . . . . . | $$ | 5 O X X . . . . . | $$ | 7 X . . . . . . . | $$ | . X . . . . . . . | $$ | X X . . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]
gets a net local score of -3 in one net play.

$$B Black first, ko, NTE
$$ ---------------------
$$ | 7 2 . 4 X . . . . |
$$ | . 1 O O X . . . . |
$$ | 3 . O X X . . . . |
$$ | 6 O O X . . . . . |
$$ | 5 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$B Black first, ko, NTE $$ --------------------- $$ | 7 2 . 4 X . . . . | $$ | . 1 O O X . . . . | $$ | 3 . O X X . . . . | $$ | 6 O O X . . . . . | $$ | 5 O X X . . . . . | $$ | . X . . . . . . . | $$ | . X . . . . . . . | $$ | X X . . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]
Now we have reached a position after one net move by Black such that either player, playing first could win the ko in two local plays by ignoring the opponent's threat.

$$W White first, NTE
$$ ---------------------
$$ | X O . O X . . . . |
$$ | 1 X O O X . . . . |
$$ | X 3 O X X . . . . |
$$ | O O O X . . . . . |
$$ | X O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$W White first, NTE $$ --------------------- $$ | X O . O X . . . . | $$ | 1 X O O X . . . . | $$ | X 3 O X X . . . . | $$ | O O O X . . . . . | $$ | X O X X . . . . . | $$ | . X . . . . . . . | $$ | . X . . . . . . . | $$ | X X . . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]
= ko threat, completes threat

After there is a local score of -7 plus a gote worth on average 0 when t > 1.

$$B Black first, NTE
$$ ---------------------
$$ | X W 1 W X . . . . |
$$ | C X W W X . . . . |
$$ | X 3 W X X . . . . |
$$ | W W W X . . . . . |
$$ | X W X X . . . . . |
$$ | C X . . . . . . . |
$$ | C X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$B Black first, NTE $$ --------------------- $$ | X W 1 W X . . . . | $$ | C X W W X . . . . | $$ | X 3 W X X . . . . | $$ | W W W X . . . . . | $$ | X W X X . . . . . | $$ | C X . . . . . . . | $$ | C X . . . . . . . | $$ | X X . . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]
= ko threat, completes threat

After there is a local score of +21.

By assumption, what each player gains by playing and completing their threat is the same, and therefore cancel out in the average, so the average value of the given position under NTE when t > 1 is the average of the value of these two politions. That value is (21 - 7)/2 = 7, 10 points better than -3.

So, under NTE conditions, the average value of the corner is (7 - 5)/2 = 1, and the miai value of a play is 6.
Though I follow your calculation the conclusion does not fit with my understanding of the miai value basic concept. For me a miai value of say "6" means that each player has to play in the local area when the temperature drops under 6.

The thing is that kos typically destroy the independence of the local area of the ko and the rest of the board. The concepts of komaster and NTE are ways of taking account of the relevance of the rest of the board. With kos we cannot always rely upon the basic idea of miai values.

Miai values rely upon average territorial values. With a ko, strictly speaking there may not be an average value, and thus, no miai value in the basic sense, either. That is why Berlekamp talked about the mast values of kos.

And indeed, if NTE conditions apply, either player should play in the corner when the global temperature drops under 6. In addition, with complex kos such as this one, altering the conditions on the rest of the board is often done instead of playing in the local ko position.

And, as we have seen, with this ko the top three plays elsewhere may be considered as local. Kos destroy the independence of the board.

Posted: **Wed Jun 09, 2021 4:22 am**

I have dubbed the ko plus other relevant regions of the board the ko ensemble.

Maybe I have goofed, but I think that the ko corner plus one humungous ko threat for Black makes Black komaster. Here is an SGF file.

The ko ensemble has a mast value of -1½, with each play gaining 6½ points on average.

Posted: **Wed Jun 09, 2021 5:53 am**

Bill Spight wrote:I have dubbed the ko plus other relevant regions of the board the ko ensemble.

Maybe I have goofed, but I think that the ko corner plus one humungous ko threat for Black makes Black komaster. Here is an SGF file.

Oops why not the above sequence? I alaways proposed to play white tenuki in the corner just after having played my new move A17. That way white can save his group.

Posted: **Wed Jun 09, 2021 6:53 am**

Very good, thanks.

