I evaluate the following carpenter's square as a local endgame on the
marked locale. I apply the modern endgame theory explained in the
books Endgame 2 - Values and Endgame 3 - Accurate Local Evaluation.

[go]
$$B Carpenter's Square
$$------------------
$$. . . C C C C C C|
$$. . X . X W C C C|
$$. . . . X W C C C|
$$. . . . X W W W C|
$$. . . . X X X X C|
$$. . . . . . . . .|
$$. . . . . . . X .|
$$. . . . . . . . .|
[/go]

For Black's start, I presume this sequence:

[go]
$$B Black's Sequence
$$------------------
$$. . . . . 4 9 a b|
$$. . X . X O 3 1 2|
$$. . . . X O . 8 6|
$$. . . . X O O O 5|
$$. . . . X X X X 7|
$$. . . . . . . . .|
$$. . . . . . . X .|
$$. . . . . . . . .|
[/go]

Furthermore, I assume this sequence and the studied variations to be dominating. Further study should verify this. If a different ko
variation should be dominating, the values might have to be corrected
slightly. White starts on the 2-2, of course.


The most important conclusions are as follows:


Initial position:

Count = 4 2/3

Move value = 11 2/3

Type = gote

Length of sequence worth playing successively = 1


In practice, it can often be correct to play the first five moves
successively because their gains are at least 10 2/3 and therefore
similar to the initial move value. The gains of the 6th and especially
7th moves are much smaller though: 5 2/3 and 1 2/3.


Moves with move value 9 2/3 (as in the created ko):

Moves 8 to 11.

In particular, it would be a mistake to derive the wrong initial
values from the sente follower after move 10 as the count 2 2/3 and
move value 9 2/3, as done in the book Yose Size List.



Black's alternating sequence

[code]
after move count move gain move value type length of successive sequence(s)

0 4 2/3 1 11 2/3 11 2/3 gote 1
1 16 1/3 2 10 2/3 5 2/3 gote 5
2 5 2/3 3 10 2/3 10 2/3 gote 3
3 16 1/3 4 11 11/12 5 2/3 gote 3
4 4 5/12 5 11 11/12 7 11/12 gote 1
5 16 1/3 6 5 2/3 5 2/3 gote 1
6 10 2/3 7 1 2/3 1 2/3 sente Black's 5
White's 4
7 12 1/3 8 9 2/3 9 2/3 sente 4
8 2 2/3 9 9 2/3 9 2/3 gote 3
9 12 1/3 10 9 2/3 9 2/3 sente 2
10 2 2/3 11 9 2/3 9 2/3 ko Black's 2
White's 1
11 12 1/3 9 2/3 ko Black's 1
White's 2
[/code]



Copy and save the following SGF file, best viewed with GoWrite:




(;SZ[19]CA[UTF-8]GM[1]FF[4]ST[2]AP[GOWrite:3.0.10]AB[mb][oe][ob][pe][qe][re][oc][od][rg]PM[2]FG[259:]C[Each
just stated value is a count.

The locale is marked.

M is the move value of the currently studied hypothesis and position.

Gb1, Gb2,... are the gains of move 1, 2,... of Black's alternating
sequence of the currently studied hypothesis and position.

Gw1, Gw2,... are the gains of move 1, 2,... of White's alternating
sequence of the currently studied hypothesis and position.

So far, we assume, but do not verify, that Black's alternating
sequence and its variations shown are dominating.

