Some endgame fun

[topazg]

Following the discussions of useful endgame analysis, I made this post in response to a post by Numsgil, and thought that, actually, some fun and very practical question exercise could be made from it. So, what's the value of "a" in the following positions:

$$Wcm1 Position A
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

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[go]$$Wcm1 Position A
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

$$Wcm1 Position B
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . W . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

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[go]$$Wcm1 Position B
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . W . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

$$Wcm1 Position C
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O B X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

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[go]$$Wcm1 Position C
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O B X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

$$Wcm1 Position D
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . C . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

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[go]$$Wcm1 Position D
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . C . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

$$Wcm1 Position E
$$ | . . O X . X X X X X X C . . . . . . . |
$$ | . . O X X X O O O O B C . X . . . . . |
$$ | . . . O O O O . . W O X . . . , . . . |
$$ | . . . . . . . . . . O X . C . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

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[go]$$Wcm1 Position E
$$ | . . O X . X X X X X X C . . . . . . . |
$$ | . . O X X X O O O O B C . X . . . . . |
$$ | . . . O O O O . . W O X . . . , . . . |
$$ | . . . . . . . . . . O X . C . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

$$Wcm1 Position F
$$ | . . O X . X X X X X X C . . . . . . . |
$$ | . . O X X X O O O O B C . X . . . . . |
$$ | . . . O O O O . . . O X . . . , . . . |
$$ | . . . . . . . . . . O X . C . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

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[go]$$Wcm1 Position F
$$ | . . O X . X X X X X X C . . . . . . . |
$$ | . . O X X X O O O O B C . X . . . . . |
$$ | . . . O O O O . . . O X . . . , . . . |
$$ | . . . . . . . . . . O X . C . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

[gaius]

My "solutions" to your problems (assuming that I did not mis-count):

So, fun final positions to think about and evaluate:

$$Wcm1
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O B X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O B X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

7 points sente for black?

$$Wcm1
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . C . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

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[go]$$Wcm1
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . C . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

I count 14 (!) points in gote

[daal]

Before attempting to answer the problems Topazg posed, I would like to share with you a sentence from Winning Go that for many is self-evident, but for me was fog-lifting:

"The reader will notice that as long as the initial X's exceed the territories won and lost, their amount is irrelevant and only provide a handy basis for comparisons."

The X's that the authors refer to, are the ones you use to count the territory in an endgame problem.

$$Super simplified example
$$ . . , . . . O X , . . . |
$$ . O . . . . O X . . . . |
$$ . . . . O . O X . X . . |
$$ . . . . . . a b . . . . |
$$ ------------------------

Click Here To Show Diagram Code

[go]$$Super simplified example
$$ . . , . . . O X , . . . |
$$ . O . . . . O X . . . . |
$$ . . . . O . O X . X . . |
$$ . . . . . . a b . . . . |
$$ ------------------------[/go]

For example, when calculating the value of a black move at "a" vs a white move at "b," we play out the sequences and then mark territory with X's. It doesn't matter if we do it like this:

$$white 12 black 11
$$ . . , . . . O X , . . . |
$$ . O x x x x O X x x x x |
$$ . x x x O x O X x X x x |
$$ . x x x x 2 1 3 x x x x |
$$ ------------------------

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[go]$$white 12 black 11
$$ . . , . . . O X , . . . |
$$ . O x x x x O X x x x x |
$$ . x x x O x O X x X x x |
$$ . x x x x 2 1 3 x x x x |
$$ ------------------------[/go]

or like this:

$$white 6 black 7
$$ . . , . . . O X , . . . |
$$ . O . . . . O X . . . . |
$$ . . x x O x O X x X x x |
$$ . . x x x 2 1 3 x x x x |
$$ ------------------------

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[go]$$white 6 black 7
$$ . . , . . . O X , . . . |
$$ . O . . . . O X . . . . |
$$ . . x x O x O X x X x x |
$$ . . x x x 2 1 3 x x x x |
$$ ------------------------[/go]

