Loops with a cycle-length of pseudo 6##### About compositions of two ko-shapes #####
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[go]$$B
$$ ? ? ? ? ? ? ? ? ?
$$ ? ? ? ? ? ? ? ? ?
$$ ? ? X X X X X ? ?
$$ ? ? X O X b X ? ?
$$ ? ? O a O X O ? ?
$$ ? ? O O O O O ? ?
$$ ? ? ? ? ? ? ? ? ?
$$ ? ? ? ? ? ? ? ? ?[/go]
DOUBLE-ko
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[go]$$B
$$ ? ? ? ? ?
$$ ? O ? b ?
$$ ? a ? X ?
$$ ? ? ? ? ?[/go]
This concentrated form will save a lot of space.
The shadowed bar in the middle is only there to achieve an odd length that suits the diagrams better than an even one.
- Click Here To Show Diagram Code
[go]$$B
$$ ? ? ? ? ?
$$ ? O ? . ?
$$ ? . ? X ?
$$ ? ? ? ? ?[/go]
In the following, for the sake of simplicity, we will dispense with explicitly specifying the markings.
##### GENUINE double-ko cycle #####
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[go]$$Bc
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . ? ? ? ? ? . . . . . . . |
$$ | . . . . . . . ? O ? W ? . . . . . . . |
$$ | . . . , . . . ? . ? . ? . . . , . . . |
$$ | . ? ? ? ? ? . ? ? ? ? ? . ? ? ? ? ? . |
$$ | . ? O ? . ? . . . . . . . ? . ? O ? . |
$$ | . ? . ? X ? . . . . . . . ? B ? . ? . |
$$ | . ? ? ? ? ? . . . . . . . ? ? ? ? ? . |
$$ | . . p . a . . . . . . . . . p . a . . |
$$ | . . s , s . . . . , . . . . s , s . . |
$$ | . ? ? ? ? ? . . . . . . . ? ? ? ? ? . |
$$ | . ? W ? . ? . . . . . . . ? . ? O ? . |
$$ | . ? . ? X ? . . . . . . . ? X ? . ? . |
$$ | . ? ? ? ? ? . . . . . . . ? ? ? ? ? . |
$$ | . . . . . . . ? ? ? ? ? . . . . . . . |
$$ | . . . , . . . ? . ? . ? . . . , . . . |
$$ | . . . . . . . ? X ? B ? . . . . . . . |
$$ | . . . . . . . ? ? ? ? ? . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+
$$ {AR M17 Q15}
$$ {AR Q12 Q9}
$$ {AR Q6 M3}
$$ {AR H3 D6}
$$ {AR D9 D12}
$$ {AR D15 H17}[/go]
This diagram shows the GENUINE double-ko cycle.
The marked stones are those that have been played last.
Please note that the two "pass" are an integral part of a GENUINE DOUBLE-ko cycle.
The player in turn does not have a ko-shape they would be able to capture in.
Therefore, it can NOT have a GENUINE length of 6.
----------------------
- Click Here To Show Diagram Code
[go]$$Bc
$$ +---------------------------------------+
$$ | . . . . . . . M M M M M . . . . . . . |
$$ | . . . . . . . ? ? ? ? ? M . . . . . . |
$$ | . . . . . . . ? O ? W ? . M M . . . . |
$$ | . . . , . . . ? . ? . ? . . . M M M . |
$$ | . ? ? ? ? ? . ? ? ? ? ? . ? ? ? ? ? M |
$$ | . ? O ? . ? . M M M M M . ? . ? O ? M |
$$ | . ? . ? X ? . . . . . . M ? B ? . ? M |
$$ | . ? ? ? ? ? . . . . . . M ? ? ? ? ? M |
$$ | . . p . a . . . . . . . . M p . a M . |
$$ | . M s , s M . . . , . . . . s , s . . |
$$ | M ? ? ? ? ? M . . . . . . ? ? ? ? ? . |
$$ | M ? W ? . ? M . . . . . . ? . ? O ? . |
$$ | M ? . ? X ? M . . . . . . ? X ? . ? . |
$$ | M ? ? ? ? ? . M M M M M . ? ? ? ? ? . |
$$ | . M M . . . . ? ? ? ? ? . . . . . . . |
$$ | . . . M M . . ? . ? . ? . . . , . . . |
$$ | . . . . . M . ? X ? B ? . . . . . . . |
$$ | . . . . . . M ? ? ? ? ? . . . . . . . |
$$ | . . . . . . . M M M M M . . . . . . . |
$$ +---------------------------------------+
$$ {AR M17 Q15}
$$ {AR Q12 Q9}
$$ {AR Q6 M3}
$$ {AR H3 D6}
$$ {AR D9 D12}
$$ {AR D15 H17}
$$ {AR Q6 Q1}
$$ {AR R6 R1}
$$ {AR S6 S1}
$$ {AR B15 B16}
$$ {AR C15 C16}
$$ {AR D15 D16}[/go]
Two of the four GENUINE moves (top, bottom) are atari at the opponent's group, so they must not be left unanswered.
