In life and death, a group needs two eyes (two real eyes) to live. If a group is totally surrounded and has only one eye (false eyes do not count), then it would be a dead group.
This group has effectively an eye at a, because White playing at b will be self-atari. However, what is not so sure is that whether the spaces at b and c is really an eye.
If White makes an atari at , then
connects and has two real eyes. Note that now White cannot play at either a or b - it is suicide.
here is also a failure. Black
captures the
stone...
... and the result is no different from the previous failure sequence.
The solution is to make a throw-in at . This is a tesuji, or technique, for destroying the eye on the outside.
Notice that the stone can be captured, but...
Suppose captures...
... then turns b into a false eye. The entire group only has an eye at a, there is no eye at b. So Black is dead.
The proof that Black is dead is that can capture...
... followed by . However, there really is no need to play
and
immediately, because White will waste two moves.
In the previous variation, we see that allowing White to play at will kill the group. What if Black plays at
?
This is what we call shortage of liberties. More precisely, Black suffers from a shortage of liberties. The reason is that can capture the four Black stones chain.
This diagram is probably unneccessary, but it clearly shows that Black is dead.
This discussion is at a higher level than the problem and may be skipped by introductory level players on their first reading.
We now turn the problem around and ask how Black can live. There are two ways, each with their pros and cons, but usually the consensus is that one of them is better.
From the previous discussion, it should be easy to deduce that makes two solid real eyes.
However, here makes two eyes and lives as well.
Of course, is a legal move, but
captures and lives.
Similarly, if here,
makes two eyes.
Now we compare both methods.
For the first way to live, Black has two points of territory at a and b. Also, White has no ko-threat.
For the second way to live, Black has three points of territory at a, b and c. However, the disadvantage is that White has one ko-threat at either b or c. On the other hand, creates a cutting-point at d, which may be used for Black's advantage in the later part of the game.
We can see that the pros of the second method significantly outweighs the cons, so the second method is better. Additionally, this is consistent with the observation that each ko-threat is probably worth 1/6 points, then at the bare minimum, we can say that the second method is 1/3 points better than the first method.
For more discussion, please refer to points or ko-threats discussion.