Solving life-and-death tsumegos by trial and error

[TIM82]

First of all: to join the chorus, yes, you are doing it right.

Here is an example, with black to move and live.

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O O . . . . .
$$ | X X X O O O . . .
$$ | a b O X X O. . . .
$$ | . X c d e O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O O . . . . .
$$ | X X X O O O . . .
$$ | a b O X X O. . . .
$$ | . X c d e O . . .
$$ ------------------[/go]

To me, on an initial review, a, b, c, d and e are the moves that look like they might be solutions, and so worth reading.

Do stronger players use any rules of thumb, or features of the position, to reduce my list of 5 down to 2 or 3?

I do not understand how, but when I looked at this problem and your a-e, I immediately dismissed all but d without even really thinking about it. Some part of me recognized it as the only reasonable shape, and there was no conscious reading. It might be that I've internalized that problem, but I think there's also some kind of unconscious shape rules-of-thumb happening here.

edit. I'll add that this isn't something I've practiced, it's just the result of having done thousands of life-and-death problems.

A never even entered my mind, and all other but d were ignored very quickly (but were considered for like 0,5 sec). Shows I'm weaker than Shaddy

Regarding rules of thumb:
When looking at the position, the very first thing that stands out is that the status of the two black stones and 1 white stone is unclear. So unless I play a move affecting those stones OR make a good enough sente move elsewhere, white will eat my two stones. If that happens -> no space for 2 eyes. I quite often (in complex problems, sometimes also in games as situation warrants) start by working out the statuses (count of liberties) of intertwined groups. It makes options clearer.

Second thing that stands out to me is the shape thing (Joaz brought it up better than I did while I was writing this).

[billywoods]

(Somehow I remember seeing this exact problem a long time ago and finding it very difficult, and I can't for the life of me now remember why. I guess that's always the way.)

[daal]

Here is an example, with black to move and live.

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O O . . . . .
$$ | X X X O O O . . .
$$ | a b O X X O. . . .
$$ | . X c d e O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O O . . . . .
$$ | X X X O O O . . .
$$ | a b O X X O. . . .
$$ | . X c d e O . . .
$$ ------------------[/go]

To me, on an initial review, a, b, c, d and e are the moves that look like they might be solutions, and so worth reading.

So, I read them all in turn, until I can read one to an irrefutable solution. This is a bit time-consuming, but I get there in the end. Sometimes I dismiss the correct move too early, then have to re-consider it when I cannot get any of the other options to work.

I am not sure if this is practical with bigger or more complicated problems, where there might be twice as many reasonable-looking moves.

Do stronger players use any rules of thumb, or features of the position, to reduce my list of 5 down to 2 or 3?

I really wish that there were rules of thumb, but what the thumb offers is more like suggestions from a drunk buddy than rules. Some things he says may be reasonable, like the first thing he invariably slurs in a slightly too loud voice: "Count your LIBerties!"

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | T O O O . . . . .
$$ | X X X O O O . . .
$$ | T T O X X O. . . .
$$ | . X . S S O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | T O O O . . . . .
$$ | X X X O O O . . .
$$ | T T O X X O. . . .
$$ | . X . S S O . . .
$$ ------------------[/go]

Here I see one string with 3 and another with 2, and in this case, that makes me aware of a key feature of this problem: Liberties are short. His next sentiment is: "Eyes, eyes, it's all about EYes!" This leads me to make my first tests: if I make an eye right away (a or b) can my opponent stop me from making a second? In this case, the answer is yes, because in either case a white d will screw up the second eye. While this piece of advice proved valuable, I must add at this point that it's not as efficient as to eliminate a and b as all of the stronger players did simply by knowing "it won't work." The next piece of advice from my drunk thumb however is questionable: "You gotta CAPture that STone!" While this suggests a wrong move (c)and a right move (d), in more complicated problems, it often just proves to be lousy advice. Somewhat better are the musings of my stoned index finger: "shapes, man, shapes." He sees what Joaz saw (though presumably by legal means):

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . ? ? ? ? ? . . .
$$ | ? ? ? O O ? ? . .
$$ | ? ? O X X O ? . . .
$$ | ? ? . d . O ? . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . ? ? ? ? ? . . .
$$ | ? ? ? O O ? ? . .
$$ | ? ? O X X O ? . . .
$$ | ? ? . d . O ? . .
$$ ------------------[/go]

and also this:

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . ? . . . . . . .
$$ | . . . . . . . . .
$$ | . ? ? ? . . . . .
$$ | ? X X O ? ? . . .
$$ | ? C O X ? ?. . . .
$$ | . X C * ? ? . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . ? . . . . . . .
$$ | . . . . . . . . .
$$ | . ? ? ? . . . . .
$$ | ? X X O ? ? . . .
$$ | ? C O X ? ?. . . .
$$ | . X C * ? ? . . .
$$ ------------------[/go]

In any case, while these digits may provide entertaining advice, success belongs to those who do the footwork.

[peppernut]

Here is an example, with black to move and live.

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O O . . . . .
$$ | X X X O O O . . .
$$ | a b O X X O. . . .
$$ | . X c d e O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O O . . . . .
$$ | X X X O O O . . .
$$ | a b O X X O. . . .
$$ | . X c d e O . . .
$$ ------------------[/go]

To me, on an initial review, a, b, c, d and e are the moves that look like they might be solutions, and so worth reading.

Do stronger players use any rules of thumb, or features of the position, to reduce my list of 5 down to 2 or 3?

I think your mistake is in how you made the list of candidates in the first place. Let me explain ....

I think what you did was saw a block of 6 black stones with some incomplete eye shape, and looked at every move inside the eye shape that makes 2 eyes. Move 'a' creates an eye in the corner. Move 'e' creates an eye at 'd'. All these moves give Black the shape of 2 eyes.

However, that's not the important feature of the problem. The problem is that Black's been cut and he needs both halves to live. One half has only 2 liberties. If Black doesn't make a move to save these stones, White plays atari next move and the group dies.

I think you need to see this before assembling your list of candidate moves. You should see what's going on first before trying to live, because when you place your next stone you need to know what its job is. This way you can avoid considering at 'a' and 'e', because they don't help capture the cutting stone before White kills you.

[PeterPeter]

Here is an example, with black to move and live.

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | f O O O . . . . .
$$ | X X X O O O . . .
$$ | a b O X X O. . . .
$$ | . X c d e O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | f O O O . . . . .
$$ | X X X O O O . . .
$$ | a b O X X O. . . .
$$ | . X c d e O . . .
$$ ------------------[/go]

Now, since you have solved this problem, let me ask you a few questions.

What are the key features of this position?

Black is very cramped, and his 2 stones on the right are in danger.

When you were solving the position, which move looked best at first? Why? What was wrong with it?

B and C were the first ones I looked at, as killing white's stone would both create space, and help connect. Both of them failed to white playing at D.

As you worked on this problem, did you pick up clues about where to make your first move?

Only after exhausting B and C, and quickly ruling out E, did I seriously consider D. On a first review, D looked like it reduced eyespace. It took a while to see that it was the only move to save the 2 stones on the right.

After Black's initial correct play, what is White's best play? (From an endgame point of view, OC. )

White should play at F.

After Black's initial correct play, are there any White moves that are obviously useless? Why?

White cannot save his stone, so any of ABC would be uselss, and E does nothing.

Having solved the problem, can you go through the main variations quickly?

Yes, playing D simplifies it a lot.

Are you confident that you have a reply for every White response?

Yes.

[Bill Spight]

Great, PeterPeter!

Here is an example, with black to move and live.

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | f O O O . . . . .
$$ | X X X O O O . . .
$$ | a b O X X O. . . .
$$ | . X c d e O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | f O O O . . . . .
$$ | X X X O O O . . .
$$ | a b O X X O. . . .
$$ | . X c d e O . . .
$$ ------------------[/go]

Now, since you have solved this problem, let me ask you a few questions.

What are the key features of this position?

Black is very cramped, and his 2 stones on the right are in danger.

You can be more specific. As others have pointed out, taking note of the exact number of dame can be helpful. (I use the term, dame, because liberty is ambiguous.) Also, in terms of life and death, being cramped is a problem because there is not much space for Black to make two eyes.

So where are Black's potential eyes?