As long as you use the same boundaries when you count the other scenario of white playing at b. So in both cases we get a white difference of 1 + a black difference of 1 = 2 pts

$$W white 13 black 10
$$ . . , . . . O X , . . . |
$$ . O x x x x O X x x x x |
$$ . x x x O x O X x X x x |
$$ . x x x x x 3 1 2 x x x |
$$ ------------------------

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[go]$$W white 13 black 10
$$ . . , . . . O X , . . . |
$$ . O x x x x O X x x x x |
$$ . x x x O x O X x X x x |
$$ . x x x x x 3 1 2 x x x |
$$ ------------------------[/go]

$$W white 7 black 6
$$ . . , . . . O X , . . . |
$$ . O . . . . O X . . . . |
$$ . . x x O x O X x X x x |
$$ . . x x x x 3 1 2 x x x |
$$ ------------------------

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[go]$$W white 7 black 6
$$ . . , . . . O X , . . . |
$$ . O . . . . O X . . . . |
$$ . . x x O x O X x X x x |
$$ . . x x x x 3 1 2 x x x |
$$ ------------------------[/go]

[gaius]

^ Following up on that post, you can simplify the procedure even further:

- Step 1: determine the results if white plays first and if black plays first, ie.

$$
$$ . . , . . . O X , . . . |
$$ . O . . . . O X . . . . |
$$ . . . . O . O X . X . . |
$$ . . . . . 2 1 3 . . . . |
$$ ------------------------

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[go]$$
$$ . . , . . . O X , . . . |
$$ . O . . . . O X . . . . |
$$ . . . . O . O X . X . . |
$$ . . . . . 2 1 3 . . . . |
$$ ------------------------[/go]

$$Wc
$$ . . , . . . O X , . . . |
$$ . O . . . . O X . . . . |
$$ . . . . O . O X . X . . |
$$ . . . . . . 3 1 2 . . . |
$$ ------------------------

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[go]$$Wc
$$ . . , . . . O X , . . . |
$$ . O . . . . O X . . . . |
$$ . . . . O . O X . X . . |
$$ . . . . . . 3 1 2 . . . |
$$ ------------------------[/go]

- Step 2: fix these results as "mental images". Then count the "plus points" and "minus points" for both players. In the example above, if black plays first, he gains one point ('a') and white loses one point ('b'). Thus, the move is worth two points.

$$Wc
$$ . . , . . . O X , . . . |
$$ . O . . . . O X . . . . |
$$ . . . . O . O X . X . . |
$$ . . . . . b . . a . . . |
$$ ------------------------

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[go]$$Wc
$$ . . , . . . O X , . . . |
$$ . O . . . . O X . . . . |
$$ . . . . O . O X . X . . |
$$ . . . . . b . . a . . . |
$$ ------------------------[/go]

I think that this way of counting is by far the fastest and easiest way there is. Probably most strong players count like this intuitively, but I think it's still worth pointing it out explicitly.

Now, as a more interesting example, let's try one of topazg's problems. Try to see how fast you can determine the value of 'a':

$$Wcm1
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Solution:

'a' is worth 14 points gote. How did I count this? Simple. If black captures, white gets minus points at 'a' and black gets plus points at 'b' (and two for ). Add everything up, and you get 14 points in gote for black. This takes me (probably) about 20 seconds to determine, plus another 10-20 seconds to verify, so it terms of time management, it's very feasible to do this whilst in byo-yomi. I think that no other method is anywhere near as fast.

$$Wcm1
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . a a a . @ X b b . . . . |
$$ | . . . . . . . a a a a . b b b . . . . |
$$ ---------------------------------------

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[go]$$Wcm1
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . a a a . @ X b b . . . . |
$$ | . . . . . . . a a a a . b b b . . . . |
$$ ---------------------------------------[/go]

ADDENDUM: this analysis is based on the following possibilities (assuming that in in diag. 1 and in diag. 2 are gote):

$$Wcm1 white plays first
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . . . . 1 O X 4 8 . . . . |
$$ | . . . . . . . . . . . 7 3 5 6 . . . . |
$$ ---------------------------------------

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[go]$$Wcm1 white plays first
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . . . . 1 O X 4 8 . . . . |
$$ | . . . . . . . . . . . 7 3 5 6 . . . . |
$$ ---------------------------------------[/go]

$$Bcm1 black plays first
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . 0 6 2 1 O X . . . . . . |
$$ | . . . . . . . 8 7 5 9 3 . . . . . . . |
$$ ---------------------------------------

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[go]$$Bcm1 black plays first
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . 0 6 2 1 O X . . . . . . |
$$ | . . . . . . . 8 7 5 9 3 . . . . . . . |
$$ ---------------------------------------[/go]

[daal]

- Step 2: fix these results as "mental images".

