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 is 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.
unkx80: Older discussion follows.
KarlKnechtel, filling in a solution.
Quickest explanation (not at all useful to those for whom this was challenging): " is tesuji.
and
are miai for the kill."
How I'd explain it to someone a bit below my level: "The throw-in at threatens to capture
. Black cannot connect because he is in shortage of liberties -
at
is answered by the capture with
at
. So Black captures, making one eye; but
destroys the other eye."
In easier terms: Black needs two eyes to live. There is already effectively one at a. White must prevent the centre space from becoming an eye. The sacrifice changes the shape of that space, achieving the goal.
This problem is small enough that we can consider all the alternatives:
Playing elsewhere is answered by Black either at a or b. (b claims an extra point of territory, at the expense of one ko threat (at a) for White[1]. So clearly White has to do something.
(Incidentally, the eye around a is an example of what I call a two-space elbow. Not that it really matters.)
One try is to threaten the existing eye; but the capture solves all of Black's problems. (On the other hand, connecting at a would be very bad for Black; White doesn't even have to capture.) This is also a throw-in, but it's the wrong one and doesn't help.
Moving to make the other eye false looks good at first, because Black is in atari to start off with. But all Black has to do is connect. Again, the two eyes are solid.
ProtoDeuteric- Don't the failure sequences prove that "a" in the first diagram below the horizontal rule is not eyespace? Shouldn't the "a" be at the point to the left?
byorgey: You're right, the point marked a in that diagram (the one titled "Reference") is not eyespace; however, that diagram doesn't claim it is. The text next to the diagram refers to a as a ko threat for white if black plays b. It is somewhat confusing having different points mared a and b in different diagrams, though...
How does in the solution destroy the eye? It looks as if a White attempt to capture at
will only start a ko fight...
-- Gregory Price
unkx80: A group requires two eyes (that is, two real eyes) to live. But the position at the end of solution sequence, which is reproduced here, shows a real eye at a and a false eye (which is not an eye) at b. So this group is dead, because White can capture at b followed by capturing at a.
I'm really new to Go (just started learn last week). I'm a little confused about the terminology "played at
". What does that mean? I guess that
is same as the numbered black stone, but what does "at
" mean?
Confused: It's a convenient shorthand for describing simple variations of a diagram. at
means, the situation if Black would have played the second move (
) at the location of the White stone at
. Here Karl Knechtel was writing about the following case.
Jasonred (I think) wrote:
Confused: It's a tesuji, although I'm not sure if that play counts as nakade too. If the definition on the nakade page is correct, I'd say it isn't one.
This is one of the reasons, why I avoid throwing around Japanese terms I don't understand completly or for which there is a good English translation. Some of those concepts are complicated enough on their own, that I don't need to confuse the matter more with the wrong Japanese names.
It's a throw-in
[1] A black play at a is safe from that, as I'd originally thought, but I managed to confuse myself somehow. Thanks SAS. :) Anyway - see Points Or Ko Threats Discussion for more information about that sort of tradeoff.