Thermography

Bill Spight · Post by **Bill Spight** » Tue Sep 29, 2020 10:19 am

Gérard TAILLE wrote:OK, Bill let me try a difference game:

$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . . X a . . O | X . . O . . . |
$$ | X X . b . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima $$ -------------------------------- $$ | . . X a . . O | X . . O . . . | $$ | X X . b . . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O O . | . X X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]
If I would try to prove that kosumi does not dominate the keima I have to find a way to win the above diference game with white, right?

I see clearly that I cannot win by beginning at "b" => no choice for my first white move:

$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . . X 1 . . O | X . . O . . . |
$$ | X X a b . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima $$ -------------------------------- $$ | . . X 1 . . O | X . . O . . . | $$ | X X a b . . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O O . | . X X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]
Now it is black turn. If black plays at "a" white plays at "b" and win => black must ansmer at "a" and the following sequence seems mandatory

$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . . X 1 4 . O | X . 6 O 7 . . |
$$ | X X 3 2 5 . O | X . . 8 . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima $$ -------------------------------- $$ | . . X 1 4 . O | X . 6 O 7 . . | $$ | X X 3 2 5 . O | X . . 8 . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O O . | . X X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]
then white can simply continue by
$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . 4 X 5 X 1 O | X . X O O 3 . |
$$ | X X O X O . O | X . . X 2 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima $$ -------------------------------- $$ | . 4 X 5 X 1 O | X . X O O 3 . | $$ | X X O X O . O | X . . X 2 O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O O . | . X X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]
winning the game.
Isn't it correct?

Very good.

A couple of notes.

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

does not threaten to kill.

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

With no dame

makes ko. If

kills Black on the right with a,

fills the ko on the left and wins by 1 pt. for the captured White stone. So White must fight the ko.

As this ko is essential to the difference game, difference games are not appropriate to this comparison.

OC, if there is no damezumari, White can win without a ko fight, and the two plays are incomparable.

Click Here To Show Diagram Code: [go]$$Wc Kosumi vs. keima $$ -------------------------------- $$ | . . X 1 4 6 O | X . 8 O 9 . . | $$ | X X 3 2 5 7 O | X . . 0 . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

Gérard TAILLE · Post by **Gérard TAILLE** » Tue Sep 29, 2020 1:46 pm

Bill Spight wrote:Well, I got carried away with the difference games, eh?

Constructing the thermograph

Truth to say, difference games are not necessary to thermography at all, and are hardly ever used. However, it was easy to show that the keima sente and the crawl sente are unnecessary. The kosumi sente dominates them. As I said, if they were gote, that would be a different matter. But anyway, all we have to consider are the kosumi sente and the large monkey jump sente. Let's review the reverse sente first.

$$Wc Reverse sente, Black continues
$$ -----------------
$$ | C 4 2 3 C C O |
$$ | X X 1 C C C O |
$$ | . X O O O O O |
$$ | . X X X O O C |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Wc Reverse sente, Black continues $$ ----------------- $$ | C 4 2 3 C C O | $$ | X X 1 C C C O | $$ | . X O O O O O | $$ | . X X X O O C | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ -----------------[/go]
blocks on the second line. Play can, and usually does, stop there. If Black replies she plays the hane-and-connect for a local score of -5 in the marked region. Each player has made the same number of plays, so the Right wall of the thermograph rises as a vertical line at that point.

Black does not reply above temperature 1, so the wall turns 45° to the left at that temperature rising one degree for each point of territory. Remember, the leftward movement is positive, the reverse the usual convention. The equation of that line is v = -6 + t, where v is the territory and t is the temperature.

Now let's look at the kosumi sente.

$$Bc Kosumi sente, Black continues
$$ -----------------
$$ | . . 1 3 4 . O |
$$ | X X 5 2 . . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Bc Kosumi sente, Black continues $$ ----------------- $$ | . . 1 3 4 . O | $$ | X X 5 2 . . O | $$ | . X O O O O O | $$ | . X X X O O . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ -----------------[/go]
The sente actually ends with but we consider what happens when Black keeps playing. The score after is -2. Black has made one more move than White, so the line from here angles up and to the right. It's equation is v = -2 - t. How far up does it go?

$$Wc Kosumi sente, White follow-up
$$ -----------------
$$ | . . B 1 . . O |
$$ | X X 2 W . . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Wc Kosumi sente, White follow-up $$ ----------------- $$ | . . B 1 . . O | $$ | X X 2 W . . O | $$ | . X O O O O O | $$ | . X X X O O . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ -----------------[/go]
If I didn't already know, it would be easy to work out that this is correct play at temperature 1. White could get an equivalent result, except for ko threats, by starting at 2. Each player has made the same number of plays, including the kosumi sente exchange, so the line rises vertically at -3. These two lines intersect at v = -3, t = 1. Above temperature 1 the prospective wall, called a scaffold, rises vertically at -3, below temperature 1 it angles down to -2 at temperature 0.

Now let's look at the monkey jump.

$$Bc Monkey jump sente 1
$$ -----------------
$$ | . . 3 4 1 . O |
$$ | X X 2 5 6 . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Bc Monkey jump sente 1 $$ ----------------- $$ | . . 3 4 1 . O | $$ | X X 2 5 6 . O | $$ | . X O O O O O | $$ | . X X X O O . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ -----------------[/go]
This is one of the main lines, with the throw-in, . White could also get here by starting at 4. That's the other main line. The usual reply is at 6. We'll consider that later.

What if Black continues?

$$Bc Monkey jump sente 1, Black continues
$$ -----------------
$$ | . 9 B 7 B 8 O |
$$ | X X W B W . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Bc Monkey jump sente 1, Black continues $$ ----------------- $$ | . 9 B 7 B 8 O | $$ | X X W B W . O | $$ | . X O O O O O | $$ | . X X X O O . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ -----------------[/go]

1 prisoner
connects the Black stones, makes a second eye, and saves the four stones. The local score is 0. Since Black has made one more move than White, the scaffold angles up from 0 to the right at one point of territory for each degree of temperature. The equation of the line is v = -t.

How high does it go? To find out, let's look at the White follow-up after the sente.

$$Wc Monkey jump sente 1, White follow-up.
$$ -----------------
$$ | . 2 B 3 B 1 O |
$$ | X X W B W . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Wc Monkey jump sente 1, White follow-up. $$ ----------------- $$ | . 2 B 3 B 1 O | $$ | X X W B W . O | $$ | . X O O O O O | $$ | . X X X O O . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ -----------------[/go]

1 prisoner

There is a ko, which White can take and win on the usual assumption of no ko threats. But comes to the same thing. After the local score is -4. Since White has made one more move than Black, the scaffold angles up from there to the left. It's equation is v = -4 + t. Note that this is the scaffold for the position after the sente, not the scaffold for the original position. The scaffolds intersect at t = 2, v = -2. The mast rises vertically from there. The scaffold for this sente is v = -2 above temperature 2 and v = -4 + t below temperature 2.

