Easy endgame problem, Dec. 2014

[Bill Spight]

(;CA[ISO8859-1]GM[1]SZ[9]ST[2]AP[GOWrite:2.3.46]FF[4]AB[ab][bb][cb][ad][ea][fa][id][hd][dd][cc][ce][gf][hf][ci][cf][bf][bg][cd][hg][hh][hi][gi][if][ag][ic][ec][ah][ge][bi]PB[ ]AW[aa][ba][db][eb][fb][hc][ga][ig][ed][gd][gc][dc][ib][fe][ef][df][cg][ch][bh][gg][gh][eg][de][hb][fi][eh][di][ih]FG[259:]GN[ ]PW[ ]PM[2]
)

A. White to play

B. Black to play

Show failure diagrams, too.

Enjoy!

[oca]

Hi Bill, what is the komi ?

[Bill Spight]

No komi. Territory scoring.

[oca]

That's allready not easy for me... so let's start slowly... here is what I see at first glimpse

$$Bc
$$ +-------------------+
$$ | O O a b X X O g . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |
$$ | X . X X O e O X X |
$$ | . . X O , O X f . |
$$ | . X X O O . X X X |
$$ | X X O . O d O X O |
$$ | X O O h O . O X O |
$$ | c X X O . O X X . |
$$ +-------------------+

Click Here To Show Diagram Code

[go]$$Bc
$$ +-------------------+
$$ | O O a b X X O g . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |
$$ | X . X X O e O X X |
$$ | . . X O , O X f . |
$$ | . X X O O . X X X |
$$ | X X O . O d O X O |
$$ | X O O h O . O X O |
$$ | c X X O . O X X . |
$$ +-------------------+[/go]

'a' to 'd' are points that looks interesting to me,
and maybe 'e' to 'h' may be somehow usefull as ko threats... and I'm pretty sure a ko fight will happend because of 'c' ...

So now I suppose I must calculate the value of each point for both black and white... but there seems to be a lot of follow up and ko so its not easy for me.. so let's try by instinct, this is the best I can do for now...

(;CA[ISO8859-1]GM[1]SZ[9]ST[2]AP[GOWrite:2.3.46]FF[4]AB[ab][bb][cb][ad][ea][fa][id][hd][dd][cc][ce][gf][hf][ci][cf][bf]
[bg][cd][hg][hh][hi][gi][if][ag][ic][ec][ah][ge][bi]PB[ ]AW[aa][ba][db][eb][fb][hc][ga][ig][ed][gd][gc][dc][ib][fe][ef][df]
[cg][ch][bh][gg][gh][eg][de][hb][fi][eh][di][ih]FG[259:]GN[ ]PW[ ]PM[2]
(;B[ca];W[da];B[fg];W[fh];B[ff];W[ai];B[bi];W[ci]
;B[ai]CR[aa][ba][ac][bc][bd][be][ae][af][he][ie][ii][ih][ig]TR[ea][fa][ha][ia][gb][fc][fd][ec][ee][dg][dh][ei]C[black : 13 + 6 = 19
white :12 + 14 = 16])
(;W[da];B[ca];W[ff]
;B[ai]CR[aa][ba][ac][bc][bd][be][ae][af][he][ie][ii][ih][ig]TR[ea][fa][ha][ia][gb][fc][fd][ee][dg][dh][fg][fh][ei][ec]C[black : 13 + 4 = 17
white : 14 + 3 = 17])
(;W[ff];B[ca];W[da]
;B[ai]C[black : 13 + 4 = 17
white : 14 + 3 = 17]))

[Kirby]

My attempt:

Endgame strategy still confuses me, but it looks like F4 is the key to this problem for black. If black can play F4 first, it looks like black can win. White can follow the move order by value, and win.

(;CA[Windows-1252]GM[1]FF[4]ST[2]PM[2]AB[bi][ab][bb][cb][hf][ea][fa][id][hd][dd]
[cc][ce][gf][ad][ci][cf][bf][bg][cd][hg][hh][hi][gi][if][ag][ic][ec][ah][ge]AW[ih]
[gg][aa][ba][db][eb][fb][hc][ga][ig][ed][gd][gc][dc][ib][fe][ef][df][cg][ch][bh]
[gh][eg][de][hb][fi][eh][di]LB[ca:A][ai:B][ff:C][fg:D]C[First, assess the point values of the marked areas.

A - 6 points gote (4 points + 1/2 chance of another 4 points).

B - 3 points gote (2 points + 1/2 chance of another 2 points)

C - 2 points (1 point gote, with privilege to gain another point)

D - 2 points gote, no follow up
]
AP[GOWrite:2.3.46]SZ[9]FG[259:]GN[ ]PB[ ]PW[ ]MULTIGOGM[1]

(;B[ca]C[Assuming black to play...