I think that this way of counting is by far the fastest and easiest way there is. Probably most strong players count like this intuitively, but I think it's still worth pointing it out explicitly.
fast.

For "stronger" players I'm sure, but even in my extremely simple example, I have difficulty comparing the two mental images. When just looking at the diagram below, I even find it hard to see that "a" and "b" are the points gained.

$$Super simplified example
$$ . . , . . . O X , . . . |
$$ . O . . . . O X . . . . |
$$ . . . . O . O X . X . . |
$$ . . . . . a . . b . . . |
$$ ------------------------

Click Here To Show Diagram Code

[go]$$Super simplified example
$$ . . , . . . O X , . . . |
$$ . O . . . . O X . . . . |
$$ . . . . O . O X . X . . |
$$ . . . . . a . . b . . . |
$$ ------------------------[/go]

If more stones are involved, keeping the mental images in my head seems utterly impossible. Keeping two numbers in my head on the other hand is not so hard.

[daal]

So, what's the value of "a" in the following positions:

$$Wcm1 Position A
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 Position A
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

$$Wcm1 For counting purposes, we can assume 2 and 3 since either are gote. w13 b4
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O x x x O X . . . , . . . |
$$ | . . . . . . . x x x O X . X . . . . . |
$$ | . . . . . . . x x x 1 O X x x . . . . |
$$ | . . . . . . . x x x x 3 2 x x . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 For counting purposes, we can assume 2 and 3 since either are gote. w13 b4
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O x x x O X . . . , . . . |
$$ | . . . . . . . x x x O X . X . . . . . |
$$ | . . . . . . . x x x 1 O X x x . . . . |
$$ | . . . . . . . x x x x 3 2 x x . . . . |
$$ ---------------------------------------[/go]

$$cm1
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . 2 1 O X . . . . . . |
$$ | . . . . . . . . . . . 3 . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$cm1
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . 2 1 O X . . . . . . |
$$ | . . . . . . . . . . . 3 . . . . . . . |
$$ ---------------------------------------[/go]

$$cm1 Will this happen? w6 b7
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O x x x O X . . . , . . . |
$$ | . . . . . . . x x x O X . X . . . . . |
$$ | . . . . . . . 6 2 O X x X x x . . . . |
$$ | . . . . . . . 4 3 1 5 X x x x . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$cm1 Will this happen? w6 b7
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O x x x O X . . . , . . . |
$$ | . . . . . . . x x x O X . X . . . . . |
$$ | . . . . . . . 6 2 O X x X x x . . . . |
$$ | . . . . . . . 4 3 1 5 X x x x . . . . |
$$ ---------------------------------------[/go]

Difference: 10 points.

Position B

5 points

[topazg]

@daal

Evaluation of endgame is not just the points, but the sente / reverse sente / gote aspect of it is vital. 10 points of which?

[daal]

@daal

Evaluation of endgame is not just the points, but the sente / reverse sente / gote aspect of it is vital. 10 points of which?

I really don't know. The first part ends in gote for black:

$$cm1
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . 2 1 O X . . . . . . |
$$ | . . . . . . . . . . . 3 . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$cm1
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . 2 1 O X . . . . . . |
$$ | . . . . . . . . . . . 3 . . . . . . . |
$$ ---------------------------------------[/go]

But since white's response would also be gote, the second part would be black's priviledge, i.e., sente for black.

$$cm1 Will this happen? w6 b7
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O x x x O X . . . , . . . |
$$ | . . . . . . . x x x O X . X . . . . . |
$$ | . . . . . . . 6 2 O X x X x x . . . . |
$$ | . . . . . . . 4 3 1 5 X x x x . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$cm1 Will this happen? w6 b7
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O x x x O X . . . , . . . |
$$ | . . . . . . . x x x O X . X . . . . . |
$$ | . . . . . . . 6 2 O X x X x x . . . . |
$$ | . . . . . . . 4 3 1 5 X x x x . . . . |
$$ ---------------------------------------[/go]

I don't know what to call it.