Refuted Hypotheses

Black's 7/9/11-move sequence

gote move value M = (12 1/3 - (-7)) / 2 = (19 1/3) / 2 = 9 2/3

M > Gb6
9 2/3 > 1 2/3

Refuted Hypotheses

Black's 8/10-move sequence

sente move value M = 2 2/3 - (-7) = 9 2/3

M > Gb6
9 2/3 > 1 2/3

Refuted Hypothesis

Black's 6-move sequence

sente move value M = 10 2/3 - (-7) = 17 2/3

M > Gb6
17 2/3 > 1 2/3

Refuted Hypotheses

Black's 3/5-move sequence

gote move value M = (16 1/3 - (-7)) / 2 = (23 1/3) / 2 = 11 2/3

M > Gb2
11 2/3 > 10 2/3

Refuted Hypothesis

Black's 4-move sequence

sente move value M = 4 5/12 - (-7) = 11 5/12

M > Gb2
11 5/12 > 10 2/3

Refuted Hypothesis

Black's 2-move sequence

sente move value M = 5 2/3 - (-7) = 12 2/3

M > Gb2
12 2/3 > 10 2/3

Confirmed Hypothesis

gote count = (16 1/3 + (-7)) / 2 = (9 1/3) / 2 = 4 2/3

gote move value M = (16 1/3 - (-7)) / 2 = (23 1/3) / 2 = 11 2/3

Gb1 = 11 2/3

Gw1 = 11 2/3

M <= Gb1
11 2/3 <= 11 2/3

M <= Gw1
11 2/3 <= 11 2/3
]PW[ ]SQ[rd][sd][se][na][oa][pa][qa][ra][sa][pb][qb][rb][sb][pc][qc][rc][sc][pd][qd]AW[rd][pc][pb][pd][qd]PB[
]GN[ ]
(
;B[rb]
;FG[259:]C[16 1/3

position after move 1

refuted hypotheses

White's 6/8/10-move sequence

sente move value M = 9 2/3

M > Gw6
9 2/3 > 1 2/3

refuted hypotheses

White's 7/9-move sequence

gote move value M = (22 - 2 2/3) / 2 = 9 2/3

M > Gw6
9 2/3 > 1 2/3

confirmed hypothesis

White's 5-move sequence

gote count = (22 + 10 2/3) / 2 = (32 2/3) / 2 = 16 1/3

gote move value M = (22 - 10 2/3) / 2 = (11 1/3) / 2 = 5 2/3

Gb1 = 5 2/3

Gw1 = 10 2/3

M <= Gb1
5 2/3 <= 5 2/3

M <= Gw1, Gw2, Gw3, Gw4, Gw5
5 2/3 <= 10 2/3, 10 2/3, 11 11/12, 11 11/12, 5 2/3]PM[2]
(
;W[sb]
;FG[259:]C[5 2/3

