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Thermography

https://www.lifein19x19.com/viewtopic.php?t=17788

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Re: Thermography

Posted: Sat Dec 12, 2020 11:38 am
by Bill Spight
Gérard TAILLE wrote:
Bill Spight wrote:
Gérard TAILLE wrote:
Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | . X X a X O . . . |
$$ | O X . X O O . . . |
$$ | . O X . X . . . . |
$$ | . O O X X . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ -----------------[/go]
if Black is komonster I do not understand why this position is like 2*. Why not simply 2 ?
The * is for filling the ko at temperature 0. And I should have written {2|-1*} for the same reason.
That was my interpretation but it is not clear : because * = {0|0} and because black is komonster the resulting position do not allow white to play towards 0. That is why I do not see a "*" here.
Black komonster: Local play at temperature 0.
Click Here To Show Diagram Code
[go]$$Bc Black first
$$ -----------------
$$ | 3 X X 1 X O . . . |
$$ | O X . X O O . . . |
$$ | 2 O X . X . . . . |
$$ | . O O X X . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ -----------------[/go]
Click Here To Show Diagram Code
[go]$$Wc White first
$$ -----------------
$$ | 5 X X 1 B O . . . |
$$ | O X . X O O . . . |
$$ | 3 O X . X . . . . |
$$ | . O O X X . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ -----------------[/go]
:b2: takes ko back, :b4: fills ko

Being komonster at territory scoring means that the winner of the ko can wait until temperature 0 to win it. She cannot leave it unsettled, she can only delay the win.
Gérard TAILLE wrote:Let's take another example:
Click Here To Show Diagram Code
[go]$$Bc
$$ -----------------
$$ | a O X . O . O |
$$ | X X X X O O O |
$$ | O O O O X X X |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------[/go]
Here again black will wait temperature drops to 0 before playing at "a" to gain a prisoner.
But it is not a {0|0}=* because white cannot play.
The Japanese rules require Black to capture the stone at temperature 0. As you point out, the position is worth 1, not 1*.

Under those rules I suppose that we could write this game as {0|}. {shrug}

Re: Thermography

Posted: Sat Dec 12, 2020 12:02 pm
by Gérard TAILLE
Bill Spight wrote:
Gérard TAILLE wrote:
Let's take another example:
Click Here To Show Diagram Code
[go]$$Bc
$$ -----------------
$$ | a O X . O . O |
$$ | X X X X O O O |
$$ | O O O O X X X |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------[/go]
Here again black will wait temperature drops to 0 before playing at "a" to gain a prisoner.
But it is not a {0|0}=* because white cannot play.
The Japanese rules require Black to capture the stone at temperature 0. As you point out, the position is worth 1, not 1*.

Under those rules I suppose that we could write this game as {0|}. {shrug}
BTW Bill, you often show that some game resolution may be made by playing at temperature -1, typically to capture opponent stones. Here in the diagram above , if I play under J89 rules (I mean no points in seki), I know as Go player that I have to capture the black stone at temperature 0. But as CGT guy playing with a tax how I know that I have to capture an opponent stone at temperature 0 rather than at temperature -1 ?

Re: Thermography

Posted: Sat Dec 12, 2020 1:06 pm
by Bill Spight
Gérard TAILLE wrote:
Bill Spight wrote:
Gérard TAILLE wrote:
Let's take another example:
Click Here To Show Diagram Code
[go]$$Bc
$$ -----------------
$$ | a O X . O . O |
$$ | X X X X O O O |
$$ | O O O O X X X |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------[/go]
Here again black will wait temperature drops to 0 before playing at "a" to gain a prisoner.
But it is not a {0|0}=* because white cannot play.
The Japanese rules require Black to capture the stone at temperature 0. As you point out, the position is worth 1, not 1*.