[topazg]

I really don't know. The first part ends in gote for black:

$$cm1
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . 2 1 O X . . . . . . |
$$ | . . . . . . . . . . . 3 . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$cm1
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . 2 1 O X . . . . . . |
$$ | . . . . . . . . . . . 3 . . . . . . . |
$$ ---------------------------------------[/go]

But since white's response would also be gote, the second part would be black's priviledge, i.e., sente for black.

$$cm1 Will this happen? w6 b7
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O x x x O X . . . , . . . |
$$ | . . . . . . . x x x O X . X . . . . . |
$$ | . . . . . . . 6 2 O X x X x x . . . . |
$$ | . . . . . . . 4 3 1 5 X x x x . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$cm1 Will this happen? w6 b7
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O x x x O X . . . , . . . |
$$ | . . . . . . . x x x O X . X . . . . . |
$$ | . . . . . . . 6 2 O X x X x x . . . . |
$$ | . . . . . . . 4 3 1 5 X x x x . . . . |
$$ ---------------------------------------[/go]

I don't know what to call it.

Because the first stage is gote for both, it's gote. As Black gets the second part in sente, it's "incorporated" into the reading of the first part as a given extension to the sequence, but it doesn't change the fact that the first part finishes in gote.

[Bill Spight]

Numsgil was interested in practical application, so I will direct my remarks to that.

$$Wcm1 Position B
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . W . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 Position B
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . W . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

As I said, memory and experience help. I happen to know that a play at "a" by either player gains 3 points.

[Bill Spight]

I happen to know this one, too. But let's pretend that I don't.

$$Wcm1 Position A
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 Position A
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

White first is easy.

$$Wcm1 Position A, White first
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . O O X . . . . . . |
$$ | . . . . . . . . . . . O X . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 Position A, White first
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . O O X . . . . . . |
$$ | . . . . . . . . . . . O X . . . . . . |
$$ ---------------------------------------[/go]

Now, Black first:

$$Wcm1 Position A, Black first. White reply. Black has 1 White prisoner
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . O X . X . . . . . . |
$$ | . . . . . . . . . O . X . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 Position A, Black first. White reply. Black has 1 White prisoner
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . O X . X . . . . . . |
$$ | . . . . . . . . . O . X . . . . . . . |
$$ ---------------------------------------[/go]

Now, I happen to know that White has a sente here, so let's show the result.

$$Wcm1 Position A, Black first. White reply, with follow-up. Black has 1 White prisoner
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . W X B X . . . . . . |
$$ | . . . . . . . . . W W X C . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 Position A, Black first. White reply, with follow-up. Black has 1 White prisoner
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . W X B X . . . . . . |
$$ | . . . . . . . . . W W X C . . . . . . |
$$ ---------------------------------------[/go]

If we compare this with the position when White plays first, we see that Black has two points more, one for the marked point and one for the stone captured where the marked stone is, and White has three points less, where the marked stones are. So if this is sente for Black, it is a 5 point sente. That is, the reverse sente by White gains 5 points. (And this diagram then shows the value of the original position. )

$$Wcm1 Position A, Black first, with follower. Black has 1 White prisoner
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . W W O X C X . . . . . . |
$$ | . . . . . . . W B X X X . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 Position A, Black first, with follower. Black has 1 White prisoner
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . W W O X C X . . . . . . |
$$ | . . . . . . . W B X X X . . . . . . . |
$$ ---------------------------------------[/go]

Assuming that White cannot fight the ko, after Black plays first she has a sente follow-up, shown here. If we compare this position with the one when White responds, we see that Black has 1 point more on the marked point and White has 4 points less where the marked stones are, for a difference of 5 points. So this is a 5 point sente.

