An L+1 group is an L group with a leg. The leg can be at the short side or the long side of the L shape. In both cases the group lives or dies according to who plays first.
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First L+1 group
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.
White can kill by playing hane at . This reduces the shape to the standard L group, which we already know is dead.
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.
First L+1 group, hane on one side
The extra hane gives a sense of security.
The throw-in at no longer works: jumps and lives. If makes a placement, connects, and White cannot connect at a due to shortage of liberties, otherwise Black captures three to make an eye.
However, White can kill by the hane at followed by the placement at . tries to live, but forces , so makes a throw in and kills. Note the importance of the stone.
First L+1 group, hane on both sides
The first L+1 group with hane on both sides is alive if it has an external liberty. However, when the group has no external liberties, White can turn it into a ten-thousand year ko.
Patrick Traill: According to Guo Juan in this lesson in her Internet Go School, the attacker has these options:
- a ten-thousand year ko in gote: a, b, c, … in problems 17 (defence) & 20 (attack);
- a sente seki: a, b, d, e, … in problems 18 (defence) & 19 (attack);
- a double ko: f, g, e, d, … in the lecture from 15:51 and problem 16 (defence).
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.
Second L+1 group
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.
Second L+1-group, hane one side
The hane at may seem to give some help in living.
The throw-in at doesn't work anymore. Black lives with : White can not connect at a because Black can capture three to make an eye.
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?
Second L+1-group, hane on both sides
The two hane stones give some degree of security.
The attachment at leads to ko.
Second L+1 group, weak leg
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.
Timm (10k) Isn't this unconditionally alive ? I can't kill it. And if it's alive, is it inferior to 'a' ?
Andy W plays at a. The points b are then miai to kill.
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.