The so called carpenter's square is one of the most complicated of the basic life and death positions. It is simple enough as far as its vital point and status are concerned - it almost always results in a ko. However there are endless ways to set up the ko, and even more ways to defend incorrectly and end up dead anyway.
It is said that anyone who completely understands the carpenter's square must be at least a shodan (tderz: professional shodan is meant here!; as explained in some Kiseido book). My personal interpretation of that gem of wisdom is that I can happily postpone complete study of the shape until I turn shodan, thank you very much.
The discussion is divided into an easy part and a difficult part. More discussion is found at the weak carpenter's square.
The black formation in this diagram is the carpenter's square. The vital point is a: Black can play there to live. White can play there to turn the corner into ko.
The best way to set up the ko depends to a large extent on the number of outside liberties of the black group. If Black has none, White starts by playing on the vital point. Black attaches on the first line, and White stretches in the other direction.
White further reduces the black eye space and after 5, the only way left for Black to live is engage in a ko.
With one outside liberty, White sets up the ko in a different way.
Black 4 in the previous diagram is forced: here's how Black dies by playing atari.
Here's why the basic method fails when there is an outside liberty. Everything goes smoothly for White until 6.
However Black squeezes White for two eyes with 2 after White first captures the ko.
If the black group has two outside liberties, White's best way towards the ko is to hane before playing on the vital 2-2 point.
Black can also turn at 2, in which case White needs to continue at the 3-2 point.
The basic attack only produces a three move approach ko that is harder for White to win.
-- Stefan
See also:
It seems that one as white, one would only need to learn the method that works with 2 outside liberties, as it will also work in 1 and 0 outside liberties. As black, the other variations I suppose have some value in knowing why they don't work (or work as well for white) in the 1 and 2 liberty versions... Or have I missed something?
(Sebastian:) The difference is that the edges have to be better protected. In diagram "Two outside liberties", you need the marked stone . With 0 liberties, you can do without.