So what do we have? A 5 point sente which threatens a 5 point sente? Or a gote that gains 5 points?

Actually, it is ambiguous. (See http://senseis.xmp.net/?AmbiguousPosition ). Anyway, either player gains 5 points.

[flOvermind]

I don't know what to call it.

Correct.

(See Bill's previous post about it being ambiguous...)

[Bill Spight]

The next one I didn't know beforehand.

$$Wcm1 Position C
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O B X . X . . . . . |
$$ | . . . O O O O . . b O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 Position C
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O B X . X . . . . . |
$$ | . . . O O O O . . b O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Does White's weakness at "b" matter?

$$Wcm1 White first
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O B X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . O O X . . . . . . |
$$ | . . . . . . . . . . . O X . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 White first
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O B X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . . O O X . . . . . . |
$$ | . . . . . . . . . . . O X . . . . . . |
$$ ---------------------------------------[/go]

White first is familiar.

$$Wcm1 Black first, White reply. Black has 1 White prisoner.
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O B X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . O X . X . . . . . . |
$$ | . . . . . . . . . O . X . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 Black first, White reply. Black has 1 White prisoner.
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O B X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . X . . . . . |
$$ | . . . . . . . . . O X . X . . . . . . |
$$ | . . . . . . . . . O . X . . . . . . . |
$$ ---------------------------------------[/go]

Here we see that White's defect is telling. It is Black who has the sente. Let's show that.

$$Wcm1 Black first, White reply, with follow-up. Black has 1 White prisoner.
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O B X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . O O X . X . . . . . |
$$ | . . . . . . . . . O X . X . . . . . . |
$$ | . . . . . . . . . O X X . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 Black first, White reply, with follow-up. Black has 1 White prisoner.
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O B X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . O O X . X . . . . . |
$$ | . . . . . . . . . O X . X . . . . . . |
$$ | . . . . . . . . . O X X . . . . . . . |
$$ ---------------------------------------[/go]

Now, since we know that without the defect this is on the borderline of sente, with the defect it is certainly sente. Now if we compare diagrams, we see that there is a difference of 7 points. So this is a 7 point sente, and the reverse sente gains 7 points.

Note:

$$Wcm1 Black first, White reply, with realistic follow-up. Black has 1 White prisoner.
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O B X . X . . . . . |
$$ | . . . O O O O . . O O X . . . , . . . |
$$ | . . . . . . . O X X O X . X . . . . . |
$$ | . . . . . . . . O O X . X . . . . . . |
$$ | . . . . . . . . . O X X . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 Black first, White reply, with realistic follow-up. Black has 1 White prisoner.
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O B X . X . . . . . |
$$ | . . . O O O O . . O O X . . . , . . . |
$$ | . . . . . . . O X X O X . X . . . . . |
$$ | . . . . . . . . O O X . X . . . . . . |
$$ | . . . . . . . . . O X X . . . . . . . |
$$ ---------------------------------------[/go]

In a real game the follow-up position will look like this, so that Black has a ko threat. But as far as the count is concerned, it is all same same. I do not have to concern myself with the exact details of the play.

[Bill Spight]

The next position is problematic.

$$Wcm1 Position D
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . C . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 Position D
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . C . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

From what we already know, Black first is easy:

$$Wcm1 Black first
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . O O O X . X . . . . . . |
$$ | . . . . . . . O X X X X . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 Black first
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . O O O X . X . . . . . . |
$$ | . . . . . . . O X X X X . . . . . . . |
$$ ---------------------------------------[/go]

But what about White first?

$$Wcm1 White first
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . . . . O O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 White first
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . . . . O O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Suppose that Black replies.

$$Wcm1 White first, Black reply, with follower
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . . . . O O X . . . . . . |
$$ | . . . . . . . . . . O X X . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 White first, Black reply, with follower
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . . . . O O X . . . . . . |
$$ | . . . . . . . . . . O X X . . . . . . |
$$ ---------------------------------------[/go]

So if this is sente, it is a 9 point sente, as a comparison of the diagrams shows.

