Normally, the easier way is to try method 1. But if method 1 does not work, then we would have to consider method 2 instead.
... and at the same time, we see that a and b are miai for life. So we can conclude applying method 1 in this problem fails.
The simply captures a stone, then kills.
What happens if White plays ? is neccessary to destroy the eye...
If you have followed the discussion so far, then you might also know that is also not a vital point. The best for answering is... I leave this as a reading exercise.
Karl Knechtel: So it seems. :) Though, doing tsumego and solving the problems in an actual game are different matters, at least for me. For big battles where the goal is to connect or to cut an opponent's connection, I seem to do fairly well for my strength, and I'm all right at destroying eye space and making good shape. But I have a horrible time of it when the group whose life is in question is isolated; if it's mine, I'll misjudge it as already alive; and if it's the opponent's, I'll mistime or mislocate my placement or throw-in.
Actually, my first experience with Go was tsumego problems - very easy ones, only one move needed to be made. They were a small puzzle game on a MUD that I played when I was 13. There was this "go player" mob in one of the rooms, and a copy of the rules for Go (which you could 'get' and 'read'), and from that you were supposed to infer the idea of two eyes. Then you would talk to the player, and he would set up a position on the board, and you would 'suggest' a move to him. Of course you could just guess randomly until you got it, although there were a handful of problems (probably five or so); but I got the idea that way.
Here's a puzzle that occurred to me this morning: eggs in a basket problem. Not really a tsumego problem in the traditional sense, but it should be interesting to work out anyway. I don't have the answer, but it shouldn't be too hard to work out.