There is another variation to consider.

$$Bc Monkey jump sente 1a
$$ -----------------
$$ | . . 3 5 1 6 O |
$$ | X X 2 . 4 . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Bc Monkey jump sente 1a $$ ----------------- $$ | . . 3 5 1 6 O | $$ | X X 2 . 4 . O | $$ | . X O O O O O | $$ | . X X X O O . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ -----------------[/go]
does not play the throw-in, but it threatens it. prevents that, with sente. As we have seen, with a different order of play, this line is incomparable with the throw-in. After the local score is -1. The scaffold rises vertically from there. OC, that is worse for White than -2, so this line is not as good as the throw-in above temperature 2. But it is better for White than 0, so we find the intersection v = -t and v = -1 to find where White switches to this line of play. The intersection is at t = 1, v = -1. So for the reply of the scaffold is vertical at -2 above temperature 2, angling down below temperature 2 to -1 at temperature 1, and vertical below that to -1 at temperature 0.

BTW, what if Black does not connect with in the previous diagram, but tries to kill?

$$Bcm5 Monkey jump sente 1a, variation
$$ -----------------
$$ | . . B 2 B 1 O |
$$ | X X W 3 W a O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Bcm5 Monkey jump sente 1a, variation $$ ----------------- $$ | . . B 2 B 1 O | $$ | X X W 3 W a O | $$ | . X O O O O O | $$ | . X X X O O . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ -----------------[/go]
Now White cannot play at a because of damezumari. Oh, for a dame!

$$Wcm8 Monkey jump sente 1a, variation
$$ -----------------
$$ | 2 1 B 3 B B O |
$$ | X X W B W 5 O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Wcm8 Monkey jump sente 1a, variation $$ ----------------- $$ | 2 1 B 3 B B O | $$ | X X W B W 5 O | $$ | . X O O O O O | $$ | . X X X O O . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ -----------------[/go]
at 8
White has no external threats, by assumption, but the play has generated a local threat for White at . With a dame White would not have needed to play the ko.

More later.

Here under the thermograph corresponding to basic sente sequences identified by Bill's post above.

My conclusion is that, under ideal environment, monkey jump is better than kosumi. BTW it was clearly identified when Bill, using difference games, shows that monkey jump is most of the time better than kosumi

Gérard TAILLE · Post by **Gérard TAILLE** » Tue Sep 29, 2020 2:12 pm

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

In the position above keima is strictly better than both kosumi and monkey jump.

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

In this one monkey jump is strictly better than both kosumi and keima.

My question is now the following : does it exist a non-ko environment for which kosumi is strictly better than both keima and monkey jump?

Bill Spight · Post by **Bill Spight** » Tue Sep 29, 2020 2:31 pm

Gérard TAILLE wrote:With your last post I believe I have found where is the misunderstanding.

The point is quite subtil: considering the environment we both exclude pure ko threats like for example:
$$B
$$ -----------------
$$ | X X X O . . . |
$$ | . a X O . . . |
$$ | X X O O . . . |
$$ | . X X O . . . |
$$ | X X O O . . . |
$$ | O O O . . . . |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$B $$ ----------------- $$ | X X X O . . . | $$ | . a X O . . . | $$ | X X O O . . . | $$ | . X X O . . . | $$ | X X O O . . . | $$ | O O O . . . . | $$ | . . . . . . . | $$ -----------------[/go]

Actually, the existence of a ko threat in an environment is not a problem, per se. The basic question is this.

$$B Zero?
$$ ------------------------------
$$ | X X X O . . . |. . . X O O O |
$$ | . a X O . . . |. . . X O . . |
$$ | X X O O . . . |. . . X X O O |
$$ | . X X O . . . |. . . X O O . |
$$ | X X O O . . . |. . . X X O O |
$$ | O O O O . . . |. . . X X X X |
$$ | . O . O . . . |. . . X . X . |
$$ -------------------------------
Click Here To Show Diagram Code
[go]$$B Zero? $$ ------------------------------ $$ | X X X O . . . |. . . X O O O | $$ | . a X O . . . |. . . X O . . | $$ | X X O O . . . |. . . X X O O | $$ | . X X O . . . |. . . X O O . | $$ | X X O O . . . |. . . X X O O | $$ | O O O O . . . |. . . X X X X | $$ | . O . O . . . |. . . X . X . | $$ -------------------------------[/go]
Does the environment and its mirror add to zero? The
existence of ko threats in the environment is not enough for that not to be so. To have a ko (or superko) fight you need a ko.

The logic of difference games has been proven only for non-ko positions and environments. However, I think that they also have a practical value if a ko fight is exceptional. For instance, this board as an environment looks OK to me, but in a difference game, while the players alternate plays, plays do not have to alternate on either board or independent region. The logic of adding to zero is that of playing mirror go, alternating between boards or regions. It would be possible, for instance, for Black to play on the left while White answered on the right, and thus for Black to create a ko on the left, thus invalidating the board on the left as an environment for difference games. While that is a theoretical possibility, it does not keep me up at night.

But you seem also to exclude from the environment "normal" sente move like for example:

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

Not so. See above. Such a ko threat is not a problem in an environment, per se. But to avoid confusion if there is an exceptional ko, best not to include one.

In fact, best to consider environments for difference games as no man's land, where plays are verboten. If a play in an environment matters, it should be part of the position or combination of positions under consideration.

This holds true for an ideal environment of thermography, as well, except for plays at its temperature. Its only significant features should be its temperature, and the fact that the temperature eventually drops in steps so small that they do not matter.

Because in my mind I accept such environment which allows to use a sente move as a ko threat our conclusions are logically different

Here is an example for the discussion kosumi vs keima

$$B
$$ -----------------
$$ | . . . . . . O |
$$ | X X . . . . O |
$$ | . X O O O O O |
$$ | . X X X X O . |
$$ | . X X b a O O |
$$ | X X . X X X O |
$$ | . X . O O . O |
$$ -----------------
Click Here To Show Diagram Code
[go]$$B $$ ----------------- $$ | . . . . . . O | $$ | X X . . . . O | $$ | . X O O O O O | $$ | . X X X X O . | $$ | . X X b a O O | $$ | X X . X X X O | $$ | . X . O O . O | $$ -----------------[/go]
You can see that after a white move at "a", white may create a sente move at "b" which can be use as a ko threat. That is the point.