Under those rules I suppose that we could write this game as {0|}. {shrug}
BTW Bill, you often show that some game resolution may be made by playing at temperature -1, typically to capture opponent stones. Here in the diagram above , if I play under J89 rules (I mean no points in seki), I know as Go player that I have to capture the black stone at temperature 0. But as CGT guy playing with a tax how I know that I have to capture an opponent stone at temperature 0 rather than at temperature -1 ?
You don't have to capture the stone at temperature 0 under CGT rules with prisoner return and a group tax. :)

With 1 play to capture the White stone and 1 play to return it, we can write the game this way:

{{{ | }|| }||| } = {{0| }|| } = {1| } = 2

The Black moves are made at temperature -1.

The J89 rules require the capture, if played, to be played at temperature 0.

Re: Thermography

Posted: Tue Dec 22, 2020 9:14 am
by Gérard TAILLE
Click Here To Show Diagram Code
[go]$$B
$$ -------------------------
$$ | X X . O . a X . . . O |
$$ | X X O . O X . X X O O |
$$ | X . X X X O X X O O O |
$$ | X X O O O O O O O . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ -----------------------[/go]
Calculating a thermograph and a miai value when each player see an interest to play in the region is quite logical. But what about a position with a ko in which each player prefers that the other one plays first?
In the above position white can begin a ko by playing at "a" but it is better to wait first for a black move doesn't it?
My question : at which temperature a player will play here (assuming a no ko threat environment). Is it the the same temperature for white and for black?

Same question for the following one in which white has a local ko threat at "b".
Click Here To Show Diagram Code
[go]$$B
$$ -------------------------
$$ | X X . O . a X . . . O |
$$ | X . O b O X . X X O O |
$$ | X . X X X O X X O O O |
$$ | X X O O O O O O O . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ -----------------------[/go]

Re: Thermography

Posted: Tue Dec 22, 2020 10:09 am
by Bill Spight
Gérard TAILLE wrote:
Click Here To Show Diagram Code
[go]$$B
$$ -------------------------
$$ | X X . O . a X . . . O |
$$ | X X O . O X . X X O O |
$$ | X . X X X O X X O O O |
$$ | X X O O O O O O O . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ -----------------------[/go]
Calculating a thermograph and a miai value when each player see an interest to play in the region is quite logical. But what about a position with a ko in which each player prefers that the other one plays first?
In the above position white can begin a ko by playing at "a" but it is better to wait first for a black move doesn't it?
Click Here To Show Diagram Code
[go]$$B
$$ -------------------------
$$ | X X . O . a X . . . O |
$$ | X X O . O X . X X O O |
$$ | X . X X X O X X O O O |
$$ | X B O O O O O O O . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ -----------------------[/go]
There is a lot going on in this position. Are the Black stones on the left immortal? If so, Black a prevents the ko and wins the semeai. If not, the problem is ill defined.

Often the player with fewer stones at stake should start the ko, but OC, each position is different.
Gérard TAILLE wrote:My question : at which temperature a player will play here (assuming a no ko threat environment). Is it the the same temperature for white and for black?
There is a temperature (sometimes more than one) at which each player will be indifferent between making a local play or not, immediately below which at least one player will prefer to make a local play.

Re: Thermography

Posted: Tue Dec 22, 2020 10:44 am
by Gérard TAILLE
Bill Spight wrote: There is a lot going on in this position. Are the Black stones on the left immortal? If so, Black a prevents the ko and wins the semeai. If not, the problem is ill defined.
Yes OC they are immortal. BTW I propose a change in the position in order that black "a" will not win the semeai.
That way I hope the problem is clearer.
Click Here To Show Diagram Code
[go]$$B
$$ -------------------------
$$ | . . . O . a X . . . O |
$$ | X X O . O X . X X O O |
$$ | X . X X X O X X O O . |
$$ | X X O O O O O O O O O |
$$ | . X . . . . . . . O . |
$$ | X X . . . . . . . O O |
$$ | . X . . . . . . . . . |
$$ | X X . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ -----------------------[/go]

Re: Thermography

Posted: Tue Dec 22, 2020 11:07 am
by Bill Spight
I thought they were immortal. :)

I'm not going to have time to analyze these positions until probably next week.

Joyeux Noël!