Strictly following the point value move order...]
;W[da];B[ai];W[ff]C[Jigo.])
(;B[ff]C[F4 is an interesting point, because it allows black to later have F3 in sente.

If white continues to follow based on point-order...]
;W[da];B[ca];W[ai];B[fg];W[fh];B[bi];W[ci];B[ai]C[Black wins.])
(;B[]C[Assuming white to play, following the point value move order...]
(;W[da];B[ca];W[ai];B[fg];W[fh];B[ff];W[bi]C[White wins.])
(;W[ff]C[If white blocks black's sente play...];B[ca];W[da];B[ai]C[Jigo.])))

Edit:

After looking at mitsun's solution, it looks like black can also win with my first variation by taking the F3 atari before connecting, so mitsun's is a better variation when starting with capturing the two stones.

That being said, is the variation I provide for black starting with F4 incorrect? This one also seems to allow black to win, but maybe I am missing a refutation.

[mitsun]

Nice problem. At the start B has 13 points of secure territory and W has 11 points of secure territory. If I understand the way you like to like to count, the unsettled position on top has an expected value of 0 by symmetry, the unsettled position in the center has an expected value of -1 for B (two plays make two points), and the unsettled position in the lower left also has an expected value of -1 for B (with a more complicated evaluation tree). So the expected value of the whole board is 0, without considering whose turn it is to play.

If the task is to find optimum play, it is not a simple problem. For the more limited problem of B to play and win, I think I have an simple answer. It may be the best sequence for B (largest win), but no guarantee. I have not looked at W to play yet.

B start with C9 (capture two stones at the top). This is sente; W has to respond at D9, capturing the corresponding two stones, since this is now the largest remaining play (4 point gote). Continue with F3 atari and connection (2 point gote), letting W finish the lower left corner (1 point sente). B beats the expected values by 1 point and wins.

[Bill Spight]

Note: White to play can win.

[DrStraw]

$$Bc
$$ +-------------------+
$$ | O O a b X X O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |
$$ | X . X X O . O X X |
$$ | . . X O , O X . . |
$$ | . X X O O . X X X |
$$ | X X O . O . O X O |
$$ | X O O . O . O X O |
$$ | . X X O . O X X . |
$$ +-------------------+

Click Here To Show Diagram Code

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

No one seems to have considered b for B's first move at the top instead of a.

$$Bc
$$ +-------------------+
$$ | O O 2 1 X X O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |
$$ | X . X X O . O X X |
$$ | . . X O , O X . . |
$$ | . X X O O . X X X |
$$ | X X O . O . O X O |
$$ | X O O . O . O X O |
$$ | . X X O . O X X . |
$$ +-------------------+

Click Here To Show Diagram Code

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

$$Bc
$$ +-------------------+
$$ | . . 5 3 4 . O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |
$$ | X . X X O . O X X |
$$ | . . X O , O X . . |
$$ | . X X O O . X X X |
$$ | X X O . O . O X O |
$$ | X O O . O . O X O |
$$ | . X X O . O X X . |
$$ +-------------------+

Click Here To Show Diagram Code

[go]$$Bc
$$ +-------------------+
$$ | . . 5 3 4 . O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |
$$ | X . X X O . O X X |
$$ | . . X O , O X . . |
$$ | . X X O O . X X X |
$$ | X X O . O . O X O |
$$ | X O O . O . O X O |
$$ | . X X O . O X X . |
$$ +-------------------+[/go]

Here each has three captures but B has one more point. It is similar if W plays at a first. However, it is gote for each so it can only be considered if the sequence is correct.

[Bill Spight]

@ Kirby

You found winning plays for each player. But there is a payoff in the failure diagrams. A few questions in the variations.

(;CA[Windows-1252]GM[1]SZ[9]ST[2]AP[GOWrite:2.3.46]FF[4]AB[ge][cc][ec][ea][fa][hg][ce][gi][hi][bi][ic][ci][bb][cd][cb][gf][dd][ab][hh][hf][if][id][bf][hd][cf][ah][bg][ad][ag]PB[ ]C[First, assess the point values of the marked areas.A - 6 points gote (4 points + 1/2 chance of another 4 points).B - 3 points gote (2 points 1/2 chance of another 2 points)C - 2 points (1 point gote, with privilege to gain another point)D - 2 points gote, no follow up]AW[aa][dc][eg][ba][cg][fe][ig][ga][gg][de][fi][gc][hc][di][df][gh][ef][ed][ih][gd][hb][ib][fb][ch][bh][eh][db][eb]FG[259:]LB[ca:A][ai:B][ff:C][fg:D]PW[ ]GN[ ]PM[2]MULTIGOGM[1]
(
;C[Assuming black to play...Strictly following the point value move order...]B[ca]
;C[*** Can Black win fr]W[da]
;B[ai]
;C[Jigo.]W[ff]
)
(
;C[F4 is an interesting point, because it allows black to later have F3 in sente.If white continues to follow based on point-order...]B[ff]
(
;C[*** See variations.]W[da]
;B[ca]
;W[ai]
;B[fg]
;W[fh]
;B[bi]
;W[ci]
;C[Black wins.]B[ai]
)
(
;C[*** Now what?]W[ai]
)