Here is a new SGF file. Edited for correctness.

So the mast value of the ko ensemble is -5 points and the average gain per move is 3 points.

Edit: As pointed out later, this calculation does not give the value of the vertical mast, which is what the unqualified term, mast value, indicates.

Posted: **Wed Jun 09, 2021 9:44 am**

Bill Spight wrote:Very good, thanks.

Here is a new SGF file. Edited for correctness.

So the average value of the ko ensemble is -5 points and the average gain per move is 3 points.

I do not understand how you count the ko variant

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

Unless the temperature is very low the answer

is very bad isn't it?
My feeling is that white must play in the corner as soon as she cannot afford to give up the corner. IOW I see a miai value far higher than yours and probably near from 6½. Are you sure white has better to play in the environment if temperature is between 3 and 6½ ?

Posted: **Thu Jun 10, 2021 9:16 am**

Gérard TAILLE wrote:
Bill Spight wrote:Very good, thanks.

Here is a new SGF file. Edited for correctness.

So the average value of the ko ensemble is -5 points and the average gain per move is 3 points.
I do not understand how you count the ko variant

$$W
$$ ---------------------
$$ | X 1 X O X X O O . |
$$ | . X O O X X O O O |
$$ | X . O X X X O O O |
$$ | 2 O O X . X O O O |
$$ | X O X X . X O O O |
$$ | . X . . . X O O . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$W $$ --------------------- $$ | X 1 X O X X O O . | $$ | . X O O X X O O O | $$ | X . O X X X O O O | $$ | 2 O O X . X O O O | $$ | X O X X . X O O O | $$ | . X . . . X O O . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]
Unless the temperature is very low the answer is very bad isn't it?

It is sente, isn't it?

To find the local territorial count you play out the sente.

If Black plays elsewhere, so that White wins the ko with more local plays than Black, we cannot find a mast value fo the ko ensemble, because then White plays as many net local plays as when White plays first. If you want to consider a Black play elsewhere at this point, it must be part of the ko ensemble.

Gérard TAILLE wrote:My feeling is that white must play in the corner as soon as she cannot afford to give up the corner. IOW I see a miai value far higher than yours and probably near from 6½. Are you sure white has better to play in the environment if temperature is between 3 and 6½ ?

When you can read the play out you don't need theory. Both thermography and the traditional way of estimating territory and the value of plays only promise estimates and heuristics. So whenever you ask me whether I am sure that a heuristic is right, regardless of the rest of the board, I have to say no.

But looking at this corner with no ko threats, we found that the three hottest positions probably belong in the ko ensemble. Maybe in this case the two hottest positions belong in the ko ensemble.

Posted: **Thu Jun 10, 2021 11:11 am**

Bill Spight wrote: It is sente, isn't it?

To find the local territorial count you play out the sente.

I do dot understand Bill.
OC it is sente but that does not mean that you have to play it to find the territorial count.
Let's take a very simple example

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

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

After

the move

is sente but looks quite bad. In order to count the local territory will you assume this sente move or will you assume a ko fight?

Posted: **Thu Jun 10, 2021 12:08 pm**

Gérard TAILLE wrote:
Bill Spight wrote: It is sente, isn't it?

To find the local territorial count you play out the sente.
I do dot understand Bill.
OC it is sente but that does not mean that you have to play it to find the territorial count.
Let's take a very simple example

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

$$W
$$ ---------------------
$$ | 1 X O X . . . . . |
$$ | X O O X . . . . . |
$$ | . O X X . . . . . |
$$ | 2 O X . . . . . . |
$$ | X X X . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$W $$ --------------------- $$ | 1 X O X . . . . . | $$ | X O O X . . . . . | $$ | . O X X . . . . . | $$ | 2 O X . . . . . . | $$ | X X X . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]
After the move is sente but looks quite bad. In order to count the local territory will you assume this sente move or will you assume a ko fight?

It's a bad play at territorial scoring, because it gains nothing. {shrug} We do not play it for that reason.

Click Here To Show Diagram Code: [go]$$W White first $$ --------------------- $$ | 1 X O X . . . . . | $$ | X O O X . . . . . | $$ | 3 O X X . . . . . | $$ | . O X . . . . . . | $$ | X X X . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ ---------------------[/go]

Result: -4 in 2 plays by White.

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

Result: +12 in 1 play by Black.