But is it sente? Does White threaten to gain more than 9 points if Black does not reply?

$$Wcm1 White first. White follower.
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X X O . . O . . |
$$ | . . . . . . . . . . O O X O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 White first. White follower.
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X X O . . O . . |
$$ | . . . . . . . . . . O O X O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ ---------------------------------------[/go]

Assuming that White lives, this is a humungous threat.

However, . . .

$$Wcm1 White first, Black reply, White follower.
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , O . . |
$$ | . . . . . . . . . . O X X O . . . . . |
$$ | . . . . . . . . . . O O X . . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 White first, Black reply, White follower.
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , O . . |
$$ | . . . . . . . . . . O X X O . . . . . |
$$ | . . . . . . . . . . O O X . . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ ---------------------------------------[/go]

That assumes that Black's reply secures the corner, and that is questionable.

$$Wcm1 White first, Black reply.
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . W . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . B . . . . . |
$$ | . . . . . . . . . . O O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 White first, Black reply.
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . W . . |
$$ | . . . O O O O . . , O X . . . , . . . |
$$ | . . . . . . . . . . O X . B . . . . . |
$$ | . . . . . . . . . . O O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Even this reply, which would make an 11 point sente, is not certain to secure the corner.

$$Wcm1 A different play?
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , W . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------

Click Here To Show Diagram Code

[go]$$Wcm1 A different play?
$$ | . . O X . X X X X X X X . . . . . . . |
$$ | . . O X X X O O O O O X . X . . . . . |
$$ | . . . O O O O . . , O X . . . , W . . |
$$ | . . . . . . . . . . O X . . . . . . . |
$$ | . . . . . . . . . . a O X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Maybe "a" is not even the right local play.

When Black has no defect, we can treat the corner independently, but not now.

Note: Books on the endgame usually contain a number of ill-defined problems. They may be realistic, but their evaluation is problematic.

[Numsgil]

Actually, it is ambiguous. (See http://senseis.xmp.net/?AmbiguousPosition ).

Ah yes, that was the sort of one I didn't know how to handle. I haven't gone over all your posts but I'm sure they'll be helpful for understanding how to approach problems.

Anyway, I think I'm uncomfortable with positions like this because I don't know how to bound my error. The normal endgame counting can fall apart if you just blithely assume you'll get all the sente plays you want. Though I think the normal endgame counting is more managable than I gave it credit for, since you can aggressively prune the decision tree by assuming that sente followups are automatic. But it does fail in some cases dealing with tedomari and ambiguous positions. But it's a good starting point.

...

So I have an abstract example (I'd try to make an actual position for it, but it's pretty tricky to work backwards from counts to a position).

Let's say I have 3 unrelated endgame sequences to resolve, and then the game is over. Assume that the point values below are all black's gain if he gets the move, and white doesn't gain any points when he plays.

Sequence A: 4 point gote play (A1) with a 1 point sente followup (A2).
Sequence B: 5 point gote play (B1) with a 2 point sente followup (B2).
Sequence C: 4 point gote play (C1) with a 3 point gote followup (C2), which itself has a 5 point sente followup (C3).

With normal endgame counting, the values would be: A:5, B:7, C:8. So we'd naively expect the play to be: Black takes C, White takes B, Black takes A. Final score: Black 13.

But if black tries that in-game:
Black takes C1 (4 points)
White prevents B1
Black takes A1 (4 points)
White uses reverse sente to prevent C2.
Black takes A2 (1 point).
Final score: Black 10. Oops...

So maybe we try a different sequence. In this case, black reevaluates using the heuristic after each move.
Black takes C1 (4 points)
White prevents B1
Black takes C2 (3 points)
White uses reverse sente and prevents C3.
Black takes A1 (4 points)
White uses reverse sente and prevents A2.

The final score here would be: Black 11. An improvement, but still 2 points off what our original heuristic gave us.

So how do we actually handle this sort of position in a way that assures us we can play optimally and find the actual final score? If we use the endgame counting heuristic, are we at least assured that the first move we play will be optimal? If so, it's just a matter of recalculating values after each move.