This is a really tough tsumego problem with a main line of 17 moves.
The challenge is to make an eye at the marked point, and still have time to make one in the corner. This is no easy task, because Black needs two moves to turn the marked point into a real eye; and a white move at a destroys any eye potential in the corner.
On the other hand, Black can threaten to capture the stones, to break through the stones or to make two eyes in the corner. With these weapons, Black has to find the proper order of moves to live.
After this , White suffers from shortage of liberties.
That is, Black can play as in the following diagram:
Now for some variations, backtracking from Black 17 to .
In response to (White 14) it is tempting to capture at , but White connects at and leaves Black with two false eyes in the corner.
If White 14 is at in this diagram, then Black plays (can also be at a). After and , White can't play at a, again because of shortage of liberties.
After (), the natural white response seems to be here. In any case, it is the vital point.
But is trapped already, and Black has time to make his second eye with and .
Of course, if White prevents that second eye, Black makes two eyes in the corner, playing himself.
What if White at ignores atari and destroys the eye potential in the corner right away with here ?
The whole meaning of and in the main line is to get in place in sente. This stone serves amongst other things to prevent a connection underneath, as demonstrated in the "variation at 8".
If Black immediately plays , White defends at , in the process destroying the eye potential there. Now Black has to make two eyes in the corner: which he can't do.
is vulgar. It destroys the possibilities and White defends at with good aji. The move at is still forcing, but now Black has run out of resources. White strikes at the vital point at and makes miai of a and b.
Now how about seeking life in the corner immediately? destroys the eye above and even after , Black doesn't have two eyes in the corner.
Now a and b are miai.
Incidentally, seems to work too: if a, then b makes an eye in sente. If tenuki from the corner then also life ? (To be continued.)