You will realise that this results in the following:
While there are FOUR different combinations of the INDIVIDUAL states of each single-ko of the compound, the DOUBLE-ko as such has ONLY TWO states.
Both players are able to leave the cycle after their opponent's "pass" only.
Please note that White's exit option in the upper left is not shown in full length for the sake of clarity.
We chose Black's exit in the lower right for an exemplary case.
----------------------
- Click Here To Show Diagram Code
[go]$$Bc
$$ +---------------------------------------+
$$ | . . . . . . . ? ? ? ? ? . . . . . . . |
$$ | . . . . . . . ? . ? O ? . . . . . . . |
$$ | . . . . . . . ? X ? . ? p a . . . . . |
$$ | . . . , . . . ? ? ? ? ? . . s . . . . |
$$ | . . . . . . . . . . . . . . . s . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . Y . . . . . . . . . . . Y . . . |
$$ | . . . . . . . . . . . . . . . . p . . |
$$ | . . . . . . . . . . . . . . . . a . . |
$$ | . . . , . . . . . , . . . . . , s . . |
$$ | . . . Q . . . ? ? ? ? ? . . . . s . . |
$$ | . . . . . . . ? W ? O ? . . . Q . . . |
$$ | . . . . . . . ? . ? . ? . . . . . . . |
$$ | . . . . . . . ? ? ? ? ? . . . . . . . |
$$ | . ? ? ? ? ? . . . . . . . . . . . . . |
$$ | . ? . ? . ? . . . , . . . M M M M M . |
$$ | . ? X ? B ? . . . . . . . M e n d M . |
$$ | . ? ? ? ? ? . . . . . . . M M M M M . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+
$$ {AR H17 D14}
$$ {AR H16 E14}
$$ {AR J16 E13}
$$ {AR L16 P13}
$$ {AR M16 P14}
$$ {AR M17 Q14}
$$ {AR Q12 Q9}
$$ {AR Q7 Q4}
$$ {AR C8 C5}
$$ {AR D8 D5}
$$ {AR E8 E5}
$$ {AR D12 H8}
$$ {AR E12 H9}
$$ {AR E13 J9}
$$ {AR P13 L9}
$$ {AR P12 M9}
$$ {AR Q12 M8}
$$ {AR D12 D10}
$$ {AR P13 E9}
$$ {AR C10 C12}[/go]
Black has two options for his move outside the cycle:
RIGHT:-- Black plays a "pass".
White has three options then:
-- She plays also a "pass". This will end status confirmation.
-- She captures into the double-ko. This returns into the cycle.
-- She plays a GENUINE move outside the cycle. This jumps to the left side of the diagram.
LEFT:-- Black plays a GENUINE move outside the cycle.
White has two options then:
-- She also plays a move outside the cycle.
Black has two options then:
-- He plays another move outside the cycle. This continues the game outside the cycle.
-- He captures into the double-ko. This returns into the cycle.
-- She captures into the double-ko. This returns into the cycle.
_________________________
EDITED
_________________
The really most difficult Go problem ever:
https://igohatsuyoron120.de/index.htmIgo Hatsuyōron #120 (really solved by KataGo)
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Gérard TAILLE
where I expected a 
?
.
exchange above white will win if Honte pass forever. That is the point you had to find.
in the lower left corner!