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O O . . . . .
$$ | X X X O O O . . .
$$ | . . W X X O. . . .
$$ | a B . b . O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O O . . . . .
$$ | X X X O O O . . .
$$ | . . W X X O. . . .
$$ | a B . b . O . . .
$$ ------------------[/go]

It looks like there is an eye at "a", thanks in part to . Where is a second possible eye?

A brief inspection suggests that the second eye should include . That being the case, Black needs to occupy or control point "b", else the eye will be false. Just thinking about eyes suggests "b" as the first move.

Does that move save the two Black stones and capture the stone, forming the second eye?

$$B Main line
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O O . . . . .
$$ | X X X O O O . . .
$$ | . . W X X O. . . .
$$ | . B 3 1 2 O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B Main line
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O O . . . . .
$$ | X X X O O O . . .
$$ | . . W X X O. . . .
$$ | . B 3 1 2 O . . .
$$ ------------------[/go]

Yes, thanks to .

Because of its significance in forming the eye in the corner and in saving the two Black stones, I would say that is a key feature of this position.

When you were solving the position, which move looked best at first? Why? What was wrong with it?

B and C were the first ones I looked at, as killing white's stone would both create space, and help connect. Both of them failed to white playing at D.

Very often, when the opponent has a good play on a point, we also have a good play on that point. (In fact, it is a go proverb. ) If I had tried "c" first and discovered White's counter at "d", "d" would have been my next try.

After Black's initial correct play, what is White's best play? (From an endgame point of view, OC. )

White should play at F.

Correct. Although correct endgame is not part of a life and death problem, I think that it can be a good exercise to figure out the best endgame play. In a real game, it is part of the problem.

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | 2 O O O . . . . .
$$ | X X X O O O . . .
$$ | . . W X X O. . . .
$$ | . B . 1 . O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | 2 O O O . . . . .
$$ | X X X O O O . . .
$$ | . . W X X O. . . .
$$ | . B . 1 . O . . .
$$ ------------------[/go]

Bonus question. Where is Black's reply?

Having solved the problem, can you go through the main variations quickly?

Yes, playing D simplifies it a lot.

The lines starting with "c" and "b" are also main lines, IMO.

Are you confident that you have a reply for every White response?

Yes.

¡Bueno!

[Bill Spight]

Here is a more difficult problem, as the Black 1-2 stone is missing.

(;ST[2]AP[GOWrite:2.2.21]GM[1]CA[ISO8859-1]FF[4]SZ[19]GN[ ]PB[ ]FG[259:]PM[2]AB[ca][cb][cc][bd][be]AW[cd][ce][dc][db][fb][cf][bf][af][bc][fb]PW[ ]
(
;B[bb]
(
;W[ae]
;B[ac]
;W[ad]
;B[aa]
;W[bc]
;B[bd]
)
(
;W[ad]
;B[ac]
;W[ae]
;B[aa]
;W[bc]
;B[bd]
)

(
;W[ac]
;B[ab]
;W[ae]
;B[ad]
)

(
;W[ab]
;B[ac]
;W[ae]
(
;B[aa]
)
(
;W[aa]
;B[ba]
;W[ad]
)

)

)
(
;B[ac]
;W[bb]
(
;B[ab]
;W[ad]
;B[ba]
;W[bc]
;B[bb]
;W[ae]
)
(
;B[ba]
;W[ad]
;B[ab]
;W[bc]
;B[bb]
;W[ae]
)

)

)

Where can Black make two eyes?

Enjoy!

[PeterPeter]

So where are Black's potential eyes?

It looks like there is an eye at "a", thanks in part to . Where is a second possible eye?

A brief inspection suggests that the second eye should include . That being the case, Black needs to occupy or control point "b", else the eye will be false. Just thinking about eyes suggests "b" as the first move.

I thought this might be a good way to approach problems: predict where the eyes will be, then work from there. However, predicting likely eyes after the connections, captures, snapbacks etc have played out is not easy, with the added complication that both eyes must be genuine. It is not very intuitive.

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O O . . . . .
$$ | X X X O O O . . .
$$ | . . W X X O . . .
$$ | C X . . . O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O O . . . . .
$$ | X X X O O O . . .
$$ | . . W X X O . . .
$$ | C X . . . O . . .
$$ ------------------[/go]

In this problem, you rightly saw that the 1-1 corner point, and the enemy stone you had to capture, would become eyes. This seems like a good starting point for similar problems.