Re: Thermography

Posted: Tue Dec 22, 2020 11:41 am
by Gérard TAILLE
Bill Spight wrote:I thought they were immortal. :)

I'm not going to have time to analyze these positions until probably next week.

Joyeux Noël!
OK Bill, Joyeux Noël for you and your family. In France it is not easy due to restriction in realtion with the coronavirus. Anyway I can always continue to study Go :) ;-)

Re: Thermography

Posted: Wed Dec 23, 2020 9:13 am
by Gérard TAILLE
Gérard TAILLE wrote:
Bill Spight wrote: There is a lot going on in this position. Are the Black stones on the left immortal? If so, Black a prevents the ko and wins the semeai. If not, the problem is ill defined.
Yes OC they are immortal. BTW I propose a change in the position in order that black "a" will not win the semeai.
That way I hope the problem is clearer.
Click Here To Show Diagram Code
[go]$$W
$$ -------------------------
$$ | . . . O . a X . . . O |
$$ | X X O . O X . X X O O |
$$ | X . X X X O X X O O . |
$$ | X X O O O O O O O O O |
$$ | . X . . . . . . . O . |
$$ | X X . . . . . . . O O |
$$ | . X . . . . . . . . . |
$$ | X X . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ -----------------------[/go]
My own analysis is the following.
First of all because locally black wins the ko and because black has no interest to lose a move in order to provoke the ko we can assume that white will play first at "a". The problem is to find at which temperature white will play at "a".
If the temperature is too high black will not answer to a white play at "a" and the play will continue by:
Click Here To Show Diagram Code
[go]$$W
$$ -------------------------
$$ | . . . O . 1 X 5 . . O |
$$ | X X O . O X 3 X X O O |
$$ | X . X X X O X X O O . |
$$ | X X O O O O O O O O O |
$$ | . X . . . . . . . O . |
$$ | X X . . . . . . . O O |
$$ | . X . . . . . . . . . |
$$ | X X . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ -----------------------[/go]
:b2: tenuki
:b4: tenuki
:b6: tenuki
and a local score -14,5
If now black choose to win the ko then it follows
Click Here To Show Diagram Code
[go]$$W
$$ -------------------------
$$ | . . . O 2 1 X . . . O |
$$ | X X O 4 O X . X X O O |
$$ | X . X X X O X X O O . |
$$ | X X O O O O O O O O O |
$$ | . X . . . . . . . O . |
$$ | X X . . . . . . . O O |
$$ | . X . . . . . . . . . |
$$ | X X . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ -----------------------[/go]
:w3: tenuki
and a local score +13,5
The deiri value of the area is 28 points for a tally equal to 4. The value of a local move is then equal to 7.
I conclude that if temperature is greater than 7 then neither side will play in this area.
What happens if the temperature is less or equal to 7? Black will never play first in the area and white is then the only player who can choose the right timing.
Because the temperature is not greater than 7 then we know that in any case black will choose to win the ko. In that case white will gain in exchange one tenuki. That means that white have to play in the area at the highest temperature which is 7 points.
The strategy of white is then the following:
if temperature is 7 then white begins the ko.
if temperature is less than 7 (say 5 or 6) then white have to look for increasing the temperature up to 7. If white cannot increase the temperature then white have to provoque the ko as soon as possible.

I have now to wait Bill's corrections!

Merry Christmas.

Re: Thermography

Posted: Wed Jan 13, 2021 1:43 pm
by Gérard TAILLE
Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | X b . . . W X X w |
$$ | X W W W W W X O . |
$$ | X X X X X O X . O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
Here is another example difficult for me to analyse.
Assume no ko threat in the environment. White may choose to play "w" in order to use the ko to give life to his marked stone but the questions are the following:
1) at which temperature white will play at "w"?
2) at which temperature black will play at "b"?