These different regions are not strictly independent, because they all involve the same string of stones which is not immortal. That may well be of little practical significance, and we may be able to treat them as independent. On a larger board it would be easy to make them independent, as long as there is no ko fight. Kos destroy independence.

It seems to me that the main difference between us is that you want to treat the environment as foreground, and I want to treat it as background.

Gérard TAILLE · Post by **Gérard TAILLE** » Wed Sep 30, 2020 11:31 am

Difference games looks now for me like a magic tool to compare the different moves.
As a consequence it seems also be an help to build thermographs.

Consider the following position

diag.1

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

And my question is how to build the thermograph of this position ?

BTW, I am not really interested by the best way to build this thermograph but I would like to know if my way of reasoning is correct!

First of all I use a difference game in order to find the best move

Black to play:

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

Firstly of all I prove that it exist an environment for which "a" if better than "b":

Black to play

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

and black wins

Secondly I prove that, white to move, white cannot win:

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

and white cannot win.

That proves black "a" is the correct move.

Now white to play:

White to play

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

Firstly of all I prove that it exist an environment for which "b" if better than "a":

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

and white wins

Secondly I prove that, black to move, black cannot win:

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

and black cannot win

That proves white "b" is the correct move.

Where are we ?

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

In the diagramm above:
if it is black to blay the sequence begin by

at "a",

at "b"
if it is white to blay the sequence begin by

at "b",

at "a"

Can I deduce the thermogrph, using this result?

diag 2

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

The thermograph of this diag2 is not difficult to draw and my point is to prove that diag1 has exactly the same than diag2 !

Here is my reasonning:
The miai value for diag2 is equal to 0.75
In diag1, as soon as the temperature drops to 0.75, then in one hand

at "a" in diag1 force the immediat answer by white

at "b", and in the other hand

at "b" in diag1 force the immediat answer by black

at "a".
My conclusion is that miai value if diag1 is also 0.75 and diag1 is completly equivalent to diag2.

Is this correct?

Bill Spight · Post by **Bill Spight** » Wed Sep 30, 2020 2:24 pm

Gérard TAILLE wrote:Difference games looks now for me like a magic tool to compare the different moves.
As a consequence it seems also be an help to build thermographs.

Consider the following position

diag.1
$$B
$$ ---------------
$$ | X O O O . . .
$$ | X . . O . . .
$$ | X . . O . . .
$$ | X O . O . . .
$$ | X O O O . . .
$$ | X X X X . . .
$$ | . . . . . . .
$$ | . . . . . . .
Click Here To Show Diagram Code
[go]$$B $$ --------------- $$ | X O O O . . . $$ | X . . O . . . $$ | X . . O . . . $$ | X O . O . . . $$ | X O O O . . . $$ | X X X X . . . $$ | . . . . . . . $$ | . . . . . . .[/go]
And my question is how to build the thermograph of this position ?

BTW, I am not really interested by the best way to build this thermograph but I would like to know if my way of reasoning is correct!

First of all I use a difference game in order to find the best move

Not necessary for deriving the thermograph, but informative.

Black to play:

$$B
$$ --------------------------------
$$ | X O O O . . . - . . . X X X O |
$$ | X a . O . . . - . . . X . . O |
$$ | X b . O . . . - . . . X . . O |
$$ | X O . O . . . - . . . X . X O |
$$ | X O O O . . . - . . . X X X O |
$$ | X X X X . . . - . . . O O O O |
$$ | . . . . . . . - . . . . . . . |
$$ | . . . . . . . - . . . . . . . |
Click Here To Show Diagram Code
[go]$$B $$ -------------------------------- $$ | X O O O . . . - . . . X X X O | $$ | X a . O . . . - . . . X . . O | $$ | X b . O . . . - . . . X . . O | $$ | X O . O . . . - . . . X . X O | $$ | X O O O . . . - . . . X X X O | $$ | X X X X . . . - . . . O O O O | $$ | . . . . . . . - . . . . . . . | $$ | . . . . . . . - . . . . . . . |[/go]
Firstly of all I prove that it exist an environment for which "a" if better than "b":

Black to play
$$B
$$ --------------------------------
$$ | X O O O . . . - . . . X X X O |
$$ | X 1 5 O . . . - . . . X . . O |
$$ | X 4 . O . . . - . . . X 3 2 O |
$$ | X O . O . . . - . . . X . X O |
$$ | X O O O . . . - . . . X X X O |
$$ | X X X X . . . - . . . O O O O |
$$ | . . . . . . . - . . . . . . . |
$$ | . . . . . . . - . . . . . . . |
Click Here To Show Diagram Code
[go]$$B $$ -------------------------------- $$ | X O O O . . . - . . . X X X O | $$ | X 1 5 O . . . - . . . X . . O | $$ | X 4 . O . . . - . . . X 3 2 O | $$ | X O . O . . . - . . . X . X O | $$ | X O O O . . . - . . . X X X O | $$ | X X X X . . . - . . . O O O O | $$ | . . . . . . . - . . . . . . . | $$ | . . . . . . . - . . . . . . . |[/go]
and black wins

After

right and left are miai, for 1 pt. for Black.

Secondly I prove that, white to move, white cannot win:

$$W
$$ --------------------------------
$$ | X O O O . . . - . . . X X X O |
$$ | X 2 4 O . . . - . . . X . . O |
$$ | X 3 . O . . . - . . . X . 1 O |
$$ | X O . O . . . - . . . X . X O |
$$ | X O O O . . . - . . . X X X O |
$$ | X X X X . . . - . . . O O O O |
$$ | . . . . . . . - . . . . . . . |
$$ | . . . . . . . - . . . . . . . |
Click Here To Show Diagram Code
[go]$$W $$ -------------------------------- $$ | X O O O . . . - . . . X X X O | $$ | X 2 4 O . . . - . . . X . . O | $$ | X 3 . O . . . - . . . X . 1 O | $$ | X O . O . . . - . . . X . X O | $$ | X O O O . . . - . . . X X X O | $$ | X X X X . . . - . . . O O O O | $$ | . . . . . . . - . . . . . . . | $$ | . . . . . . . - . . . . . . . |[/go]
and white cannot win.

That proves black "a" is the correct move.

