If White is allowed to capture simply at , Black dies.
Since White a will atari the stones,
will be neccessary sooner or later, after which
connects and kills Black by either almost filling into farmer's hat or bulky five (with a White stone at b).
For Black to save himself, Black must not allow White to play both and
in the analysis diagrams. The only way is to fight the ko with
. If Black wins this ko, he lives by playing at a.
Scartol: Should I leave these so that actual beginners can have a crack at them?
Okay, it looks like the best black can do is get a ko. White has to fight it; white connecting at A only leads to black life at B.
Would it work better if black goes at B?
Scartol is stupid
This was my initial response, but I realized (after I posted it -- stupid!) that white can connect after capturing the marked stone, and black is left with a dead shape once he captures the white stones.
Leads to bulky five and black dies. -Xelloss
This is the vital point I think. Once Black moves here, White can't play at 'a' because of a Black snapback, and can't play at 'b' because it's lacking a liberty. Black can form two eyes and live by playing at 1.
Black can respond with 1. Once Black moves to 1, if White tries to move to 'a', Black can capture White's stones by playing above at 'b'. If White chooses to respond at 'b' instead and capture the marked stone, Black can snapback and capture White.
If White plays at 'b' instead, Black can respond at 1, forcing White to capture at 2, then Black can play at 3. White is missing the liberty needed to capture Black. When Black plays at 3, White will have no choice but to use the move as sente. Black doesn't need to play there unless White tries to play at 'a' or 'b', in which case Black can simply capture the group of White stones.
unkx80: After , Black is dead.
The reason is because if , then
connects and even if Black plays at a, Black still dies in a bulky five nakade.
Now W can close ko, making B take a. If B takes over ko, he lives. In other case, W closes ko, B plays a. If then W takes c, B extends to the right, and if W doesn't play c, white can take it, close it and live
is the vital point. Variations lead to life for Black.
Kayaq: This appears to be wrong; see my example three boards down.
unkx80: What happens after ?
Here, the twisted four lives.
unkx80: Thanks to the kamikaze of , Black lives, but White could just tenuki after
and Black will still be dead with a bent three.
Kayaq: above seems not to work!
After , black cannot connect at a because of shortage of liberties.
After , White can play for ko or seki but cannot kill.
unkx80: captures, Black dies.