Re: Thermography

Posted: Wed Jan 13, 2021 4:43 pm
by Bill Spight
Gérard TAILLE wrote:
Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | X b . . . W X X w |
$$ | X W W W W W X O . |
$$ | X X X X X O X . O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
Here is another example difficult for me to analyse.
Assume no ko threat in the environment. White may choose to play "w" in order to use the ko to give life to his marked stone but the questions are the following:
1) at which temperature white will play at "w"?
2) at which temperature black will play at "b"?
Click Here To Show Diagram Code
[go]$$W Sacrifice
$$ -----------------
$$ | X b . . . W X X 1 |
$$ | X W W W W W X O 2 |
$$ | X X X X X O X 4 O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
:w3: elsewhere

The result after :b4: is 26 - t, where :w3: gains t points.

White can play :w1: when he threatens to win the ko.
Click Here To Show Diagram Code
[go]$$W White wins ko
$$ -----------------
$$ | X b . . . W X X 1 |
$$ | X W W W W W X O . |
$$ | X X X X X O X 3 O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
:b2:, :b4: elsewhere

The result after :b4: is -17⅛ + 2t, on average, counting -2⅛ for the corridor.

White threatens to win the ko when t ≤ 14⅜. So that is when White will throw in to make the ko, as a rule.

Black has no inclination to play at b, because that will reduce Black's potential result by 1 point.

But what happens at the end of the game? By Japanese rules Black must play first to keep this from becoming 0 points. Black will have to play twice to kill the White stones, so the final score will be 24.

That means that, as a rule, White will have no incentive to play elsewhere when doing so gains less than 2 points. So White will normally play the throw-in when 2 ≤ t ≤ 14⅜.

Edit: Hmmm. I may well be wrong about the Japanese rules, as White has no chance to live because Black wins the ko. In which case Black does not have to spend any play to kill White and White will normally make the ko when t ≤ 14⅜,

Re: Thermography

Posted: Thu Jan 14, 2021 5:28 am
by Gérard TAILLE
Bill Spight wrote:
Gérard TAILLE wrote:
Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | X b . . . W X X w |
$$ | X W W W W W X O . |
$$ | X X X X X O X . O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
Here is another example difficult for me to analyse.
Assume no ko threat in the environment. White may choose to play "w" in order to use the ko to give life to his marked stone but the questions are the following:
1) at which temperature white will play at "w"?
2) at which temperature black will play at "b"?
Click Here To Show Diagram Code
[go]$$W Sacrifice
$$ -----------------
$$ | X b . . . W X X 1 |
$$ | X W W W W W X O 2 |
$$ | X X X X X O X 4 O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
:w3: elsewhere

The result after :b4: is 26 - t, where :w3: gains t points.

White can play :w1: when he threatens to win the ko.
Click Here To Show Diagram Code
[go]$$W White wins ko
$$ -----------------
$$ | X b . . . W X X 1 |
$$ | X W W W W W X O . |
$$ | X X X X X O X 3 O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
:b2:, :b4: elsewhere

The result after :b4: is -17⅛ + 2t, on average, counting -2⅛ for the corridor.

White threatens to win the ko when t ≤ 14⅜. So that is when White will throw in to make the ko, as a rule.

Black has no inclination to play at b, because that will reduce Black's potential result by 1 point.
I do not understand why black has not to play at "b" if the temperature is low. Hasn't black to avoid the sequence
Click Here To Show Diagram Code
[go]$$W
$$ -----------------
$$ | X 3 . 5 . O X X 1 |
$$ | X O O O O O X O 2 |
$$ | X X X X X O X 4 O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]

Re: Thermography

Posted: Thu Jan 14, 2021 6:42 am
by Bill Spight
Gérard TAILLE wrote:
Bill Spight wrote:
Gérard TAILLE wrote:
Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | X b . . . W X X w |
$$ | X W W W W W X O . |
$$ | X X X X X O X . O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
Here is another example difficult for me to analyse.
Assume no ko threat in the environment. White may choose to play "w" in order to use the ko to give life to his marked stone but the questions are the following:
1) at which temperature white will play at "w"?
2) at which temperature black will play at "b"?
Click Here To Show Diagram Code
[go]$$W Sacrifice
$$ -----------------
$$ | X b . . . W X X 1 |
$$ | X W W W W W X O 2 |
$$ | X X X X X O X 4 O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
:w3: elsewhere

The result after :b4: is 26 - t, where :w3: gains t points.

