Dear Robert, here come the promised diagrams to explain some of my ideas to you.
For the sake of not interfering with anything you might have in mind when reading "eye", let us call points, which must not be taken by one player, because that move would leave a chain of him with no lifeline left, as "
taboo".
- Click Here To Show Diagram Code
[go]$$
$$ . . . . .
$$ . O O O .
$$ . O 1 O .
$$ . O O O .
$$ . . . . .[/go]
The move of 1 is not allowed for Black, ...
- Click Here To Show Diagram Code
[go]$$
$$ . . . . .
$$ . O O O .
$$ . O M O .
$$ . O O O .
$$ . . . . .[/go]
... so the marked point is a "taboo-point" of White's chain.
"Evaluate" is done for each chain of stones separately, starting with a move of the opponent (may be a Pass).
Status "2-eyed" has three sub-statuses:(I) The chain already has at least 2 "taboo-points". Black must not play at each of them. So the chain cannot be taken off the board. Just to give a help where to look, these "taboo-points" lay "inside" the chain, "inside" not to be taken literally, as the second example shows.
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |M W M W X .
$$ |W W W W X .
$$ |X X X X X .
$$ |. . . . . .[/go]
- Click Here To Show Diagram Code
[go]$$
$$ |----------
$$ |M W W X .
$$ |O M W X .
$$ |O O X X .
$$ |X X X . .
$$ |. . . . .[/go]
- Click Here To Show Diagram Code
[go]$$
$$ |--------------------
$$ |M W M W . X . W X .
$$ |W W W W W W W W X .
$$ |X X X X X X X X X .
$$ |. . . . . . . . . .[/go]
(II) The chain has not 2 "taboo-points" yet. Whatever Black tries, the chain will be transformed into sub-status (I).
- Click Here To Show Diagram Code
[go]$$
$$ |----------------
$$ |. . W . . W X .
$$ |W W W W W W X .
$$ |X X X X X X X .
$$ |. . . . . . . .[/go]
White's chain has no "taboo-points" yet.
- Click Here To Show Diagram Code
[go]$$
$$ |----------------
$$ |M 3 O M 1 O X .
$$ |O O O O O O X .
$$ |X X X X X X X .
$$ |. . . . . . . .[/go]
Now the chain has 2 "taboo-points".
- Click Here To Show Diagram Code
[go]$$
$$ |--------------
$$ |. O . O X . .
$$ |O O O O X X .
$$ |. . . . W X .
$$ |W W W W W X .
$$ |X X X X X X .
$$ |. . . . . . .[/go]
White's marked chain has no "taboo-points" yet.
- Click Here To Show Diagram Code
[go]$$
$$ |--------------
$$ |M O M O X . .
$$ |O O O O X X .
$$ |. 2 1 . W X .
$$ |W W W W W X .
$$ |X X X X X X .
$$ |. . . . . . .[/go]
Now the chain has 2 "taboo-points".
Chains of sub-status (II) cannot be taken off the board, too.
(III) The chain can be captured by the opponent. Thereafter, each of the primary points of this chain will be transformed into sub-status (I).
- Click Here To Show Diagram Code
[go]$$
$$ |--------------
$$ |X X O . O . O
$$ |X . O O O O O
$$ |X W X . . . .
$$ |O X X X X . .
$$ |O O O O X . .
$$ |. O . O X . .
$$ |O O O O X . .
$$ |. . . . . . .[/go]
White's marked stone can be captured by Black.
- Click Here To Show Diagram Code
[go]$$
$$ |--------------
$$ |X X O . O . O
$$ |X 1 O O O O O
$$ |X W X . . . .
$$ |O X X X X . .
$$ |O O O O X . .
$$ |. O . O X . .
$$ |O O O O X . .
$$ |. . . . . . .[/go]
- Click Here To Show Diagram Code
[go]$$
$$ |--------------
$$ |X X O . O . O
$$ |X X O O O O O
$$ |X 2 X . . . .
$$ |O X X X X . .
$$ |O O O O X . .
$$ |. O . O X . .
$$ |O O O O X . .
$$ |. . . . . . .[/go]
- Click Here To Show Diagram Code
[go]$$
$$ |--------------
$$ |. . O . O . O
$$ |. 3 O O O O O
$$ |4 W X . . . .
$$ |O X X X X . .
$$ |O O O O X . .
$$ |M O M O X . .
$$ |O O O O X . .
$$ |. . . . . . .[/go]
Now White's chain has 2 "taboo-points".
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |W W . X O .
$$ |X X X X O .
$$ |O O O O O .
$$ |. O . O O .
$$ |O O O O O .
$$ |. . . . . .[/go]
White's marked chain can be captured.