The ko has a mast value when

-4 + 2t = 12 - t, that is, when

t = 5⅓ , and the mast value, m, is

m = 6⅔

Posted: **Thu Jun 10, 2021 1:16 pm**

OK, let's at the simple gote, U = {u|-u}, u > 0, to the ko ensemble. U is not shown on the board.

Click Here To Show Diagram Code: [go]$$W White first, var. 1 $$ --------------------- $$ | . . . . X X O O . | $$ | 1 . O O X X O O O | $$ | . . O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O a . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]

= u

If White plays in the ko corner, Black replies in U when u > 1. That yields a sente sequence with the result, u - 8.

If White plays first in U, then the mast value is -5 - u in gote. Playing first in U dominates when

u - 8 ≥ -5 - u , that is, when

u ≥ 1½

Next, let Black play first.

Click Here To Show Diagram Code: [go]$$Wm10 Black first, var. 1 $$ --------------------- $$ | X 1 X O X X O O . | $$ | 3 X O O X X O O O | $$ | X 5 O X X X O O O | $$ | 2 O O X . X O O O | $$ | X O X X . X O O O | $$ | 4 X . . . X O O . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]

= u

Note that

gains 1 point at territorial go.

The result is u - 5 in 1 net Black play.

Click Here To Show Diagram Code: [go]$$Wm10 Black first, var. 2 $$ --------------------- $$ | X 1 X O X X O O . | $$ | 3 X O O X X O O O | $$ | X 5 O X X X O O O | $$ | . O O X . X O O O | $$ | X O X X . X O O O | $$ | 4 X . . . X O O . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]

= u

Note that

gains 1 point at territorial go.

The result is u - 6 in sente.

The play in U dominates when

u - 5 ≥ u - 6 , that is, always.

Last, suppose that Black plays in U first. Then White will play in the top left corner and the result will be u - 8 in sente. Obviously, variation 2 dominates that, as well.

Now to find the mast value of the ko ensemble.

When u > 1½ the mast value of the ko ensemble happens when

u - 5 = -5 - u , that is, when

u = 0 , that is, never.

So let 1½ > u > 1. The the mast value of the ko ensemble occurs when

u - 5 = u - 8, that is, never.

Maybe I have goofed again, but it seems like adding a simple gote to the ko ensemble does not produce a mast value for it. {shrug}

Give me a while and I'll try adding two simple gote to the ko ensemble.

Posted: **Thu Jun 10, 2021 1:48 pm**

Before analysing your last post I give you my own analysis:

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

white plays first in the corner:

Click Here To Show Diagram Code: [go]$$W White first $$ --------------------- $$ | . . . . X X O O . | $$ | 1 . O O X X O O O | $$ | . . O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O . . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]

tenuki
scoreWhiteFirst = -7 + t

Black plays first in the corner and white saves his corner:

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

Click Here To Show Diagram Code: [go]$$Bcm9 Black first, ko $$ --------------------- $$ | X 2 X O X X O O . | $$ | 4 X O O X X O O O | $$ | X 6 O X X X O O O | $$ | 8 O O X . X O O O | $$ | 1 O X X . X O O O | $$ | 7 X . . . X O O . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]

tenuki

tenuki
scoreBlackFirstKo = -8 + 2t

Black plays first in the corner and white gives up his corner:

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

tenuki

tenuki
scoreBlackFirstWhiteKilled = +18 - 3t

The three scores are:
scoreWhiteFirst = -7 + t
scoreBlackFirstKo = -8 + 2t
scoreBlackFirstWhiteKilled = +18 - 3t

Obviously scoreBlackFirstKo is not good for white if t > 1.
The two other scores have to be compared:
scoreWhiteFirst = scoreBlackFirstWhiteKilled <=>
-7 + t = +18 - 3t <=>
t = 6¼ and m = -¾

In the last sequence you can argue that

may be omitted because black's ko threat is still available. If it is playable then
scoreBlackFirstWhiteKilledBis = +18 - 2t and
scoreWhiteFirst = scoreBlackFirstWhiteKilledBis <=>
-7 + t = +18 - 2t <=>
t = 8⅓ and m = 1⅓

The miai value is probably somewhere between 6¼ and 8⅓.

Posted: **Thu Jun 10, 2021 2:52 pm**

Many thanks, Gérard.