Now white to play:

White to play
$$W
$$ --------------------------------
$$ | X O O O . . . - . . . X X X O |
$$ | X a . O . . . - . . . X . . O |
$$ | X b . O . . . - . . . X . . O |
$$ | X O . O . . . - . . . X . X O |
$$ | X O O O . . . - . . . X X X O |
$$ | X X X X . . . - . . . O O O O |
$$ | . . . . . . . - . . . . . . . |
$$ | . . . . . . . - . . . . . . . |
Click Here To Show Diagram Code
[go]$$W $$ -------------------------------- $$ | X O O O . . . - . . . X X X O | $$ | X a . O . . . - . . . X . . O | $$ | X b . O . . . - . . . X . . O | $$ | X O . O . . . - . . . X . X O | $$ | X O O O . . . - . . . X X X O | $$ | X X X X . . . - . . . O O O O | $$ | . . . . . . . - . . . . . . . | $$ | . . . . . . . - . . . . . . . |[/go]
Firstly of all I prove that it exist an environment for which "b" if better than "a":

$$W
$$ --------------------------------
$$ | X O O O . . . - . . . X X X O |
$$ | X 4 6 O . . . - . . . X . 2 O |
$$ | X 1 7 O . . . - . . . X 5 3 O |
$$ | X O . O . . . - . . . X . X O |
$$ | X O O O . . . - . . . X X X O |
$$ | X X X X . . . - . . . O O O O |
$$ | . . . . . . . - . . . . . . . |
$$ | . . . . . . . - . . . . . . . |
Click Here To Show Diagram Code
[go]$$W $$ -------------------------------- $$ | X O O O . . . - . . . X X X O | $$ | X 4 6 O . . . - . . . X . 2 O | $$ | X 1 7 O . . . - . . . X 5 3 O | $$ | X O . O . . . - . . . X . X O | $$ | X O O O . . . - . . . X X X O | $$ | X X X X . . . - . . . O O O O | $$ | . . . . . . . - . . . . . . . | $$ | . . . . . . . - . . . . . . . |[/go]
and white wins

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

I'm sure you saw this. Just for completeness.

Secondly I prove that, black to move, black cannot win:

$$B
$$ --------------------------------
$$ | X O O O . . . - . . . X X X O |
$$ | X 3 . O . . . - . . . X . 1 O |
$$ | X 2 . O . . . - . . . X . 4 O |
$$ | X O . O . . . - . . . X . X O |
$$ | X O O O . . . - . . . X X X O |
$$ | X X X X . . . - . . . O O O O |
$$ | . . . . . . . - . . . . . . . |
$$ | . . . . . . . - . . . . . . . |
Click Here To Show Diagram Code
[go]$$B $$ -------------------------------- $$ | X O O O . . . - . . . X X X O | $$ | X 3 . O . . . - . . . X . 1 O | $$ | X 2 . O . . . - . . . X . 4 O | $$ | X O . O . . . - . . . X . X O | $$ | X O O O . . . - . . . X X X O | $$ | X X X X . . . - . . . O O O O | $$ | . . . . . . . - . . . . . . . | $$ | . . . . . . . - . . . . . . . |[/go]
and black cannot win

That proves white "b" is the correct move.

Very good.

Where are we ?

$$B
$$ ---------------
$$ | X O O O . . .
$$ | X a . O . . .
$$ | X b . O . . .
$$ | X O . O . . .
$$ | X O O O . . .
$$ | X X X X . . .
$$ | . . . . . . .
$$ | . . . . . . .
Click Here To Show Diagram Code
[go]$$B $$ --------------- $$ | X O O O . . . $$ | X a . O . . . $$ | X b . O . . . $$ | X O . O . . . $$ | X O O O . . . $$ | X X X X . . . $$ | . . . . . . . $$ | . . . . . . .[/go]
In the diagramm above:
if it is black to blay the sequence begin by at "a", at "b"
if it is white to blay the sequence begin by at "b", at "a"

Can I deduce the thermogrph, using this result?

Strictly speaking, no. We know that a is Black's best move, but we have not shown that b is White's best response. OC, it obviously is, and we can prove that (with the ko fight caveat).

And, OC, after

at b, a is Black's only response.

diag 2
$$B
$$ ---------------
$$ | X O O O . . .
$$ | X X . O . . .
$$ | X O . O . . .
$$ | X O . O . . .
$$ | X O O O . . .
$$ | X X X X . . .
$$ | . . . . . . .
$$ | . . . . . . .
Click Here To Show Diagram Code
[go]$$B $$ --------------- $$ | X O O O . . . $$ | X X . O . . . $$ | X O . O . . . $$ | X O . O . . . $$ | X O O O . . . $$ | X X X X . . . $$ | . . . . . . . $$ | . . . . . . .[/go]
The thermograph of this diag2 is not difficult to draw and my point is to prove that diag1 has exactly the same than diag2 !

Here is my reasonning:
The miai value for diag2 is equal to 0.75
In diag1, as soon as the temperature drops to 0.75, then in one hand at "a" in diag1 force the immediat answer by white at "b", and in the other hand at "b" in diag1 force the immediat answer by black at "a".
My conclusion is that miai value if diag1 is also 0.75 and diag1 is completly equivalent to diag2.

Is this correct?

If White b is the answer to Black a, and vice versa then they are miai and, indeed, the thermograph for diagram 1 is the same as the thermograph for diagram 2.

That's an excellent insight.

And, since there are no kos in this position, we can go further. The two positions are equivalent, subject to the ko fight warning.

We can show that with a difference game. Difference games are not just about comparing plays. We can also compare positions.

Click Here To Show Diagram Code: [go]$$Bc Black to play cannot win $$ -------------------------------- $$ | X O O O . . . - . . . X X X O | $$ | X B . O . . . - . . . X . 2 O | $$ | X W . O . . . - . . . X . 1 O | $$ | X O . O . . . - . . . X . X O | $$ | X O O O . . . - . . . X X X O | $$ | X X X X . . . - . . . O O O O | $$ | . . . . . . . - . . . . . . . | $$ | . . . . . . . - . . . . . . . |[/go]

If

,

makes a mirror position, for jigo.

Click Here To Show Diagram Code: [go]$$Bc Black to play cannot win $$ -------------------------------- $$ | X O O O . . . - . . . X X X O | $$ | X B 1 O . . . - . . . X 4 2 O | $$ | X W . O . . . - . . . X . 3 O | $$ | X O . O . . . - . . . X . X O | $$ | X O O O . . . - . . . X X X O | $$ | X X X X . . . - . . . O O O O | $$ | . . . . . . . - . . . . . . . | $$ | . . . . . . . - . . . . . . . |[/go]

makes a mirror position, as well.