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |W W 1 X O .
$$ |X X X X O .
$$ |O O O O O .
$$ |. O . O O .
$$ |O O O O O .
$$ |. . . . . .[/go]
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |2 3 X X O .
$$ |X X X X O .
$$ |O O O O O .
$$ |. O . O O .
$$ |O O O O O .
$$ |. . . . . .[/go]
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |4 X X X O .
$$ |X X X X O .
$$ |O O O O O .
$$ |. O . O O .
$$ |O O O O O .
$$ |. . . . . .[/go]
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |O 6 . . O .
$$ |8 . . . O .
$$ |O O O O O .
$$ |M O M O O .
$$ |O O O O O .
$$ |. . . . . .[/go]
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |W W . . O .
$$ |O . . . O .
$$ |O O O O O .
$$ |M O M O O .
$$ |O O O O O .
$$ |. . . . . .[/go]
Each primary point now belongs to a chain with 2 "taboo-points".
Status "stable" has two sub-statuses:(I) The chain does not have 2 "taboo-points", but will not be captured by the opponent.
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |W W W W X .
$$ |X 1 2 W X X
$$ |X X X X O X
$$ |O O O O O X
$$ |. O . O . X
$$ |O O O O . X
$$ |. . . . . .[/go]
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |O O O O X .
$$ |. . O O X X
$$ |. . . . O X
$$ |O O O O O X
$$ |. O . O . X
$$ |O O O O . X
$$ |. . . . . .[/go]
Black cannot capture White's chain.
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |. X W . O X
$$ |X . W . O X
$$ |W W W W X X
$$ |X X X X X .
$$ |. . . . . .[/go]
White's marked chain ...
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |. X W 2 O X
$$ |X . W 1 O X
$$ |W W W W X X
$$ |X X X X X .
$$ |. . . . . .[/go]
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |. X O O O X
$$ |X . O M O X
$$ |O O O O X X
$$ |X X X X X .
$$ |. . . . . .[/go]
... will not get 2 "taboo-points", so it remains in status "stable".
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |1 W W X . .
$$ |X W W X X X
$$ |O X X X . .
$$ |O O O O . .
$$ |. O . O . .
$$ |O O O O . .
$$ |. . . . . .[/go]
White's marked chain can be captured by Black.
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |X 4 . X . .
$$ |X 2 3 X X X
$$ |O X X X . .
$$ |O O O O . .
$$ |. O . O . .
$$ |O O O O . .
$$ |. . . . . .[/go]
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |. O 5 X . .
$$ |6 O X X X X
$$ |O X X X . .
$$ |O O O O . .
$$ |. O . O . .
$$ |O O O O . .
$$ |. . . . . .[/go]
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |M W B X . .
$$ |O W B X X X
$$ |O X X X . .
$$ |O O O O . .
$$ |M O M O . .
$$ |O O O O . .
$$ |. . . . . .[/go]
Only 2 of the primary 4 points are part of a group with at least 2 "taboo-points". So the chain cannot get the status "2-eyed". But will be "stable".
Status "removable" contains, what cannot reach neither status "2-eyed" nor "stable":
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |W W 1 X . .
$$ |X X X X . .
$$ |. X . X . .
$$ |X X X X . .
$$ |. . . . . .[/go]
White's marked chain can be captured by Black, ...
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |C C X X . .
$$ |X X X X . .
$$ |. X . X . .
$$ |X X X X . .
$$ |. . . . . .[/go]
... and none of its primary points will become part of a "2-eyed" White chain.
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |3 W X . X O
$$ |O 1 X 2 X O
$$ |X X X X O O
$$ |O O O O O .
$$ |. . . . . .[/go]
White's marked chain can be captured by Black ...
- Click Here To Show Diagram Code
[go]$$
$$ |------------
$$ |. B X . X O
$$ |X . X O X O
$$ |X X X X O O
$$ |O O O O O .
$$ |. . . . . .[/go]
... and its primary point will not become part of a "2-eyed" White chain.
- Click Here To Show Diagram Code
[go]$$
$$ |--------------
$$ |. O X 2 X O .
$$ |O O O X X O .
$$ |1 O X . X O .
$$ |W X X X X O .
$$ |X X O O O O .
$$ |O O O . . . .
$$ |. . . . . . .[/go]
White's marked chain can be captured by Black ...
- Click Here To Show Diagram Code
[go]$$
$$ |--------------
$$ |. O 5 O X O .
$$ |O O O X X O .
$$ |X O X . X O .
$$ |4 X X X X O .
$$ |X X O O O O .
$$ |O O O . . . .
$$ |. . . . . . .[/go]
- Click Here To Show Diagram Code
[go]$$
$$ |--------------
$$ |. O X 8 X O .
$$ |O O O X X O .
$$ |7 O X . X O .
$$ |O X X X X O .
$$ |X X O O O O .
$$ |O O O . . . .
$$ |. . . . . . .[/go]
... and its primary point will not become part of a "2-eyed" White chain.
_________________
The really most difficult Go problem ever:
https://igohatsuyoron120.de/index.htmIgo Hatsuyōron #120 (really solved by KataGo)