Gérard TAILLE wrote:Before analysing your last post I give you my own analysis:

$$W
$$ ---------------------
$$ | . . . . X X O O . |
$$ | . . O O X X O O O |
$$ | . . O X X X O O O |
$$ | . O O X . X O O O |
$$ | . O X X . X O O O |
$$ | . X . . . X O . . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$W $$ --------------------- $$ | . . . . X X O O . | $$ | . . O O X X O O O | $$ | . . O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O . . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]
white plays first in the corner:
$$W White first
$$ ---------------------
$$ | . . . . X X O O . |
$$ | 1 . O O X X O O O |
$$ | . . O X X X O O O |
$$ | . O O X . X O O O |
$$ | . O X X . X O O O |
$$ | . X . . . X O . . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$W White first $$ --------------------- $$ | . . . . X X O O . | $$ | 1 . O O X X O O O | $$ | . . O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O . . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]
tenuki
scoreWhiteFirst = -7 + t

I think you misremembered that the average value of the top left corner is -5. The average value for the ko ensemble is -5 - 3 = -8. Result: -8 + t.

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

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

Gérard TAILLE wrote:Black plays first in the corner and white saves his corner:
$$B Black first, ko
$$ ---------------------
$$ | 5 2 7 8 X X O O . |
$$ | 4 1 O O X X O O O |
$$ | 3 . O X X X O O O |
$$ | . O O X . X O O O |
$$ | . O X X . X O O O |
$$ | . X . . . X O 6 . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$B Black first, ko $$ --------------------- $$ | 5 2 7 8 X X O O . | $$ | 4 1 O O X X O O O | $$ | 3 . O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O 6 . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]

$$Bcm9 Black first, ko
$$ ---------------------
$$ | X 2 X O X X O O . |
$$ | 4 X O O X X O O O |
$$ | X 6 O X X X O O O |
$$ | 8 O O X . X O O O |
$$ | 1 O X X . X O O O |
$$ | 7 X . . . X O O . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$Bcm9 Black first, ko $$ --------------------- $$ | X 2 X O X X O O . | $$ | 4 X O O X X O O O | $$ | X 6 O X X X O O O | $$ | 8 O O X . X O O O | $$ | 1 O X X . X O O O | $$ | 7 X . . . X O O . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]
tenuki
tenuki
scoreBlackFirstKo = -8 + 2t

Again, your counting seems off by 1.

White has 6 points of territory and has captured 4 stones, for -10, and Black has 1 point of territory and has captured 2 stones, for +3. Result -7 + 2t.

Gérard TAILLE wrote:Black plays first in the corner and white gives up his corner:
$$B black first, white killed
$$ ---------------------
$$ | 5 2 7 9 X X O O . |
$$ | 4 1 O O X X O O O |
$$ | 3 . O X X X O O O |
$$ | . O O X . X O O O |
$$ | . O X X . X O O O |
$$ | . X . . . X O . . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$B black first, white killed $$ --------------------- $$ | 5 2 7 9 X X O O . | $$ | 4 1 O O X X O O O | $$ | 3 . O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O . . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]
tenuki
tenuki
tenuki
scoreBlackFirstWhiteKilled = +18 - 3t

Here we agree on the score.

The three scores are:
scoreWhiteFirst = -7 + t
scoreBlackFirstKo = -8 + 2t
scoreBlackFirstWhiteKilled = +18 - 3t

Gérard TAILLE wrote:Obviously scoreBlackFirstKo is not good for white if t > 1.
The two other scores have to be compared:
scoreWhiteFirst = scoreBlackFirstWhiteKilled <=>
-7 + t = +18 - 3t <=>
t = 6¼ and m = -¾

That should be, I think,

countWhiteFirst = -8 + t
scoreBlackFirstKo = -7 + 2t
scoreBlackFirstWhiteKilled = +18 - 3t

When t > 1 then scoreBlackFirstKo is better for White than countWhiteFirst. That is, in the BlackFirstKo sequence Black has made a mistake or White has made a mistake in the countWhiteFirst sequence.

The temperature of indifference between the WhiteFirst sequence and the BlackFirstWhiteKilled sequence occurs when

-8 + t = +18 - 3t , that is, when

t = 6½.

Posted: **Thu Jun 10, 2021 3:14 pm**

Bill Spight wrote:Many thanks, Gérard.

