John Fairbairn started several threads on Ô Meien's book on Yose. There are heavy discussions on the mathematical background of his method, which I do not evaluate to be something extraordinary.
I would like to give you a special view on the diagrams Ô Meien uses to visualize the result. These diagrams seem to be somewhat surprising at first sight.
But before, I have to mention that chapter 1.1 of Ô Meien's book (presented so far by John) deals with some special kind of Yose only. I would like to call it two-step-endgame, using the term generally known with Ko. Black can finish the Yose with only one move. If it is White's turn, there may be a second move, either by Black or by White. This second move finishes the Yose, too.
Perhaps Ô Meien had in mind to give examples for a simplified way to get these results.
In this kind of positions it will be possible to get the result by just identifying the type of stones or unoccupied points, without playing the position out.
I will follow mostly the flow of the diagrams in the book and give explanations in-between.
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[go]$$ Diagram A
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O . . . . . . . . . . . . . O . . |
$$ | O . . . . . . . . . . . . . . . . . O |
$$ | . O O . . . . . . , . . . . . , O O . |
$$ | O O X . X . . . . . . . . . X . X O O |
$$ | X X X . . X . . . . . . . X . . X X . |
$$ | . X . . . . . . . . . . . . . . . X Q |
$$ ------------------------------------------[/go]
1 = 1 (first-step-stone)
I will call a stone that can only be captured with the first move a "first-step-stone". An example is the marked White stone in the diagram above.
A "first-step-stone" has a value of 1.
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[go]$$ Diagram B
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | X X X . X . . . . . . . . . X . X X X |
$$ | X . X . . . . . . . . . . . . . X Q X |
$$ | X . X , . . . . . , . . . . . , X Q . |
$$ | O X X O . O . . . . . . . O . O X X O |
$$ | O O O . O . . . . . . . . . O . O O O |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ----------------------------------------[/go]
2 = 2 (first-step-stones)
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[go]$$ Diagram C
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O . . . . . . . . . . . . . O . . |
$$ | O O . . . . . . . . . . . . . . . O O |
$$ | O X X , X . . . . , . . . . X , X X O |
$$ | X . X . . . . . . . . . . . . . X Q . |
$$ | . . X . . . . . . . . . . . . . X Q Q |
$$ | . . X . . . . . . . . . . . . . X Q Q |
$$ ----------------------------------------[/go]
5 = 5 (first-step-stones)
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[go]$$ Diagram F
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O . . . . . . . . . . . . . O . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O O O , . . . . . , . . . . . , O O O |
$$ | X X O X . X . . . . . . . X . X O Z . |
$$ | Z . X . X . . . . . . . . . X . X Q Q |
$$ | X X X . . . . . . . . . . . . . X X X |
$$ ----------------------------------------[/go]
1 = 2 (first-step-stones) - 1 (solid-capturable-stone)
I will call a Black stone that can be captured by White and providing an eye-point, a "solid-capturable-stone".
A "solid-capturable-stone" has a value of -1, if captured in step 1.
- Click Here To Show Diagram Code
[go]$$ Diagram F-mod
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . O . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O O O O . . . . . , . . . . . O O O O |
$$ | X X X O . X . . . . . . . X . O Z Z . |
$$ | . Z Z X X . X . . . . . X . X X Q Q Q |
$$ | . X X X . . . . . . . . . . . X X X Q |
$$ ----------------------------------------[/go]
2 = 4 (first-step-stones) - 2 (solid-capturable-stones)
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[go]$$ Diagram D
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . O . |
$$ | . . O . . . . . . . . . . . . . O . . |
$$ | O O . , . . . . . , . . . . . , . O O |
$$ | O X X . X . . . . . . . . . X . X X O |
$$ | X . X . . . . . . . . . . . . . X Q . |
$$ | . . X . . . . . . . . . . . . . X Q T |
$$ ----------------------------------------[/go]
3 = 2 (first-step-stones) + 1 (first-step-point)
Here you have a point that could be secured in step 1.
Such a "first-step-point" has a value of 1.
- Click Here To Show Diagram Code
[go]$$ Diagram D-mod
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . O . |
$$ | . . O . . . . . . . . . . . . . O . . |
$$ | O O . , . . . . . , . . . . . , . O O |
$$ | O X X . X . . . . . . . . . X . X X O |
$$ | X . X . . . . . . . . . . . . . X Q . |
$$ | . . X . . . . . . . . . . . . . X Q Q |
$$ ----------------------------------------[/go]
3 = 3 (first-step-stones)
Diagram D is equivalent to this one.