Click Here To Show Diagram Code: [go]$$Bc Black to play cannot win $$ -------------------------------- $$ | X O O O . . . - . . . X X X O | $$ | X B 1 O . . . - . . . X . 2 O | $$ | X W 4 O . . . - . . . X 3 . O | $$ | X O . O . . . - . . . X . X O | $$ | X O O O . . . - . . . X X X O | $$ | X X X X . . . - . . . O O O O | $$ | . . . . . . . - . . . . . . . | $$ | . . . . . . . - . . . . . . . |[/go]

White makes jigo this way.

Click Here To Show Diagram Code: [go]$$Bc Black to play cannot win $$ -------------------------------- $$ | X O O O . . . - . . . X X X O | $$ | X B 1 O . . . - . . . X . 2 O | $$ | X W 3 O . . . - . . . X 4 . O | $$ | X O . O . . . - . . . X . X O | $$ | X O O O . . . - . . . X X X O | $$ | X X X X . . . - . . . O O O O | $$ | . . . . . . . - . . . . . . . | $$ | . . . . . . . - . . . . . . . |[/go]

And, finally, another jigo.

White to play cannot win, either.

Click Here To Show Diagram Code: [go]$$Wc White to play cannot win $$ -------------------------------- $$ | X O O O . . . - . . . X X X O | $$ | X B 1 O . . . - . . . X 4 3 O | $$ | X W . O . . . - . . . X . 2 O | $$ | X O . O . . . - . . . X . X O | $$ | X O O O . . . - . . . X X X O | $$ | X X X X . . . - . . . O O O O | $$ | . . . . . . . - . . . . . . . | $$ | . . . . . . . - . . . . . . . |[/go]

Black makes jigo.

Click Here To Show Diagram Code: [go]$$Wc White to play cannot win $$ -------------------------------- $$ | X O O O . . . - . . . X X X O | $$ | X B . O . . . - . . . X . 1 O | $$ | X W . O . . . - . . . X . 2 O | $$ | X O . O . . . - . . . X . X O | $$ | X O O O . . . - . . . X X X O | $$ | X X X X . . . - . . . O O O O | $$ | . . . . . . . - . . . . . . . | $$ | . . . . . . . - . . . . . . . |[/go]

makes a mirror position, for jigo.

So Diagram 1 and Diagram 2 are equal, with the ko fight caveat. (White b in Diagram 1 is a possible ko threat.)

Recognizing such miai positions can make the endgame easier in actual games.

We may regard each of the following sequences of play as a unit, just as we do the hane-and-connect.

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

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

Either player has a as a follow-up, to gain ½ pt.

Roughly, the thermograph looks like this.

Code: Select all

               | t = ¾
              / \ t = ½
             |   \
             |    \ t = 0
          ------------
            -1 -1¼ -2

As you say, it's easy to draw, but not everybody knows it.

Gérard TAILLE · Post by **Gérard TAILLE** » Thu Oct 01, 2020 7:44 am

To avoid damezumari, the best is to take your suggestion with the following difference game:

Click Here To Show Diagram Code: [go]$$B Kosumi vs. keima + monkey jump $$ -------------------------------- $$ | . . a . . . O | X . b c . . . | $$ | X X . . . . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

In order to have a still better understanding of difference games I tried to go further with the challenge of proving that kosumi is dominated by the couple (monkey jump + keima).

In other words the challenge is to prove that is does not exist an environment for which kosumi is strictly better than both keima and monkey jump.

Click Here To Show Diagram Code: [go]$$W Kosumi vs. keima + monkey jump $$ ----------------- $$ | . . X 1 . . O | $$ | X X 2 . . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]

The first part of the proof is to consider all the environments in which

is (one of) the best move in answer to

For such environment the kosumi cannot dominate the keima because of the following difference game showing black cannot win:

Click Here To Show Diagram Code: [go]$$B Kosumi vs. keima + monkey jump $$ -------------------------------- $$ | . . B 4 . . O | X . 1 W 2 8 . | $$ | X X 5 6 . . O | X . . 3 7 O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

Let's now take all the other environments.

Click Here To Show Diagram Code: [go]$$W Kosumi vs. keima + monkey jump $$ ----------------- $$ | . . X 1 . . O | $$ | X X a 2 . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]

For these remaining environments remenber that, if white plays at

black MUST answer at

because a play "a" cannot be the best move

For these remaining environments white chooses the monkey jump and the challenge is again to prove black cannot win

Click Here To Show Diagram Code: [go]$$B Kosumi vs. keima + monkey jump $$ -------------------------------- $$ | . . B . . . O | X . W a . . . | $$ | X X . . . . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

In order to win black must play on the right diagram but what sequence she must choose?
Surprisingly the answer is quite simple because black MUST avoid to play at "a" during the sente sequence as the example here after shows clearly:

Click Here To Show Diagram Code: [go]$$B Kosumi vs. keima + monkey jump $$ -------------------------------- $$ | . . B 6 . . O | X . W 3 2 . . | $$ | X X a b . . O | X . 1 4 5 O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

and now because a black move at "a" is not allowed, black must play at "b" with an obvious jigo : with the two stones

and

we easily reach a mirror position.

As a consequence the only possibility for black is to play

Click Here To Show Diagram Code: [go]$$B Kosumi vs. keima + monkey jump $$ -------------------------------- $$ | . . B . . . O | X 3 W . 2 . . | $$ | X X . . . . O | X . 1 . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

and here again black cannot win because of the following answer:

Click Here To Show Diagram Code: [go]$$W Kosumi vs. keima + monkey jump $$ -------------------------------- $$ | . . B 4 5 . O | X B W 2 W . . | $$ | X X 6 1 . . O | X . B 7 3 O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

My conclusion:
Though kosumi and keima are incomparable, and kosumi and monkey jump are incomparable, it is possible to prove that the couple (keima + monkey jump) dominates kosumi.

Can you tell me if the reasonning is correct Bill ?

Of course this proof does not mean that we can forget the kosumi move. It proves only that the kosumi move may be the best move ONLY for ko fight reasons which looks a quite interesting information isn'it ?

Bill Spight · Post by **Bill Spight** » Thu Oct 01, 2020 11:22 am

Gérard TAILLE wrote:To avoid damezumari, the best is to take your suggestion with the following difference game:

$$B Kosumi vs. keima + monkey jump
$$ --------------------------------
$$ | . . a . . . O | X . b c . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$B Kosumi vs. keima + monkey jump $$ -------------------------------- $$ | . . a . . . O | X . b c . . . | $$ | X X . . . . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]
In order to have a still better understanding of difference games I tried to go further with the challenge of proving that kosumi is dominated by the couple (monkey jump + keima).