Gérard TAILLE wrote:Before analysing your last post I give you my own analysis:

$$W
$$ ---------------------
$$ | . . . . X X O O . |
$$ | . . O O X X O O O |
$$ | . . O X X X O O O |
$$ | . O O X . X O O O |
$$ | . O X X . X O O O |
$$ | . X . . . X O . . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$W $$ --------------------- $$ | . . . . X X O O . | $$ | . . O O X X O O O | $$ | . . O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O . . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]
white plays first in the corner:
$$W White first
$$ ---------------------
$$ | . . . . X X O O . |
$$ | 1 . O O X X O O O |
$$ | . . O X X X O O O |
$$ | . O O X . X O O O |
$$ | . O X X . X O O O |
$$ | . X . . . X O . . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$W White first $$ --------------------- $$ | . . . . X X O O . | $$ | 1 . O O X X O O O | $$ | . . O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O . . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]
tenuki
scoreWhiteFirst = -7 + t
I think you misremembered that the average value of the top left corner is -5. The average value for the ko ensemble is -5 - 3 = -8. Result: -8 + t.

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

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

Gérard TAILLE wrote:Black plays first in the corner and white saves his corner:
$$B Black first, ko
$$ ---------------------
$$ | 5 2 7 8 X X O O . |
$$ | 4 1 O O X X O O O |
$$ | 3 . O X X X O O O |
$$ | . O O X . X O O O |
$$ | . O X X . X O O O |
$$ | . X . . . X O 6 . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$B Black first, ko $$ --------------------- $$ | 5 2 7 8 X X O O . | $$ | 4 1 O O X X O O O | $$ | 3 . O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O 6 . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]

$$Bcm9 Black first, ko
$$ ---------------------
$$ | X 2 X O X X O O . |
$$ | 4 X O O X X O O O |
$$ | X 6 O X X X O O O |
$$ | 8 O O X . X O O O |
$$ | 1 O X X . X O O O |
$$ | 7 X . . . X O O . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$Bcm9 Black first, ko $$ --------------------- $$ | X 2 X O X X O O . | $$ | 4 X O O X X O O O | $$ | X 6 O X X X O O O | $$ | 8 O O X . X O O O | $$ | 1 O X X . X O O O | $$ | 7 X . . . X O O . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]
tenuki
tenuki
scoreBlackFirstKo = -8 + 2t
Again, your counting seems off by 1. White has 6 points of territory and has captured 4 stones, for -10, and Black has 1 point of territory and has captured 2 stones, for +3. Result -7 + 2t.

Gérard TAILLE wrote:Black plays first in the corner and white gives up his corner:
$$B black first, white killed
$$ ---------------------
$$ | 5 2 7 9 X X O O . |
$$ | 4 1 O O X X O O O |
$$ | 3 . O X X X O O O |
$$ | . O O X . X O O O |
$$ | . O X X . X O O O |
$$ | . X . . . X O . . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$B black first, white killed $$ --------------------- $$ | 5 2 7 9 X X O O . | $$ | 4 1 O O X X O O O | $$ | 3 . O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O . . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]
tenuki
tenuki
tenuki
scoreBlackFirstWhiteKilled = +18 - 3t
Here we agree on the score.

The three scores are:
scoreWhiteFirst = -7 + t
scoreBlackFirstKo = -8 + 2t
scoreBlackFirstWhiteKilled = +18 - 3t

Gérard TAILLE wrote:Obviously scoreBlackFirstKo is not good for white if t > 1.
The two other scores have to be compared:
scoreWhiteFirst = scoreBlackFirstWhiteKilled <=>
-7 + t = +18 - 3t <=>
t = 6¼ and m = -¾
That should be, I think,

countWhiteFirst = -8 + t
scoreBlackFirstKo = -7 + 2t
scoreBlackFirstWhiteKilled = +18 - 3t

When t > 1 then scoreBlackFirstKo is better for White than countWhiteFirst. That is, in the BlackFirstKo sequence Black has made a mistake or White has made a mistake in the countWhiteFirst sequence.

The temperature of indifference between the WhiteFirst sequence and the BlackFirstWhiteKilled sequence occurs when

-8 + t = +18 - 3t , that is, when

t = 6½.

Thank you for correcting my count Bill.
with the t = 6½ conclusion I am satisfied.
As we can see the situation with a black ko threat is very different. When no black ko threat white will not play in the corner (unless temperature is very low) but with one black ko threat white must play in the corner as soon as temperature drops under t = 6½.
BTW with more than one black ko threat, black is komaster and again white must play in the corner when temperature drops under t = 6½.
Finally with only one ko threat black is "almost" komaster isn't she?