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[go]$$ Diagram E
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O O O O . . . . . . . . . . . O O O O |
$$ | O X X . . . . . . . . . . . . . X X O |
$$ | X . X , . . . . . , . . . . . , X Q B |
$$ | . . X . . X . . . . . . . X . . X Q S |
$$ | X . X . . . . . . . . . . . . . X Q . |
$$ | X X X . . . . . . . . . . . . . X X X |
$$ ----------------------------------------[/go]
3 = 3 (first-step-stones) + .5 (point-behind) - .5 (capturable-stone)
An unoccupied point "behind" some Black or White stone(s) will be called a "point-behind", marked with a square.
The marked Black stone could be captured by White in step 1, but will not become an eye-point. This kind of stone is called "capturable-stone", marked with a circle.
A "point-behind" has a value of .5.A "capturable-stone" has a value of -.5.
- Click Here To Show Diagram Code
[go]$$ Diagram M
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O O O O . . . . . . . . . . . O O O O |
$$ | O X X . . . . . . . . . . . . . X X O |
$$ | X . X , . . . . . , . . . . . , X Q B |
$$ | . . X . . X . . . . . . . X . . X Q S |
$$ | X . X . . . . . . . . . . . . . X Q . |
$$ | . X X . . . . . . . . . . . . . X X S |
$$ ----------------------------------------[/go]
3.5 = 3 (first-step-stones) + 2 * .5 (point-behind) - .5 (capturable-stone)
- Click Here To Show Diagram Code
[go]$$ Diagram G
$$ | . . O . . . . . . . . . . . . . O . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O O O . . . . . . . . . . . . . O O O |
$$ | X X . , . . . . . , . . . . . , . X . |
$$ | . X . X . X . . . . . . . X . X . X Q |
$$ | . X . . . . . . . . . . . . . . . X S |
$$ | . X . . . . . . . . . . . . . . . X @ |
$$ ----------------------------------------[/go]
3 = 1 (first-level-stone) + .5 (point-behind) + 1.5 (second-level-stone)
A "second-level-stone" is a stone that could be captured by Black in step 2, if White started the Yose-sequence. These stones are marked with a square.
A "second-level-stone" has a value of 1.5.Please be aware that Ô Meien does not use unoccupied points behind a "second-level-stone", if there is no Black stone to become captured in step 2. Such an unoccupied point "behind-behind" would have a value of .25.
- Click Here To Show Diagram Code
[go]$$ Diagram G-mod
$$ | O O O . . . . . . . . . . . . . O . . |
$$ | . X O . . . . . . . . . . . . . . . . |
$$ | . X O . . . . . . . . . . . . . O O O |
$$ | X X X , . . . . . , . . . . . , X X . |
$$ | . . X . X . X . . . . . X . X . X Q Q |
$$ | . X X . . . . . . . . . . . . . X X S |
$$ | . . X . . . . . . . . . . . . . X @ @ |
$$ ----------------------------------------[/go]
5.5 = 2 (first-level-stones) + .5 (point-behind) + 2 * 1.5 (second-level-stone)
- Click Here To Show Diagram Code
[go]$$ Diagram H
$$ | O O O O O . . . . . . . . . . . O . . |
$$ | . X . X . . . . . . . . . . . . . . . |
$$ | . X . X . . . . . . . . . . . . O O O |
$$ | X X X X . . . . . . . . . . . . X X . |
$$ | . . X , . . . . . , . . . . . , X Q Q |
$$ | . X X . . . . . . . . . . . . . X X S |
$$ | . . X . . . . . . . . . . . . . X @ @ |
$$ | . X X . . . . . . . . . . . . . X X @ |
$$ ----------------------------------------[/go]
6 = 2 (first-level-stones) + .5 (point-behind) + 3 * 1.5 (second-level-stone)
- Click Here To Show Diagram Code
[go]$$ Diagram I
$$ | . O . . . . . . . . . . . . . . . O . |
$$ | . . O . . . . . . . . . . . . . O . . |
$$ | O O X X . . . . . . . . . . . X X O O |
$$ | X X . , . . . . . , . . . . . , . X . |
$$ | . X X . . . . . . . . . . . . . X X Q |
$$ | . . X . . . . . . . . . . . . . X Q Q |
$$ | X . X . . . . . . . . . . . . . X S Z |
$$ ----------------------------------------[/go]
3 = 3 (first-level-stones) + .5 (point-behind) - .5 (solid-capturable-second-level-stone)
- Click Here To Show Diagram Code
[go]$$ Diagram J
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O O O . . . . . . . . . . . . . O O O |
$$ | X X . . . . . . . . . . . . . . . X . |
$$ | . X . , . . . . . , . . . . . , . X Q |
$$ | . X X X . . . . . . . . . . . X X X Q |
$$ | X . . X . . . . . . . . . . . X @ @ X |
$$ | . . X X . . . . . . . . . . . X X S S |
$$ ----------------------------------------[/go]
6 = 2 (first-level-stones) + 2 * .5 (points-behind) + 2 * 1.5 (second-level-stones)
The unmarked Black stone, which could be captured by White in step 2, has a value of 0.