In other words the challenge is to prove that is does not exist an environment for which kosumi is strictly better than both keima and monkey jump.

Not sure what you mean by that.

$$W Kosumi vs. keima + monkey jump
$$ -----------------
$$ | . . X 1 . . O |
$$ | X X 2 . . . O |
$$ | . X O O O O O |
$$ | . X X X O . . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ ----------------
Click Here To Show Diagram Code
[go]$$W Kosumi vs. keima + monkey jump $$ ----------------- $$ | . . X 1 . . O | $$ | X X 2 . . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]
The first part of the proof is to consider all the environments in which is (one of) the best move in answer to

Ah! The raises the question. Is the position after

is at least as good for White as the original? IOW, can Black to play win this difference game? If not, then the left side is at least as good for White as the right side.

Click Here To Show Diagram Code: [go]$$Bc Can Black to play win? $$ -------------------------------- $$ | . . B W . . O | X . . . . . . | $$ | X X . . . . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

Click Here To Show Diagram Code: [go]$$Bc White makes jigo $$ -------------------------------- $$ | . 3 B W . . O | X . . 5 4 6 . | $$ | X X 2 . . . O | X . . . 1 O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

The reverse sente,

, does not win. Black has nothing better after

Click Here To Show Diagram Code: [go]$$Bc Capture :wc: $$ -------------------------------- $$ | . . B W 3 . O | X 9 4 8 6 . . | $$ | X X 2 1 0 . O | X . 7 . 5 O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

and

transpose to a position starting with the large monkey jump. To avoid a mirror position, after

and later, Black avoids playing at 8.

holds White to 1 point in the corner.

Click Here To Show Diagram Code: [go]$$Bcm11 Black wins $$ -------------------------------- $$ | . 3 X 1 X 2 O | X X O O O . . | $$ | X X O X O . O | X . X . X O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

1

prisoner

connects to win by 1 point. The captured stone makes the difference.

Instead of playing the large monkey jump at

, let White play the small monkey jump (keima).

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

Click Here To Show Diagram Code: [go]$$Wcm8 Black wins $$ -------------------------------- $$ | . 4 X 2 X 3 O | X . X O O . . | $$ | X X O X 1 . O | X . . X 5 O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

The keima does not take away enough territory. Black wins by 2 points.

So in the original zero position for the difference game, if

is the kosumi, the jump attachment for

lets Black win by capturing the

stone. That does not mean that if White makes the jump attachment in later play that Black is forced to capture the stone. If we consider the original position on the right to be an environment for the position on the left after

, that does not mean the correct play with that environment is the same as correct play with a different environment.

Bill Spight · Post by **Bill Spight** » Thu Oct 01, 2020 12:34 pm

BTW, for the position after

plays the kosumi and

plays the jump attachment, the solid connection first wins the difference game against the atari.

Difference game setup

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

Click Here To Show Diagram Code: [go]$$Bc Connection first beats the atari $$ -------------------------------- $$ | . . B W . . O | X 7 4 B W 8 . | $$ | X X 1 6 . . O | X . 5 2 3 O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

captures two stones and wins by one point.

Click Here To Show Diagram Code: [go]$$Wcm6 Variation for :w6: $$ -------------------------------- $$ | . . B W 6 7 O | X 2 O 1 W 3 . | $$ | X X X 4 5 . O | X . X O X O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

1

prisoner

connects to win by one point.

Gérard TAILLE · Post by **Gérard TAILLE** » Thu Oct 01, 2020 1:05 pm

It seems you do not really take my point because ... I agree 100% with your last post !

Bill Spight wrote:
Gérard TAILLE wrote: In other words the challenge is to prove that is does not exist an environment for which kosumi is strictly better than both keima and monkey jump.
Not sure what you mean by that.

OK let me try to clarify the challenge.

In my previous post viewtopic.php?p=260193#p260193 I showed two positions: one for which the only correct move is the keima move, and the other for which the only correct move is the monkey jump move.

After having failed to find a position in which the only correct move would have been the kosumi move I chose to try to prove that it is simply impossible (in absence of ko fight). This is my challenge.

Bill Spight wrote:
$$Bc Can Black to play win?
$$ --------------------------------
$$ | . . B W . . O | X . . . . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$Bc Can Black to play win? $$ -------------------------------- $$ | . . B W . . O | X . . . . . . | $$ | X X . . . . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

You proved black can win the above difference game but I had no doubts about that!

Click Here To Show Diagram Code: [go]$$W Kosumi vs. keima + monkey jump $$ ----------------- $$ | . . X 1 . . O | $$ | X X a . . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]

I never claimed that

is in any case the correct move against kosumi.
I claimed that

is the correct move against kosumi ONLY if black, due to the environment, has to answer with a move at "a".

My strategy with white is to play either the keima or the monkey jump but how do I make the choice?
It is a little subtil: I imagine the following sequence :

Click Here To Show Diagram Code: [go]$$B Kosumi vs. keima + monkey jump $$ ----------------- $$ | . . 1 2 . . O | $$ | X X a b . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]

and I find the best move for black (depending of the environment):
if the best move for black is a move at "a" then in the intial position I choose the keima
otherwise I choose the monkey jump.

The purpose of my last post was to analyse this strategy to prove it is correct providing there are no ko fight.

Obviously it is not easy to explain but that is my challenge.

Bill Spight · Post by **Bill Spight** » Thu Oct 01, 2020 3:19 pm

Gérard TAILLE wrote:To avoid damezumari, the best is to take your suggestion with the following difference game:

$$B Kosumi vs. keima + monkey jump
$$ --------------------------------
$$ | . . a . . . O | X . b c . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$B Kosumi vs. keima + monkey jump $$ -------------------------------- $$ | . . a . . . O | X . b c . . . | $$ | X X . . . . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]
In order to have a still better understanding of difference games I tried to go further with the challenge of proving that kosumi is dominated by the couple (monkey jump + keima).

In other words the challenge is to prove that is does not exist an environment for which kosumi is strictly better than both keima and monkey jump.

I do not believe that, given three positions (A, B, C) which are mutually incomparable, you can show by means of difference games that there does not exist a non-ko environment (E) such that A + E > C + E and B + E > C + E. In fact, using difference games you cannot even show that A + E > C + E. The reason is that the difference game is A + E - C - E = A - C, which we already know is incomparable with zero.

You may be able to show that by other means, however.

Edit: I see I must say that you cannot prove that by difference games alone.