Posted: **Thu Jun 10, 2021 4:20 pm**

Gérard TAILLE wrote:
Bill Spight wrote:Many thanks, Gérard.

Gérard TAILLE wrote:Before analysing your last post I give you my own analysis:

$$W
$$ ---------------------
$$ | . . . . X X O O . |
$$ | . . O O X X O O O |
$$ | . . O X X X O O O |
$$ | . O O X . X O O O |
$$ | . O X X . X O O O |
$$ | . X . . . X O . . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$W $$ --------------------- $$ | . . . . X X O O . | $$ | . . O O X X O O O | $$ | . . O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O . . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]
white plays first in the corner:
$$W White first
$$ ---------------------
$$ | . . . . X X O O . |
$$ | 1 . O O X X O O O |
$$ | . . O X X X O O O |
$$ | . O O X . X O O O |
$$ | . O X X . X O O O |
$$ | . X . . . X O . . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$W White first $$ --------------------- $$ | . . . . X X O O . | $$ | 1 . O O X X O O O | $$ | . . O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O . . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]
tenuki
scoreWhiteFirst = -7 + t
I think you misremembered that the average value of the top left corner is -5. The average value for the ko ensemble is -5 - 3 = -8. Result: -8 + t.

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

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

Gérard TAILLE wrote:Black plays first in the corner and white saves his corner:
$$B Black first, ko
$$ ---------------------
$$ | 5 2 7 8 X X O O . |
$$ | 4 1 O O X X O O O |
$$ | 3 . O X X X O O O |
$$ | . O O X . X O O O |
$$ | . O X X . X O O O |
$$ | . X . . . X O 6 . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$B Black first, ko $$ --------------------- $$ | 5 2 7 8 X X O O . | $$ | 4 1 O O X X O O O | $$ | 3 . O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O 6 . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]

$$Bcm9 Black first, ko
$$ ---------------------
$$ | X 2 X O X X O O . |
$$ | 4 X O O X X O O O |
$$ | X 6 O X X X O O O |
$$ | 8 O O X . X O O O |
$$ | 1 O X X . X O O O |
$$ | 7 X . . . X O O . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$Bcm9 Black first, ko $$ --------------------- $$ | X 2 X O X X O O . | $$ | 4 X O O X X O O O | $$ | X 6 O X X X O O O | $$ | 8 O O X . X O O O | $$ | 1 O X X . X O O O | $$ | 7 X . . . X O O . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]
tenuki
tenuki
scoreBlackFirstKo = -8 + 2t
Again, your counting seems off by 1. White has 6 points of territory and has captured 4 stones, for -10, and Black has 1 point of territory and has captured 2 stones, for +3. Result -7 + 2t.

Gérard TAILLE wrote:Black plays first in the corner and white gives up his corner:
$$B black first, white killed
$$ ---------------------
$$ | 5 2 7 9 X X O O . |
$$ | 4 1 O O X X O O O |
$$ | 3 . O X X X O O O |
$$ | . O O X . X O O O |
$$ | . O X X . X O O O |
$$ | . X . . . X O . . |
$$ | . X . . . X X O O |
$$ | X X . . . . X X X |
$$ | . . . . . . . . . |
$$ ---------------------
Click Here To Show Diagram Code
[go]$$B black first, white killed $$ --------------------- $$ | 5 2 7 9 X X O O . | $$ | 4 1 O O X X O O O | $$ | 3 . O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O . . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]
tenuki
tenuki
tenuki
scoreBlackFirstWhiteKilled = +18 - 3t
Here we agree on the score.

The three scores are:
scoreWhiteFirst = -7 + t
scoreBlackFirstKo = -8 + 2t
scoreBlackFirstWhiteKilled = +18 - 3t

Gérard TAILLE wrote:Obviously scoreBlackFirstKo is not good for white if t > 1.
The two other scores have to be compared:
scoreWhiteFirst = scoreBlackFirstWhiteKilled <=>
-7 + t = +18 - 3t <=>
t = 6¼ and m = -¾
That should be, I think,

countWhiteFirst = -8 + t
scoreBlackFirstKo = -7 + 2t
scoreBlackFirstWhiteKilled = +18 - 3t

When t > 1 then scoreBlackFirstKo is better for White than countWhiteFirst. That is, in the BlackFirstKo sequence Black has made a mistake or White has made a mistake in the countWhiteFirst sequence.

