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The first L+1 group lives or dies according to who plays first.
Black should live by playing on the 1-2 point as shown.
Black could also live by playing at the 2-1 point, but white could then close at a in sente. Plus, if white needed to play b, this would be sente as it threatens to turn the corner into a ko.
Other possible defenses end up in ko or seki, depending on black's choice, as the next two diagrams show.
White 2 is the only move. White 2 at the vital point fails to Black 2 at 4.
Yes, white still can kill. After 7 - Bent four in the corner
The extra hane gives a sense of security.
The key tesuji here is .
The best moves by both players lead to a ten-thousand year ko.
Black can also get a multi step ko (for white) sequence, which should be better than ten-thousand year ko.
Black can get rid of ko threats before continuing. If white wants to continue, he must play yet another ko to force the atari.
Only now does the real ko for the life of the corner start. Black usually should be able to win this or receive ample compensation elsewhere in the steps white took.
If the leg is on the other side, again the position depends on who plays first.
Now the 2-2 point is the vital point for Black.
As for the first L+1 group, black could live like this, but then one of the two yose moves would be sente for white.
The hane at may seem to give some help in living.
To kill, can go straight to the 2-2 point. After , Black is dead. See the addendum of the rectangular six in the corner page. Study: Would the marked White stone still be needed if the Black hane was at the top?
The two hane stones give some degree of security.
The second L+1 group with a weak leg is also unsettled. White can kill at a or Black can live by playing at a.
fractic: That's not correct Andy.
fractic: Black lives like this. However is inferior to the other 1-2 point. After white can play hane at in sente reducing Black's territory to at most 4 points. Whereas with the other 1-2 point White needs to play 2 gote hanes to reduce Black to 4 points. So is worse pointwise. Also White playing at threatens ko but there are no such sente moves with the other 1-2 point.
fractic: This ko is what happens if Black ignores the descent of . So all in all does live but it has two downsides.
Timm: Thanks for the explanations ! I am still unconfident reading these subtle issues. I've seen it now ! I've added the diagram along with the reason why it's incorrect.