Bill Spight · Post by **Bill Spight** » Thu Oct 01, 2020 7:00 pm

Gérard TAILLE wrote:To avoid damezumari, the best is to take your suggestion with the following difference game:

$$B Kosumi vs. keima + monkey jump
$$ --------------------------------
$$ | . . a . . . O | X . b c . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$B Kosumi vs. keima + monkey jump $$ -------------------------------- $$ | . . a . . . O | X . b c . . . | $$ | X X . . . . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]
In order to have a still better understanding of difference games I tried to go further with the challenge of proving that kosumi is dominated by the couple (monkey jump + keima).

In other words the challenge is to prove that is does not exist an environment for which kosumi is strictly better than both keima and monkey jump.

$$W Kosumi vs. keima + monkey jump
$$ -----------------
$$ | . . X 1 . . O |
$$ | X X 2 . . . O |
$$ | . X O O O O O |
$$ | . X X X O . . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ ----------------
Click Here To Show Diagram Code
[go]$$W Kosumi vs. keima + monkey jump $$ ----------------- $$ | . . X 1 . . O | $$ | X X 2 . . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]
The first part of the proof is to consider all the environments in which is (one of) the best move in answer to
For such environment the kosumi cannot dominate the keima because of the following difference game showing black cannot win:

$$B Kosumi vs. keima + monkey jump
$$ --------------------------------
$$ | . . B 4 . . O | X . 1 W 2 8 . |
$$ | X X 5 6 . . O | X . . 3 7 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$B Kosumi vs. keima + monkey jump $$ -------------------------------- $$ | . . B 4 . . O | X . 1 W 2 8 . | $$ | X X 5 6 . . O | X . . 3 7 O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

Maybe this is more like your idea.

Click Here To Show Diagram Code: [go]$$Bc Kosumi sequence vs. keima $$ -------------------------------- $$ | . . 1 2 . . O | X . . 4 . . . | $$ | X X 3 . . . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

Can Black to play win?

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

We know that Black must prevent a kill on the right, so

and

are forced. Then

nails down the left and

takes the sente on the right. The result is jigo. Therefore, Black cannot win and the keima is at least as good as the kosumi sequence.

What if White plays first in the difference game?

Click Here To Show Diagram Code: [go]$$Wc Kosumi sequence vs. keima $$ -------------------------------- $$ | . . X O 4 5 O | X . 1 O . . . | $$ | X X X 2 3 . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

White kills on the right and lives on the left, so the keima dominates the kosumi sequence.

What about the large monkey jump?

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

captures two stones and wins by 1 point.

Click Here To Show Diagram Code: [go]$$Wcm6 Variation for :w6: $$ -------------------------------- $$ | . . X O 6 7 O | X 2 O 1 O 3 . | $$ | X X X 4 5 . O | X . X O X O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

1

prisoner

connects and wins by 1 point.

Since the monkey jump threatens to kill, we know that White first will win by killing on the right.

So the kosumi sequence and the large monkey jump are incomparable.

For fun, how does the kosumi sequence compare to the crawl?

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

Black to play wins.

The kosumi sequence is incomparable with the crawl, as well.

Bill Spight · Post by **Bill Spight** » Thu Oct 01, 2020 8:17 pm

Now let's look at kosumi sequence 2, the capture.

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

Can Black to play win?

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

1

prisoner

Black wins by 1 point. The captured stone makes the difference. (If White plays

at 5 Black throws in at 4 and lives by oshitsubushi. No damezumari for Black.)

What if White plays first?

Click Here To Show Diagram Code: [go]$$Wc Kosumi sequence 2 vs. large monkey jump $$ -------------------------------- $$ | . . X . X . O | X . O 4 3 . . | $$ | X X O X 1 . O | X . 6 5 2 O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

1

prisoner

If

, then

-

makes a mirror position, for jigo. OC, if

kills on the right,

kills on the left for jigo.

Black to play wins, but White to play does not, so the kosumi sequence 2 dominates the large monkey jump.

Gérard TAILLE · Post by **Gérard TAILLE** » Fri Oct 02, 2020 4:45 am

Bill Spight wrote:
Maybe this is more like your idea.

$$Bc Kosumi sequence vs. keima
$$ --------------------------------
$$ | . . 1 2 . . O | X . . 4 . . . |
$$ | X X 3 . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$Bc Kosumi sequence vs. keima $$ -------------------------------- $$ | . . 1 2 . . O | X . . 4 . . . | $$ | X X 3 . . . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]
Can Black to play win?

$$Bc Kosumi sequence vs. keima
$$ --------------------------------
$$ | . . X O . . O | X . 1 O 2 6 . |
$$ | X X X 4 . . O | X . . 3 5 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$Bc Kosumi sequence vs. keima $$ -------------------------------- $$ | . . X O . . O | X . 1 O 2 6 . | $$ | X X X 4 . . O | X . . 3 5 O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]
We know that Black must prevent a kill on the right, so and are forced. Then nails down the left and takes the sente on the right. The result is jigo. Therefore, Black cannot win and the keima is at least as good as the kosumi sequence.

What if White plays first in the difference game?

$$Wc Kosumi sequence vs. keima
$$ --------------------------------
$$ | . . X O 4 5 O | X . 1 O . . . |
$$ | X X X 2 3 . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$Wc Kosumi sequence vs. keima $$ -------------------------------- $$ | . . X O 4 5 O | X . 1 O . . . | $$ | X X X 2 3 . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]
White kills on the right and lives on the left, so the keima dominates the kosumi sequence.

Yes Bill you clearly quite understand the strategy I described for choosing my first move :

Click Here To Show Diagram Code: [go]$$B Kosumi vs. keima + monkey jump $$ ----------------- $$ | . . . a b . O | $$ | X X . . . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]

My strategy with white is to play either the keima or the monkey jump but how do I make the choice?
It is a little subtil: I imagine the following sequence :

Click Here To Show Diagram Code: [go]$$B Kosumi vs. keima + monkey jump $$ ----------------- $$ | . . 1 2 . . O | $$ | X X a b . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]

and I find the best move for black (depending of the environment):
if the best move for black is a move at "a" then in the intial position I choose the keima
otherwise I choose the monkey jump.
[/i]

Surely you took into account 50% of my strategy, but for the time being, the remaining 50% are not still taken into account.
In other words you take into account the white strategy to choose the very first move (keima or monkey jump rather than the kosumi) but you do not take into account the white strategy in answer to an hypothetical kosumi.

Let my try to explain this remaining 50%.

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

My strategy with white is to play either at "a" or "b" but how do I make the choice?
Here again I use exactly the same subtility: I imagine the following sequence :

Click Here To Show Diagram Code: [go]$$B answer to the kosumi $$ ----------------- $$ | . . . . 1 . O | $$ | X X . . . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]

and I find the best move for white (depending of the environment):

Click Here To Show Diagram Code: [go]$$B answer to the kosumi $$ ----------------- $$ | . . . 2 B . O | $$ | X X . . . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]

if

is (one of) the best move for white then, against the kosumi, I choose the tsuke "a"
otherwise I choose the move at "b".