- Click Here To Show Diagram Code
[go]$$ Diagram J-mod2
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O O O . . . . . . . . . . . . . O O O |
$$ | X X . . . . . . . . . . . . . . . X . |
$$ | . X . , . . . . . , . . . . . , . X Q |
$$ | . X X X . . . . . . . . . . X X X X Q |
$$ | X . . X . . . . . . . . . . X @ @ @ X |
$$ | . . X X . . . . . . . . . . X X X S S |
$$ ----------------------------------------[/go]
7.5 = 2 (first-level-stones) + 2 * .5 (points-behind) + 3 * 1.5 (second-level-stones)
- Click Here To Show Diagram Code
[go]$$ Diagram J-mod
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O O O . . . . . . . . . . . . . O O O |
$$ | X X . . . . . . . . . . . . . . . X . |
$$ | . X . , . . . . . , . . . . . , . X Q |
$$ | . X X X . . . . . . . . . . . X X X Q |
$$ | . . . X . . . . . . . . . . . X @ @ S |
$$ | X . X X . . . . . . . . . . . X X @ Z |
$$ ----------------------------------------[/go]
6.5 = 2 (first-level-stones) + .5 (point-behind) + 3 * 1.5 (second-level-stones) - .5 (solid-capturable-second-level-stone)
- Click Here To Show Diagram Code
[go]$$ Diagram J-mod-3
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O O O . . . . . . . . . . . . . O O O |
$$ | X X . . . . . . . . . . . . . . . X . |
$$ | . X . , . . . . . , . . . . . , . X Q |
$$ | . X X X X X X . . . . . . . X X X X Q |
$$ | X . . . X . X . . . . . . . X @ @ @ S |
$$ | X . X X X . X . . . . . . . X X X @ Z |
$$ ----------------------------------------[/go]
8 = 2 (first-level-stones) + .5 (point-behind) + 4 * 1.5 (second-level-stones) - .5 (solid-capturable-second-level-stone)
- Click Here To Show Diagram Code
[go]$$ Diagram L
$$ | O O O . . . . . . . . . . . . . . . . |
$$ | . X . . . . . . . . . . . . . . . . . |
$$ | . X . . . . . . . . . . . . . . O O O |
$$ | X X . , . . . . . , . . . . . , . X . |
$$ | . X . . X . . . . . . . . . X . . X Q |
$$ | . X . . . . . . . . . . . . . . . X Q |
$$ | X X . . . . . . . . . . . . . . . X S |
$$ ----------------------------------------[/go]
2.5 = 2 (first-level-stones) + .5 (point-behind)
- Click Here To Show Diagram Code
[go]$$ Diagram N
$$ | O O O O . . . . . . . . . . . . . . . |
$$ | . X . . . . . . . . . . . . . . . . . |
$$ | . X . . . . . . . . . . . . . O O O O |
$$ | X X X . . . . . . . . . . . . . X X . |
$$ | . . X , . . . . . , . . . . . , X Q Q |
$$ | . X X . . X . . . . . . . X . . X X Q |
$$ | X . X . . . . . . . . . . . . . X @ X |
$$ | . X X . . . . . . . . . . . . . X S S |
$$ ----------------------------------------[/go]
5.5 = 3 (first-level-stones) + 2 * .5 (point-behind) + 1.5 (second-level-stone)
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