[billywoods]

predicting likely eyes after the connections, captures, snapbacks etc have played out is not easy, with the added complication that both eyes must be genuine. It is not very intuitive.

I think Bill did something subtly different from predicting likely eyes. Let's remove all the enemy stones and just look at the eyespace:

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | C C C X X . . . .
$$ | C X C C . . . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | C C C X X . . . .
$$ | C X C C . . . . .
$$ ------------------[/go]

Here's a couple of ways of making eyes in it:

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | B C C X X . . . .
$$ | C X C B . . . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | B C C X X . . . .
$$ | C X C B . . . . .
$$ ------------------[/go]

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | C B C X X . . . .
$$ | C X B C . . . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | C B C X X . . . .
$$ | C X B C . . . . .
$$ ------------------[/go]

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | C B C X X . . . .
$$ | B X C B . . . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | C B C X X . . . .
$$ | B X C B . . . . .
$$ ------------------[/go]

It should be fairly easy to see that all of the sensible formations you can make with two eyes involve this point:

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | . . C X X . . . .
$$ | . X . . . . . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | . . C X X . . . .
$$ | . X . . . . . . .
$$ ------------------[/go]

Bill's slight leap of logic went: "so I need to capture the stone at circle. Ah, but not all captured stones become eyes! It's a real eye for black if and only if black controls all or all but one of the corners":

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X Y X T . . . . .
$$ | . . C X X . . . .
$$ | . Y . T . . . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X Y X T . . . . .
$$ | . . C X X . . . .
$$ | . Y . T . . . . .
$$ ------------------[/go]

Finally, let's overlay this onto the real diagram:

$$B
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O O . . . . .
$$ | X Y X Q O O . . .
$$ | . . W X X O . . .
$$ | . Y . T . O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O O . . . . .
$$ | X Y X Q O O . . .
$$ | . . W X X O . . .
$$ | . Y . T . O . . .
$$ ------------------[/go]

The marked point (where there is a white stone) must become an eye. So black must capture the white stone. Additionally, for it to become an eye, black must occupy at least three of the four triangle points (aka for white to make that eye false, white must occupy the remaining triangle point). This immediately tells you the only moves worth considering for both black and white: the point you both really want to occupy is the remaining triangle point, so try it first. But if you happened to try another point first in an attempt to capture the white stone, white's refutation probably starts by occupying that triangle point too.

I also liked peppernut's explanation: black's been cut. As you can see, there are lots of ways to think about the same problem.

[PeterPeter]

It should be fairly easy to see that all of the formations you can make with two eyes involve this point:

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | . . C X X . . . .
$$ | . X . . . . . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | . . C X X . . . .
$$ | . X . . . . . . .
$$ ------------------[/go]

I am not sure about this. Is this not an equally valid way of getting 2 eyes?

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | C C B X X . . . .
$$ | C X C B . . . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | C C B X X . . . .
$$ | C X C B . . . . .
$$ ------------------[/go]

I think a bigger clue is the fact that a white stone starts off occupying that point.

The marked point (where there is a white stone) must become an eye. So black must capture the white stone. Additionally, for it to become an eye, black must occupy at least three of the four triangle points (aka for white to make that eye false, white must occupy the remaining triangle point). This immediately tells you the only moves worth considering for both black and white: the point you both really want to occupy is the remaining triangle point, so try it first.

That could be a really good clue: look at the diagonal points of each enemy stone in your eyespace area. I like it!

[Bill Spight]

That could be a really good clue: look at the diagonal points of each enemy stone in your eyespace area. I like it!

Actually, that's not a bad heuristic, even on a larger scale. The point is, OC, that you are attacking the stone or stones. Points that are diagonally related to defensive stones are often important. And it is not just the adjacent diagonals. For instance, after a one space jump from a single stone, the two stones are related along diagonal lines. (There is another go proverb that says that a one space jump is for defense, a knight's move for attack.)

Getting back to life and death problems, diagonals to your owns stones can be important, as well, because of eye potential.

[billywoods]

It should be fairly easy to see that all of the formations you can make with two eyes involve this point:

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | . . C X X . . . .
$$ | . X . . . . . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | . . C X X . . . .
$$ | . X . . . . . . .
$$ ------------------[/go]

I am not sure about this. Is this not an equally valid way of getting 2 eyes?