The temperature of indifference between the WhiteFirst sequence and the BlackFirstWhiteKilled sequence occurs when

-8 + t = +18 - 3t , that is, when

t = 6½.
Thank you for correcting my count Bill.
with the t = 6½ conclusion I am satisfied.
As we can see the situation with a black ko threat is very different. When no black ko threat white will not play in the corner (unless temperature is very low) but with one black ko threat white must play in the corner as soon as temperature drops under t = 6½.
BTW with more than one black ko threat, black is komaster and again white must play in the corner when temperature drops under t = 6½.
Finally with only one ko threat black is "almost" komaster isn't she?

I am sorry, my statement about one player or other, given the results, having made a mistake, is wrong.

After

Black can kill the corner without playing

, so

is a mistake. Black should wait until White eliminates the ko threat to play

. That yields scoreBlackFirstWhiteKilled = +19 - 2t. The temperature of indifference occurs when

-8 + t = +19 - 2t , that is, when

t = 9.

When should

eliminate the ko threat after

? If, indeed, White's best play afterwards is the sequence in BlackFirstKo, it is when

-7 + 2t < +19 - 2t , that is, when

t < 6½.

So, given that assumption, if

does not eliminate the ko threat,

should not kill the corner unless t < 9, and

should not eliminate the ko threat until t < 6½. Between those two temperatures Black wins the ko, even though White can prevent it, a case of tunneling.

Posted: **Thu Jun 10, 2021 5:40 pm**

Our discussion of Gérard's last post has revealed that the question of

filling the ko threat is open with a larger ko ensemble, so let me revisit it when we add a simple gote, U = {u|-u}, u > 0.

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

= -u

Result: +19 - u + t

There are a number of transpositions to get that result.

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

= -u

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

= u

Click Here To Show Diagram Code: [go]$$Wm10 Black first, var. 2 $$ --------------------- $$ | X 1 X O X X O O . | $$ | 3 X O O X X O O O | $$ | X 5 O X X X O O O | $$ | . O O X . X O O O | $$ | . O X X . X O O O | $$ | . X . . . X O O . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]

= u

Result when t > 0: u - 8 + 3t

Alternate results after

.

Click Here To Show Diagram Code: [go]$$Wm10 Black first, var. 2 $$ --------------------- $$ | X 1 X O X X O O . | $$ | 3 X O O X X O O O | $$ | X 5 O X X X O O O | $$ | . O O X . X O O O | $$ | 2 O X X . X O O O | $$ | . X . . . X O O . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]

= u

Click Here To Show Diagram Code: [go]$$Wm10 Black first, var. 2 $$ --------------------- $$ | X 1 X O X X O O . | $$ | 3 X O O X X O O O | $$ | X 5 O X X X O O O | $$ | . O O X . X O O O | $$ | 2 O X X . X O O O | $$ | . X . . . X O O . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]

= u

Result when t > 1: u - 7 + 2t

Click Here To Show Diagram Code: [go]$$Wm10 Black first, var. 3 $$ --------------------- $$ | X 1 X O X X O O . | $$ | 3 X O O X X O O O | $$ | X 5 O X X X O O O | $$ | 4 O O X . X O O O | $$ | 2 O X X . X O O O | $$ | 6 X . . . X O O . | $$ | . X . . . X X O O | $$ | X X . . . . X X X | $$ | . . . . . . . . . | $$ ---------------------[/go]

= u

Result when t > 1: u - 6 + t

OC, the best result when Black takes U and t > 1 is the first.

So it looks like, even with the humungous threat, we want to add 3 simple gote to the ko ensemble, U = {u|-u}, V = {v|-v}, and W = {w|-w}, u ≥ v ≥ w > 1.

Life In 19x19

This 'n' that

Re: This 'n' that

Re: This 'n' that

Re: This 'n' that

Re: This 'n' that

Re: This 'n' that

Re: This 'n' that

Re: This 'n' that

Re: This 'n' that

Re: This 'n' that

Re: This 'n' that

Re: This 'n' that

Re: This 'n' that

Re: This 'n' that

Re: This 'n' that

Re: This 'n' that