position after move 2

refuted hypotheses

Black's 5/7/9-move sequence and White's 1/3-move sequence

gote move value M = (12 1/3 - (-5)) / 2 = (17 1/3) / 2 = 8 2/3

M > Gb5
8 2/3 > 1 2/3

refuted hypotheses

Black's 5/7/9-move sequence and White's 2-move sequence

sente move value M = 12 1/3 - 0 = 12 1/3

M > Gb5
12 1/3 > 1 2/3

refuted hypotheses

Black's 6/8-move sequence and White's 1/3-move sequence

sente move value M = 2 2/3 - (-5) = 7 2/3

M > Gb5
7 2/3 > 1 2/3

refuted hypotheses

Black's 4-move sequence and White's 1/3-move sequence

sente move value M = 10 2/3 - (-5) = 15 2/3

M > Gb4
15 2/3 > 5 2/3

refuted hypothesis

Black's 1/3-move sequence and White's 3-move sequence

gote move value M = (16 1/3 - (-5)) / 2 = (21 1/3) / 2 = 10 2/3

M > Gw3
10 2/3 > 5

refuted hypothesis

Black's 3-move sequence and White's 2-move sequence

sente move value M = 16 1/3 - 0 = 16 1/3

M > Gw2
16 1/3 > 5

confirmed hypothesis

Black's 3-move sequence and White's 1-move sequence

gote count = (16 1/3 + (-5)) / 2 = (11 1/3) / 2 = 5 2/3

gote move value M = (16 1/3 - (-5)) / 2 = (21 1/3) / 2 = 10 2/3

Gb1 = 10 2/3

Gw1 = 10 2/3

M <= Gb1, Gb2, Gb3
10 2/3 <= 10 2/3, 11 11/12, 11 11/12

M <= Gw1
10 2/3 <= 10 2/3]PM[2]
(
;B[qb]
;FG[259:]C[16 1/3

position after move 3

refuted hypotheses

White's 4/6/8-move sequence

sente move value M = 22 - 12 1/3 = 9 2/3

M > Gw4
9 2/3 > 1 2/3

refuted hypotheses

White's 5/7-move sequence

gote move value M = (22 - 2 2/3) / 2 = (19 1/3) / 2 = 9 2/3

M > Gw4
9 2/3 > 1 2/3

confirmed hypothesis

White's 3-move sequence

gote count = (22 + 10 2/3) / 2 = (32 2/3) / 2 = 16 1/3

gote move value M = (22 - 10 2/3) / 2 = (11 1/3) / 2 = 5 2/3

Gb1 = 5 2/3

Gw1 = 16 1/3 - 4 5/12 = 11 11/12

M <= Gb1
5 2/3 <= 5 2/3

M <= Gw1, Gw2, Gw3
5 2/3 <= 11 11/12, 11 11/12, 5 2/3]PM[2]
(
;C[ ]W[pa]
;FG[259:]C[4 5/12

position after move 4

refuted hypotheses

Black's 3/5/7-move sequence and White's 4/2-move sequence

sente move value M = 12 1/3 - 0 = 12 1/3

M > Gb2
12 1/3 > 1 2/3

refuted hypotheses

Black's 3/5/7-move sequence and White's 3-move sequence

gote move value M = (12 1/3 - (-5)) / 2 = (17 1/3) / 2 = 8 2/3

M > Gb2
8 2/3 > 1 2/3

refuted hypotheses

Black's 3/5/7-move sequence and White's 1-move sequence

gote move value M = (12 1/3 - (-3 1/2)) / 2 = (15 5/6) / 2 = 7 11/12

M > Gb2
7 11/12 > 1 2/3

refuted hypotheses

Black's 6/4-move sequence and White's 3-move sequence

sente move value M = 2 2/3 - (-5) = 7 2/3

M > Gb2
7 2/3 > 1 2/3

refuted hypotheses

Black's 6/4-move sequence and White's 1-move sequence

sente move value M = 2 2/3 - (-3 1/2) = 6 1/6

M > Gb2
6 1/6 > 1 2/3

refuted hypothesis

Black's 2-move sequence and White's 3-move sequence

sente move value M = 10 1/2 - (-5) = 15 1/2

M > Gb2
15 1/2 > 1 2/3

refuted hypothesis

Black's 2-move sequence and White's 1-move sequence

gote move value M = 10 1/2 - (-3 1/2) = 14

M > Gb2
14 > 1 2/3

refuted hypotheses

Black's 1-move sequence and White's 4/2-move sequence

sente move value M = 12 1/3 - 0 = 12 1/3

M > Gw2
12 1/3 > 3 1/2

refuted hypothesis

Black's 1-move sequence and White's 3-move sequence

gote move value M = (12 1/3 - (-5)) / 2 = (17 1/3) / 2 = 8 2/3

M > Gw2
8 2/3 > 3 1/2

confirmed hypothesis

Black's 1-move sequence and White's 1-move sequence

gote count = (12 1/3 + (-3 1/2)) / 2 = (8 5/6) / 2 = 4 5/12

gote move value M = (12 1/3 - (-3 1/2)) / 2 = (15 5/6) / 2 = 7 11/12

Gb1 = 7 11/12

Gw1 = 7 11/12

M <= Gb1
7 11/12 <= 7 11/12

M <= Gw1
7 11/12 <= 7 11/12
]PM[2]
(
;B[sd]
;FG[259:]C[16 1/3

position after move 5

Hypothesis 1

White's long sente with White's 6-move sequence

sente count = 12 1/3

sente move value M = 22 - 12 1/3 = 9 2/3

Gb1 = 9 2/3

Gw1 = 1 2/3

refuting Hypothesis 1\:

M > Gw1
9 2/3 > 1 2/3

Hypothesis 2

White's long gote with White's 5-move sequence

gote count = (22 + 2 2/3) / 2 = (24 2/3) / 2 = 12 1/3

gote move value M = (22 - 2 2/3) / 2 = (19 1/3) / 2 = 9 2/3

Gb1 = 9 2/3

Gw1 = 1 2/3

refuting Hypothesis 2\:

M > Gw1
9 2/3 > 1 2/3

Hypothesis 3

White's long sente with White's 4-move sequence

sente count = 12 1/3

sente move value M = 22 - 12 1/3 = 9 2/3

Gb1 = 9 2/3

Gw1 = 1 2/3

refuting Hypothesis 3\:

M > Gw1
9 2/3 > 1 2/3

Hypothesis 4

White's long gote with White's 3-move sequence

gote count = (22 + 2 2/3) / 2 = (24 2/3) / 2 = 12 1/3

gote move value M = (22 - 2 2/3) / 2 = (19 1/3) / 2 = 9 2/3

Gb1 = 9 2/3

Gw1 = 1 2/3

refuting Hypothesis 4\:

M > Gw1
9 2/3 > 1 2/3

Hypothesis 5

White's local sente with White's 2-move sequence

sente count = 12 1/3

sente move value M = 22 - 12 1/3 = 9 2/3

Gb1 = 9 2/3

Gw1 = 1 2/3

refuting Hypothesis 5\:

M > Gw1
9 2/3 > 1 2/3

Hypothesis 6

White's local gote with White's 1-move sequence

gote count = (22 + 10 2/3) / 2 = (32 2/3) / 2 = 16 1/3

gote move value M = (22 - 10 2/3) / 2 = (11 1/3) / 2 = 5 2/3

Gb1 = 5 2/3

Gw1 = 5 2/3

confirming Hypothesis 6\:

M <= Gb1
5 2/3 <= 5 2/3

M <= Gw1
5 2/3 <= 5 2/3]PM[2]
(
;W[sc]
;FG[259:]C[position after move 6

Hypothesis 1

White's long sente with Black's 5-move sequence and White's 4-move
sequence

sente count = 10 2/3

sente move value M = 12 1/3 - 10 2/3 = 1 2/3

Gb1 = 1 2/3

Gw1 = 9 2/3

confirming Hypothesis 1\:

M <= Gb1, Gb2, Gb3, Gb4, Gb5
1 2/3 <= 1 2/3, 9 2/3, 9 2/3, 9 2/3, 9 2/3

M <= Gw1, Gw2, Gw3, Gw4
9 2/3 <= 9 2/3, 9 2/3, 9 2/3, 9 2/3]PM[2]
(
;B[se]
;FG[259:]C[position after move 7

Hypothesis 1

White's long sente

sente count = 12 1/3

sente move value M = 22 - 12 1/3 = 9 2/3

Gb1 = 9 2/3

Gw1 = 9 2/3

confirming Hypothesis 1\:

M <= Gb1
9 2/3 <= 9 2/3

M <= Gw1, Gw2, Gw3, Gw4
9 2/3 <= 9 2/3, 9 2/3, 9 2/3, 9 2/3]PM[2]
(
;W[rc]
;FG[259:]C[position after move 8

Hypothesis 1

Black's long gote

gote count = (12 1/3 + (-7)) / 2 = (5 1/3) / 2 = 2 2/3

gote move value M = (12 1/3 - (-7)) / 2 = (19 1/3) / 2 = 9 2/3

Gb1 = 9 2/3

Gw1 = 9 2/3

confirming Hypothesis 1\:

M <= Gb1, Gb2, Gb3
9 2/3 <= 9 2/3, 9 2/3, 9 2/3

M <= Gw1
9 2/3 <= 9 2/3]PM[2]
(
;B[qa]
;FG[259:]C[position after move 9

Hypothesis 1

White's local sente

sente count = 12 1/3

sente move value M = 22 - 12 1/3 = 9 2/3

Gb1 = 9 2/3

Gw1 = 9 2/3

confirming Hypothesis 1\:

M <= Gb1
9 2/3 <= 9 2/3

M <= Gw1, Gw2
9 2/3 <= 9 2/3, 9 2/3]PM[2]
(
;W[ra]
;FG[259:]C[position after move 10

ordinary ko

move value
(22 - (-7)) / 3 = 29/3 = 9 2/3

count
-7 + 1 * 9 2/3 = 2 2/3]PM[2]
(
;B[sa]
;FG[259:]PM[2]MN[1]C[position after move 11

ordinary ko

move value
(22 - (-7)) / 3 = 29/3 = 9 2/3

count
22 - 1 * 9 2/3 = 12 1/3]
(
;W[ra];B[tt];C[-7]W[qc]
)
(
;FG[259:]PM[2];B[ra]C[22]
)

)
(
;FG[259:]MN[1]PM[2]
(
;B[sa];W[tt];B[ra]C[22]
)
(
;FG[259:]PM[2];C[-7]W[qc]
)

)

)
(
;FG[259:]MN[1]PM[2]
(
;B[ra]C[22]
)
(
;FG[259:]PM[2];C[2 2/3]W[ra]
;B[sa]C[12 1/3

Gw2 = 9 2/3]
)

)

)
(
;FG[259:]MN[1]PM[2]
(
;B[qa]C[12 1/3]
;C[2 2/3

Gb2 = 9 2/3]W[ra]
;B[sa]C[12 1/3

Gb3 = 9 2/3]
)
(
;FG[259:]PM[2];C[-7]W[ra]
)

)

)
(
;FG[259:]MN[1]PM[2]
(
;B[rc]C[22]
)
(
;FG[259:]PM[2];C[2 2/3]W[rc]
;B[qa]C[12 1/3

Gw2 = 9 2/3]
;C[2 2/3

Gw3 = 9 2/3]W[ra]
;B[sa]C[12 1/3

Gw4 = 9 2/3]
)

)

)
(
;FG[259:]MN[1]PM[2]
(
;B[se]C[12 1/3]
;C[2 2/3

Gb2 = 9 2/3]W[rc]
;B[qa]C[12 1/3

Gb3 = 9 2/3]
;C[2 2/3

Gb4 = 9 2/3]W[ra]
;B[sa]C[12 1/3

Gb5 = 9 2/3]
)
(
;FG[259:]PM[2]
;C[Hypothesis 1

Black's long gote

gote count = (10 2/3 + (-8 2/3)) / 2 = 2/2 = 1

gote move value M = (10 2/3 - (-8 2/3)) / 2 = (19 1/3) / 2 = 9 2/3

Gb2 = 11 1/3

Gw2 = 9 2/3

confirming Hypothesis 1\:

M <= Gb2, Gb3, Gb4
9 2/3 <= 9 2/3, 9 2/3, 9 2/3

M <= Gw2
9 2/3 <= 9 2/3]W[ra]
(
;B[qa]C[10 2/3

Gw2 = 9 2/3]
;C[best because W avoids approach ko

2 2/3 + (-1 2/3) = 1

Gw3 = 9 2/3]W[rc]
;B[sa]C[10 2/3

Gw4 = 9 2/3

After dissolution of the ko, the count of the remaining endgame with
W's privilege on the right side, with a locale temporarily expanded by
one intersection, is\: -2/3

Shrinking the expanded locale means modifying the count by -1.

Accounted for the initial locale, the the remaining endgame has the
adjusted count -1 2/3.

In the initial locale, the count is 12 1/3.

In the initial locale, the total count including the remaining local
endgame is 12 1/3 + (-1 2/3) = 10 2/3.]
)
(
;FG[259:]MN[2]PM[2]
;C[simply speaking, this is the best move

-7 + (-1 2/3) = -8 2/3]W[rc]
)

)

)

)
(
;FG[259:]MN[1]PM[2]
(
;B[sc]C[22]
)
(
;FG[259:]C[2 2/3

Gw3 = 9 2/3]PM[2]
;C[10 2/3]W[sc]
;B[se]C[12 1/3

Gw2 = 1 2/3]
;C[2 2/3

Gw3 = 9 2/3]W[rc]
;B[qa]C[12 1/3

Gw4 = 9 2/3]
;C[2 2/3

Gw5 = 9 2/3]W[ra]
;B[sa]C[12 1/3

Gw6 = 9 2/3]
)