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | C C B X X . . . .
$$ | C X C B . . . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | X X X . . . . . .
$$ | C C B X X . . . .
$$ | C X C B . . . . .
$$ ------------------[/go]

I think a bigger clue is the fact that a white stone starts off occupying that point.

Yes, fair point. The main thing to take away from what I said was: if you don't capture the white stone, you haven't got space for two eyes. You're right that I phrased it badly.

[Bill Spight]

Some comments on my problem.

$$B Black to play
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | . . W X X O. . . .
$$ | . . . . . O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B Black to play
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | . . W X X O. . . .
$$ | . . . . . O . . .
$$ ------------------[/go]

Where can Black make eyes?

That is not so obvious. We know from the previous problem that White can prevent the point from becoming an eye. That does not leave much space.

Those who have seen examples of the smallest groups with two eyes may see the point.

$$B Main line
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | . 1 W X X O. . . .
$$ | 5 . 3 4 2 O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B Main line
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | . 1 W X X O. . . .
$$ | 5 . 3 4 2 O . . .
$$ ------------------[/go]

- capture . Then takes away the potential eye at , but makes two eyes in the corner.

$$B Main line (continued)
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | . X 6 B X O. . . .
$$ | X a X O O O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B Main line (continued)
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | . X 6 B X O. . . .
$$ | X a X O O O . . .
$$ ------------------[/go]

takes the two Black stones and threatens to make the eye at "a" false, but takes back at , retaining control of the point at .

$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | . B a X . O. . . .
$$ | B . B O O O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | . B a X . O. . . .
$$ | B . B O O O . . .
$$ ------------------[/go]

Note that Black's first three moves are diagonally related. (The last one is, too, but the point is the capture, not the diagonal relationship.) That underscores the point that diagonals are important in forming eyes.

The fact that it takes White two moves to eliminate the potential eye at "a" gives Black time to make the eyes.

$$B Failure
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | 5 2 W X X O. . . .
$$ | . 3 1 4 . O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B Failure
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | 5 2 W X X O. . . .
$$ | . 3 1 4 . O . . .
$$ ------------------[/go]

The other atari, in this diagram, gives White the opportunity to play on the main diagonal, spoiling the eye shape. ( and are interchangeable.)

$$B Failure (continued)
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | X 7 6 X X O. . . .
$$ | . X X O 8 O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B Failure (continued)
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | X 7 6 X X O. . . .
$$ | . X X O 8 O . . .
$$ ------------------[/go]

After Black threatens to make a different set of eyes. The throw-in of , on a different diagonal, takes away the potential eye at , and then makes the eye at false.

And finally:

$$B White error
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | . 1 W X X O. . . .
$$ | . 3 2 . . O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B White error
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | . 1 W X X O. . . .
$$ | . 3 2 . . O . . .
$$ ------------------[/go]

here also prevents Black from playing on the main diagonal. Unfortunately for White, it allows Black to make an eye at .

Gee, when you explain everything, there is a lot to this little problem, isn't there? I wonder if it is 10 kyu or so.

[PeterPeter]

$$B Black to play
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | . . O X X O. . . .
$$ | a b . . . O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B Black to play
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | . . O X X O. . . .
$$ | a b . . . O . . .
$$ ------------------[/go]

Interesting that having a black stone on A is essential here, but was not even a candidate move when there was a black stone starting on B. No wonder that I sometimes stare at these puzzles for a while, not knowing where to start.

[Bill Spight]

$$B Black to play
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | . . O X X O. . . .
$$ | a b . . . O . . .
$$ ------------------

Click Here To Show Diagram Code

[go]$$B Black to play
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .
$$ | . O O . . . . . .
$$ | X X X O O O . . .
$$ | . . O X X O. . . .
$$ | a b . . . O . . .
$$ ------------------[/go]

Interesting that having a black stone on A is essential here, but was not even a candidate move when there was a black stone starting on B. No wonder that I sometimes stare at these puzzles for a while, not knowing where to start.

In such a tight space, if Black to live occupies either "a" or "b", the other one almost has to be an eye, or part of one.

My main point in talking about these relationships is not that I think you should know them before tackling these problems, but that you can learn them from doing the problems, or from playing games. Especially if you review them afterwards.