Yet another old problem

mw42 · Post by **mw42** » Sun Feb 06, 2011 5:45 am

@daal

Bill Spight · Post by **Bill Spight** » Sun Feb 06, 2011 10:34 am

A deceptive problem.

Solution:

Click Here To Show Diagram Code: [go]$$Wc Tempting $$ ------------------- $$ | O . O 1 2 . X O . | $$ | . O O X . X X O O | $$ | O O X X X X X O . | $$ | . X X O X O O O O | $$ | . X X . O O X X X | $$ | . X O O O X . X . | $$ | . X X X O X X X X | $$ | . X O O O O O O O | $$ | . X . . . . . . . | $$ -------------------[/go]

The sente of

is very tempting. If all of the plays were independent, it would be the dominant play.

But the plays are not independent.

The problem is that

makes two eyes, rendering the Black group invulnerable. That reduces the size of the plays in the center and on the left side.

is aji keshi.

Click Here To Show Diagram Code: [go]$$Wc Aji Keshi $$ ------------------- $$ | O . O 1 2 . X O . | $$ | . O O X . X X O O | $$ | O O X X X X X O . | $$ | . X X O X O O O O | $$ | . X X 3 O O X X X | $$ | . X O O O X . X . | $$ | . X X X O X X X X | $$ | . X O O O O O O O | $$ | . X 4 . . . . . . | $$ -------------------[/go]

After

,

is best. But then

gets jigo. The rest is miai.

Click Here To Show Diagram Code: [go]$$Wc Solution $$ ------------------- $$ | O . O . . . X O . | $$ | . O O X . X X O O | $$ | O O X X X X X O . | $$ | . X X O X O O O O | $$ | . X X 1 O O X X X | $$ | . X O O O X . X . | $$ | . X X X O X X X X | $$ | . X O O O O O O O | $$ | . X . . . . . . . | $$ -------------------[/go]

takes away Black's potential eye in the center. Now Black will have to make an eye on the left side. OC, Black will have no trouble doing so, but that makes plays on the left side slightly bigger than in the first diagram. That slight difference is enough for the win.

Click Here To Show Diagram Code: [go]$$Wc Canonical play $$ ------------------- $$ | O . O 2 . . X O . | $$ | . O O X . X X O O | $$ | O O X X X X X O . | $$ | 3 X X O X O O O O | $$ | 4 X X 1 O O X X X | $$ | . X O O O X . X . | $$ | . X X X O X X X X | $$ | . X O O O O O O O | $$ | . X 5 . . . . . . | $$ -------------------[/go]

This diagram shows technically correct play. White gets

to win by 1 point.

Click Here To Show Diagram Code: [go]$$Wc Stubborn resistance $$ ------------------- $$ | O . O . . . X O . | $$ | . O O X . X X O O | $$ | O O X X X X X O . | $$ | 3 X X O X O O O O | $$ | . X X 1 O O X X X | $$ | . X O O O X . X . | $$ | . X X X O X X X X | $$ | . X O O O O O O O | $$ | . X 2 . . . . . . | $$ -------------------[/go]

may be technically inferior, but it puts up a stubborn resistance. White still must avoid the aji keshi on the top side.

Click Here To Show Diagram Code: [go]$$Wc Stubborn resistance, continued $$ ------------------- $$ | O . O . . . X O . | $$ | . O O X . X X O O | $$ | O O X X X X X O . | $$ | 3 X X O X O O O O | $$ | 5 X X 1 O O X X X | $$ | 7 X O O O X . X . | $$ | 9 X X X O X X X X | $$ | 0 X O O O O O O O | $$ | . X 2 4 6 8 . . . | $$ -------------------[/go]

Black continues to resist, but

is sente.

Click Here To Show Diagram Code: [go]$$Bcm10 Death $$ ------------------- $$ | O . O . 2 . X O . | $$ | . O O X . X X O O | $$ | O O X X X X X O . | $$ | O X X O X O O O O | $$ | O X X O O O X X X | $$ | O X O O O X . X . | $$ | O X X X O X X X X | $$ | . X O O O O O O O | $$ | . X X X X X 1 . . | $$ -------------------[/go]

If

pushes again,

kills.

Click Here To Show Diagram Code: [go]$$Bcm10 Stubborn resistance, finis $$ ------------------- $$ | O . O 2 3 . X O . | $$ | . O O X . X X O O | $$ | O O X X X X X O . | $$ | O X X O X O O O O | $$ | O X X O O O X X X | $$ | O X O O O X . X . | $$ | O X X X O X X X X | $$ | 1 X O O O O O O O | $$ | . X X X X X 4 . . | $$ -------------------[/go]

So

makes the eye,

plays sente on the top side, and then

wins by 1 point.

Bill Spight · Post by **Bill Spight** » Sun Feb 06, 2011 12:24 pm

Now to count.

Click Here To Show Diagram Code: [go]$$Wc White to play and win $$ ------------------- $$ | O . O a . . X O . | $$ | . O O X . X X O O | $$ | O O X X X X X O . | $$ | b X X O X O O O O | $$ | . X X c O O X X X | $$ | . X O O O X . X . | $$ | . X X X O X X X X | $$ | . X O O O O O O O | $$ | . X d . . . . . . | $$ -------------------[/go]

To count this, let's first review closed corridors.

Click Here To Show Diagram Code: [go]$$ Closed corridors $$ ---------------------- $$ . O X X X . . . . . . $$ . O u . X . . . . . . $$ . O X X X X . . . . . $$ . O v . . X . . . . . $$ . O X X X X X . . . . $$ . O w . . . X . . . . $$ . O X X X X X . . . . $$ . O . . . . . . . . .[/go]

Corridor "u" is worth 1/2 point for Black, corridor "v" is worth 1 1/4 points for Black,
corridor "w" is worth 2 1/8 points for Black, etc.