)

)
(
;FG[259:]MN[1]PM[2]
(
;B[sd]C[12 1/3]
;C[10 2/3

Gb2 = 1 2/3]W[sc]
;B[se]C[12 1/3

Gb3 = 1 2/3]
;C[2 2/3

Gb4 = 9 2/3]W[rc]
;B[qa]C[12 1/3

Gb5 = 9 2/3]
;C[2 2/3

Gb6 = 9 2/3]W[ra]
;B[sa]C[12 1/3

Gb7 = 9 2/3]
)
(
;FG[259:]PM[2]
;C[-3 1/2

best choice because possible seki is better than the kos

In locale, white follower's count -8.

In expanded locale, white follower's count -7.

In locale with accounting remaining endgame of expanded locale to the
locale, white follower's count -7.

Hypothesis 1

Black's long gote

gote count = -3 1/2

gote move value M = 3 1/2

Gb2 = 3 1/2
Gb3 = 12
Gb4 = 5

Gw2 = 3 1/2

confirming Hypothesis 1\:

M <= Gb2, Gb3, Gb4
3 1/2 <= 3 1/2, 12, 5

M <= Gw2
3 1/2 <= 3 1/2]W[rc]
;B[ra]C[best

sente count 0

Gw2 = 3 1/2]
;C[sente seki is best

with expanded locale\:

gote count -5

Gw3 = 12]W[qa]
;B[sd]C[0

Gw4 = 5]
)

)

)
(
;FG[259:]MN[1]PM[2]
(
;B[pa]C[22]
)
(
;FG[259:]PM[2];C[4 5/12]W[pa]
;B[sd]C[16 1/3

Gw2 = 11 11/12]
;C[10 2/3

Gw3 = 5 2/3]W[sc]
;B[se]C[12 1/3

Gw4 = 1 2/3]
;C[2 2/3

Gw5 = 9 2/3]W[rc]
;B[qa]C[12 1/3

Gw6 = 9 2/3]
;C[2 2/3

Gw7 = 9 2/3]W[ra]
;B[sa]C[12 1/3

Gw8 = 9 2/3]
)

)

)
(
;FG[259:]MN[1]PM[2]
;B[qb]C[16 1/3]
;C[4 5/12

Gb2 = 11 11/12]W[pa]
;B[sd]C[16 1/3

Gb3 = 11 11/12]
;C[10 2/3

Gb4 = 5 2/3]W[sc]
;B[se]C[12 1/3

Gb5 = 1 2/3]
;C[2 2/3

Gb6 = 9 2/3]W[rc]
;B[qa]C[12 1/3

Gb7 = 9 2/3]
;C[2 2/3

Gb8 = 9 2/3]W[ra]
;B[sa]C[12 1/3

Gb9 = 9 2/3]
)

(
;FG[259:]MN[1]PM[2]
;C[-5

because

Black next sente result -5

White next result -6]W[rc]
;B[sd]C[0

because Black next sente result 0

Gw2 = 5]
;C[-5

Gw3 = 5]W[ra]
)

)
(
;FG[259:]MN[1]PM[2]
;B[sd]C[22]
)

(
;FG[259:]MN[1]PM[2]
;C[5 2/3]W[sb]
;B[qb]C[16 1/3

Gw2 = 10 2/3]
;C[4 5/12

Gw3 = 11 11/12]W[pa]
;B[sd]C[16 1/3

Gw4 = 11 11/12]
;C[10 2/3

Gw5 = 5 2/3]W[sc]
;B[se]C[12 1/3

Gw6 = 1 2/3]
;C[2 2/3

Gw7 = 9 2/3]W[rc]
;B[qa]C[12 1/3

Gw8 = 9 2/3]
;C[2 2/3

Gw9 = 9 2/3]W[ra]
;B[sa]C[12 1/3

Gw10 = 9 2/3]
)

)
(
;FG[259:]PM[2];C[-7]W[rb]
)

)

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