White can play :w1: when he threatens to win the ko.
Click Here To Show Diagram Code
[go]$$W White wins ko
$$ -----------------
$$ | X b . . . W X X 1 |
$$ | X W W W W W X O . |
$$ | X X X X X O X 3 O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
:b2:, :b4: elsewhere

The result after :b4: is -17⅛ + 2t, on average, counting -2⅛ for the corridor.

White threatens to win the ko when t ≤ 14⅜. So that is when White will throw in to make the ko, as a rule.

Black has no inclination to play at b, because that will reduce Black's potential result by 1 point.
I do not understand why black has not to play at "b" if the temperature is low. Hasn't black to avoid the sequence
Click Here To Show Diagram Code
[go]$$W
$$ -----------------
$$ | X 3 . 5 . O X X 1 |
$$ | X O O O O O X O 2 |
$$ | X X X X X O X 4 O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
Thanks. :) I replied too quickly. I thought I could just assume that Black was komaster. Given the source, I should have known better. :lol:
Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | X B . . . W X X w |
$$ | X W W W W W X O . |
$$ | X X X X X O X . O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
After :bc: White can play first and lose the ko for a result of 25 - t when t ≤ 13¾.

That means that the left scaffold for the thermograph is 25 - 2t. (Edit: At that temperature range, OC. ;)) I'll go to work on the right scaffold now, considering the local ko threat. :)

Re: Thermography

Posted: Thu Jan 14, 2021 7:01 am
by Gérard TAILLE
Gérard TAILLE wrote: I do not understand why black has not to play at "b" if the temperature is low. Hasn't black to avoid the sequence
Click Here To Show Diagram Code
[go]$$W
$$ -----------------
$$ | X 3 . 5 . O X X 1 |
$$ | X O O O O O X O 2 |
$$ | X X X X X O X 4 O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
Bill Spight wrote: Thanks. :) I replied too quickly. I thought I could just assume that Black was komaster. Given the source, I should have known better. :lol:
yes Bill due to this sequence it is not so obvious to analyse the situation.
OC, as I said, I assume there are no ko threat in the environment but localy it exists one ko threat (not as big as 14⅜ but it exists).
I understand your count : (26 + 17⅛) / 3 = 14⅜ but this count assume that black can kill all white stone by winning the ko but this is not true because white is able to choose to save some of his stones instead of playing in the environment.
Due to this white possibility my feeling is that a white play at "w" is not as big as 14⅜ and here is my difficulty when looking for the temperature at which white has to play at "w".

Re: Thermography

Posted: Thu Jan 14, 2021 7:47 am
by Bill Spight
OK. Let's try again. ;)
Click Here To Show Diagram Code
[go]$$Bc Black first
$$ -----------------
$$ | X 1 . . . W X X 2 |
$$ | X W W W W W X O 3 |
$$ | X X X X X O X 5 O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
:w4:, :w6: elsewhere

Result: 25 - 2t
Click Here To Show Diagram Code
[go]$$Wc White first
$$ -----------------
$$ | X 3 . 5 . W X X 1 |
$$ | X W W W W W X O 2 |
$$ | X X X X X O X 4 O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
:b6: elsewhere

Result: 4 + t

Assuming that these are the sequences of play when t is small enough, we find that t by solving the equation,

25 - 2t = 4 + t

t = 21/3 = 7

And we can verify that when Black plays first, White throws in to make the ko when t ≤ 13¾ , and when White plays first and Black takes the ko, White plays the ko threat when t ≤ 11. So these are the normal sequences of play when t ≤ 7. :)

The mast value of the position is 11.

If I haven't goofed, that is. :lol:

Edit: We need to check the case when 26 - t < 11. I.e., when t > 15. White does not threaten to make and win the ko when t > 14⅜, so we're OK. :)
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