A corridor of length, L, is worth L - 2 + 2^(1 - L). A play in the corridor gains 1 - 2^(1 - L).

In the problem diagram, corridor "d" has a length of 7, so we count it as 5 1/64 points for White, and a play there gains 63/64.

Corridor "b" has a length of 6, but it is not independent.

Click Here To Show Diagram Code: [go]$$Bc Black first $$ ------------------- $$ | O . O 2 3 . X O . | $$ | . O O X . X X O O | $$ | O O X X X X X O . | $$ | . X X W X O O O O | $$ | . X X 1 O O X X X | $$ | . X O O O X . X . | $$ | . X X X O X X X X | $$ | . X O O O O O O O | $$ | . X . . . . . . . | $$ -------------------[/go]

First, suppose that Black takes

. Now Black has an eye in the center and one on the top side, and does not need one on the left side. That makes the left side independent, and we can count it as 4 1/32 for Black.

is sente, so we can count the top side as 2. With the center, that gives Black a local count of 8 1/32.

Click Here To Show Diagram Code: [go]$$Wc White first $$ ------------------- $$ | O . O 5 6 . X O . | $$ | . O O X . X X O O | $$ | O O X X X X X O . | $$ | 3 X X O X O O O O | $$ | 4 X X 1 O O X X X | $$ | . X O O O X . X . | $$ | . X X X O X X X X | $$ | . X O O O O O O O | $$ | . X . . . . . . . | $$ -------------------[/go]

Next, suppose that White plays first. After

Black must make an eye on the left side, and plays there gain 1 point.

and

each gain 1 point, so the count remains the same. The same goes for

and

. Black has a local count of 6.

Putting it all together, the Black group in the original diagram has a local count of 7 1/64. A move there gains 1 1/64.

We may count the left side corridor as the average of 4 1/32 and 4, or 4 1/64.

Click Here To Show Diagram Code: [go]$$Wc Solution $$ ------------------- $$ | O . O . . . X O . | $$ | . O O X . X X O O | $$ | O O X X X X X O . | $$ | . X X O X O O O O | $$ | . X X 1 O O X X X | $$ | . X O O O X . X . | $$ | . X X X O X X X X | $$ | . X O O O O O O O | $$ | . X . . . . . . . | $$ -------------------[/go]

In the original position Black has an overall count of 9 1/64 and White also has an overall count of 9 1/64. The board is even.

plays to a count of -1 1/64. At this point the best Black can do is to "round up" to -1, which is a win for White.

Click Here To Show Diagram Code: [go]$$Wc Solution $$ ------------------- $$ | O . O 2 . . X O . | $$ | . O O X . X X O O | $$ | O O X X X X X O . | $$ | 3 X X O X O O O O | $$ | 4 X X 1 O O X X X | $$ | . X O O O X . X . | $$ | . X X X O X X X X | $$ | . X O O O O O O O | $$ | . X 5 . . . . . . | $$ -------------------[/go]

Then

,

, and

each gain 1 point, yielding a count of -1/64.

"rounds down" to a net score of -1.

Click Here To Show Diagram Code: [go]$$Wc Aji keshi $$ ------------------- $$ | O . O 1 2 . X O . | $$ | . O O X . X X O O | $$ | O O X X X X X O . | $$ | . X X O X O O O O | $$ | . X X . O O X X X | $$ | . X O O O X . X . | $$ | . X X X O X X X X | $$ | . X O O O O O O O | $$ | . X . . . . . . . | $$ -------------------[/go]

After

and

the center and left side are independent, with local counts of 1 and 4 1/32, respectively. The overall count for Black is 9 1/32. That makes the overall count of the board 1/64. (

is a losing sente, losing 1/64.) Now the best White can do is to "round down" to 0.

mw42 · Post by **mw42** » Sun Feb 06, 2011 3:56 pm

I think you are a wizard.

Thanks for your great post. After reading it, I feel insecure about my simplistic way of thinking.

Numsgil · Post by **Numsgil** » Wed Feb 09, 2011 12:46 pm

Bill Spight wrote:
If you do decide to write that book I will definitely get me a copy
Many thanks. I do plan to write it.[/hide]

I'd also get one, for the record. The "Elementary Series of Go" book on the endgame is all there is at present, and I feel like there's wide room for improvement in presentation. I'd also say that you shouldn't feel like the problems are boring if they don't have any special tesujis; even handling straightforward whole-board endgame problems is more than most SDKs can handle! No need to complicate things.

Bill Spight · Post by **Bill Spight** » Wed Feb 09, 2011 1:23 pm

mw42 wrote:I think you are a wizard. Thanks for your great post. After reading it, I feel insecure about my simplistic way of thinking.

Thanks for the compliment.

But really, most dan players could see the aji keshi pretty quickly. (Not they would in a real game, since it would rarely matter.) The rest is just arithmetic and experience.

Bill Spight · Post by **Bill Spight** » Wed Feb 09, 2011 1:29 pm

Numsgil wrote:
Bill Spight wrote:
If you do decide to write that book I will definitely get me a copy
Many thanks. I do plan to write it.[/hide]
I'd also get one, for the record. The "Elementary Series of Go" book on the endgame is all there is at present, and I feel like there's wide room for improvement in presentation. I'd also say that you shouldn't feel like the problems are boring if they don't have any special tesujis; even handling straightforward whole-board endgame problems is more than most SDKs can handle! No need to complicate things.

Thank you for the encouragement.

And many thanks for your feedback in these threads. It has helped me figure out what kind of materal to put in the book.

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