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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 hane 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.
also kills. After , a and b are miai. However in practice, White's outside wall is often weaker than the diagram (for example the circled White stone is missing) and Black may later choose a or b and exploit outside aji around c or d. In life and death situations in general, attacks from outside should be considered first, inside plays last.
What happens if White plays on Black's vital point? "Your opponent's good move is your good move" does apply here as well? Yes, white still can kill. After , it is a Bent four in the corner. But this method is generally inferior because if White's outside wall have lost eyes, he must start the ko in the corner.
The extra hane gives a sense of security.
The key tesuji here is .
This sequence leads to a ten-thousand year ko, however white can do better by getting a multi-step ko as shown below.
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.
Black can also live with at , but if white plays both a and b later (perhaps as a result of fighting on the outside) then Black will need an extra defensive move, or else white can kill.
Black can't play a because of shortage of liberties. If at , then at is atari.
As for the first L+1 group, black could live like this, but then white a is sente, so black's corner is only worth four points instead of six points in the first diagram above.
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.