Tapir: This situation occurred recently (i don't have a game record, but as far as i remember) between Xuefen Lin (W) and Huiren Yang in the endgame, and I didn't understand the moves.
Why does work? And why is (if it is) necessary? Why extending at a with don't work? How is the result to evaluate against the simple one with at a, at and at ?
Herman Hiddema: Lets see what happens if black stretches...
Herman Hiddema: Here we can see why stretching doesn't work. black cannot connect with 15, because then everything dies. So stretching is useless, because black cannot defend those stones later.
Tapir: I see that black can't connect... what if he doesn't? Defending with at instead? How big is the difference to the game? White captures 5 stones but played 4 stones into his own territory. Black captures one additional stone. So there is no difference in points, but white has no ko threats in the corner. (The 1 point white won in the whole situation against simple atari + connect is due to as far as i understand, not due to . But while i understand now, i don't understand .)
Herman Hiddema: The throw in at 4 is required to make move 2 work. If white plays like this, he gains nothing. Now, black can simply respond to the stretch at a with a move at b. The throw in takes a vital liberty in this situation. Tapir: Thank you very much. I will try this next time. So white gains 1 point + x (the ko) in this sequence. Only the white ko threats in the corner become smaller this way, not threatening to take the whole corner anymore, don't they? (This seems to be of minor importance.)
Herman Hiddema: This is the situation in the game after . Black still cannot stretch at a, because he still cannot connect at b after an atari. If black plays b in this position, a becomes sente. So basically, a is now white's privilege. White will either be able to defend her territory in sente (if black plays b), or be able to play ko. Also, black must, after the situation around the ko has been resolved, play at c to capture.
If we compare the game situation to this one, and count points, we can see that there are two black stones less (b, and c) and two white prisoners less (a and the throw in at ). But in the original situation, black still had to play a move at d, so actually, there are three less black stones in his territory, versus only two extra prisoners, so the game is one point better for white, plus the possibility of another point through the ko.
Symplicity (2k): Here is my guess. Firstly, we note that Black cannot play at a. If Black does...
Moreover, if White plays a, Black cannot connect, because again, White can pull out at b. Black still owes a move at b, so compared to the simple hane-and-connect, assuming that white gets a (which is very likely), White is a point better . And if White plays here again first at a, Black cannot connect, so White has the chance for a point in ko.