The difficulty for me is to manage to take into account these two parts of white strategy. Because you manage to take into account the first part of white strategy, I am quite confident (

) you will be able to take into account whole white strategy!

Bill Spight · Post by **Bill Spight** » Fri Oct 02, 2020 6:27 am

Gérard TAILLE wrote:
Bill Spight wrote:
Maybe this is more like your idea.

$$Bc Kosumi sequence vs. keima
$$ --------------------------------
$$ | . . 1 2 . . O | X . . 4 . . . |
$$ | X X 3 . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$Bc Kosumi sequence vs. keima $$ -------------------------------- $$ | . . 1 2 . . O | X . . 4 . . . | $$ | X X 3 . . . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]
Can Black to play win?

$$Bc Kosumi sequence vs. keima
$$ --------------------------------
$$ | . . X O . . O | X . 1 O 2 6 . |
$$ | X X X 4 . . O | X . . 3 5 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$Bc Kosumi sequence vs. keima $$ -------------------------------- $$ | . . X O . . O | X . 1 O 2 6 . | $$ | X X X 4 . . O | X . . 3 5 O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]
We know that Black must prevent a kill on the right, so and are forced. Then nails down the left and takes the sente on the right. The result is jigo. Therefore, Black cannot win and the keima is at least as good as the kosumi sequence.

What if White plays first in the difference game?

$$Wc Kosumi sequence vs. keima
$$ --------------------------------
$$ | . . X O 4 5 O | X . 1 O . . . |
$$ | X X X 2 3 . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------
Click Here To Show Diagram Code
[go]$$Wc Kosumi sequence vs. keima $$ -------------------------------- $$ | . . X O 4 5 O | X . 1 O . . . | $$ | X X X 2 3 . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]
White kills on the right and lives on the left, so the keima dominates the kosumi sequence.
Yes Bill you clearly quite understand the strategy I described for choosing my first move :

$$B Kosumi vs. keima + monkey jump
$$ -----------------
$$ | . . . a b . O |
$$ | X X . . . . O |
$$ | . X O O O O O |
$$ | . X X X O . . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ ----------------
Click Here To Show Diagram Code
[go]$$B Kosumi vs. keima + monkey jump $$ ----------------- $$ | . . . a b . O | $$ | X X . . . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]
My strategy with white is to play either the keima or the monkey jump but how do I make the choice?
It is a little subtil: I imagine the following sequence :
$$B Kosumi vs. keima + monkey jump
$$ -----------------
$$ | . . 1 2 . . O |
$$ | X X a b . . O |
$$ | . X O O O O O |
$$ | . X X X O . . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ ----------------
Click Here To Show Diagram Code
[go]$$B Kosumi vs. keima + monkey jump $$ ----------------- $$ | . . 1 2 . . O | $$ | X X a b . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]
and I find the best move for black (depending of the environment):
if the best move for black is a move at "a" then in the intial position I choose the keima
otherwise I choose the monkey jump.
[/i]

Surely you took into account 50% of my strategy, but for the time being, the remaining 50% are not still taken into account.
In other words you take into account the white strategy to choose the very first move (keima or monkey jump rather than the kosumi) but you do not take into account the white strategy in answer to an hypothetical kosumi.

Let my try to explain this remaining 50%.

$$B answer to the kosumi
$$ -----------------
$$ | . . 1 a . . O |
$$ | X X . b . . O |
$$ | . X O O O O O |
$$ | . X X X O . . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ ----------------
Click Here To Show Diagram Code
[go]$$B answer to the kosumi $$ ----------------- $$ | . . 1 a . . O | $$ | X X . b . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]
My strategy with white is to play either at "a" or "b" but how do I make the choice?
Here again I use exactly the same subtility: I imagine the following sequence :

$$B answer to the kosumi
$$ -----------------
$$ | . . . . 1 . O |
$$ | X X . . . . O |
$$ | . X O O O O O |
$$ | . X X X O . . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ ----------------
Click Here To Show Diagram Code
[go]$$B answer to the kosumi $$ ----------------- $$ | . . . . 1 . O | $$ | X X . . . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]
and I find the best move for white (depending of the environment):
$$B answer to the kosumi
$$ -----------------
$$ | . . . 2 B . O |
$$ | X X . . . . O |
$$ | . X O O O O O |
$$ | . X X X O . . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ ----------------
Click Here To Show Diagram Code
[go]$$B answer to the kosumi $$ ----------------- $$ | . . . 2 B . O | $$ | X X . . . . O | $$ | . X O O O O O | $$ | . X X X O . . | $$ | . . . X O O O | $$ | . . . X X X X | $$ | . . . . . . . | $$ ----------------[/go]
if is (one of) the best move for white then, against the kosumi, I choose the tsuke "a"
otherwise I choose the move at "b".

OK, let's set up the difference game for that.

Click Here To Show Diagram Code: [go]$$Bc Kosumi 2 vs. Large monkey jump 1 $$ -------------------------------- $$ | . . 1 4 . . O | X . 2 3 . . . | $$ | X X . . . . O | X . . . . O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

Can Black to play win?

Click Here To Show Diagram Code: [go]$$Bc Kosumi 2 vs. Large monkey jump 1 $$ -------------------------------- $$ | . . X O . . O | X 7 O X 4 8 . | $$ | X X 1 6 . . O | X . 5 2 3 O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

captures two stones to win by 1 point.

Click Here To Show Diagram Code: [go]$$Wcm6 Variation for :w6: $$ -------------------------------- $$ | . . X O 6 7 O | X 2 O 1 O 3 . | $$ | X X X 4 5 . O | X . X O X O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O . . | . . X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

1

prisoner

connects to win by 1 point.

OC, White to play can also win. So the two sequences are incomparable.

The difficulty for me is to manage to take into account these two parts of white strategy. Because you manage to take into account the first part of white strategy, I am quite confident ( ) you will be able to take into account whole white strategy!

I suspect that the two parts of the White strategy are incompatible, at least for difference games. In this difference game if Black plays

in the first diagram, then you play

. However, that is not good enough to keep Black from winning, because Black can play the solid connection.

The other part of the White strategy is to play

at

and then if Black plays the solid connection, to play the keima on the right side instead of the large monkey jump. That's a different strategy from playing on the right side and then deciding, based upon Blacks' reply, how to play on the left side.

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