(
;C[*** Now what?]W[fg]
)

)

(
;C[Assuming white to play, following the point value move order...]B[tt]
(
;C[*** See variation.]W[da]
;B[ca]
;W[ai]
;B[fg]
;W[fh]
;B[ff]
;C[White wins.]W[bi]
)
(
;C[If white blocks black's sente play...]W[ff]
;B[ca]
;W[da]
;C[Jigo.]B[ai]
)

(
;C[*** Now what?]W[ai]
)

)

)

[Kirby]

A few additional sequences.

(;CA[Windows-1252]GM[1]FF[4]ST[2]PM[2]AB[ag][ge][cc][ec][ea][fa][hg][ce][gi][hi]
[bi][ic][ci][bb][cd][cb][gf][dd][ab][hh][hf][if][id][bf][hd][cf][ah][bg][ad]AW[eh]
[db][eb][aa][dc][eg][ba][cg][fe][ig][ga][gg][de][fi][gc][hc][di][df][gh][ef][ed]
[ih][gd][hb][ib][fb][ch][bh]LB[ca:A][ai:B][ff:C][fg:D]C[First, assess the point values of the marked areas.A - 6 points gote (4 points + 1/2 chance of another 4 points).B - 3 points gote (2 points 1/2 chance of another 2 points)C - 2 points (1 point gote, with privilege to gain another point)D - 2 points gote, no follow up]
AP[GOWrite:2.3.46]SZ[9]FG[259:]GN[ ]PB[ ]PW[ ]MULTIGOGM[1]

(;B[ca]C[Assuming black to play...Strictly following the point value move order...]
;W[da];B[fg];W[fh];B[ff];W[ai];B[bi]C[Black wins.])
(;B[ff]C[F4 is an interesting point, because it allows black to later have F3 in sente.If white continues to follow based on point-order...]
(;W[da]C[*** See variations.];B[ca];W[ai];B[fg];W[fh];B[bi];W[ci];B[ai]C[Black wins.])
(;W[ai]C[*** Now what?];B[ca]
(;W[da]C[If white plays D9, it reverts to the other branch.];B[fg];W[fh];B[bi])
(;W[fg]C[Even if white gets both F3 and B1, the profit from D9 is too large, so black still wins.]
;B[da];W[bi]))
(;W[fg]C[*** Now what?];B[ca];W[da]C[White must capture, because A1 can gain a maximum of 3 points.]
;B[ai]C[Black wins.]))
(;B[tt]C[Assuming white to play, following the point value move order...]
(;W[da]C[*** See variation.];B[ca];W[ai];B[fg];W[fh];B[ff];W[bi]C[White wins.])
(;W[ff]C[If white blocks black's sente play...];B[ca];W[da];B[ai]C[Jigo.])
(;W[ai]C[*** Now what?];B[ca]C[If white gets both F4 and B1, it is 4 points. So the maximum white can get by not playing D9 is 4 points.

Therefore, D9 is at least as good as optimal play.]
;W[da];B[fg];W[fh];B[ff];W[bi]C[White wins.])))

But there is a payoff in the failure diagrams.

Do you mean that there is value in providing failure diagrams?

[Bill Spight]

A few additional sequences.

(;CA[Windows-1252]GM[1]FF[4]ST[2]PM[2]AB[ag][ge][cc][ec][ea][fa][hg][ce][gi][hi]
[bi][ic][ci][bb][cd][cb][gf][dd][ab][hh][hf][if][id][bf][hd][cf][ah][bg][ad]AW[eh]
[db][eb][aa][dc][eg][ba][cg][fe][ig][ga][gg][de][fi][gc][hc][di][df][gh][ef][ed]
[ih][gd][hb][ib][fb][ch][bh]LB[ca:A][ai:B][ff:C][fg:D]C[First, assess the point values of the marked areas.A - 6 points gote (4 points + 1/2 chance of another 4 points).B - 3 points gote (2 points 1/2 chance of another 2 points)C - 2 points (1 point gote, with privilege to gain another point)D - 2 points gote, no follow up]
AP[GOWrite:2.3.46]SZ[9]FG[259:]GN[ ]PB[ ]PW[ ]MULTIGOGM[1]

(;B[ca]C[Assuming black to play...Strictly following the point value move order...]
;W[da];B[fg];W[fh];B[ff];W[ai];B[bi]C[Black wins.])
(;B[ff]C[F4 is an interesting point, because it allows black to later have F3 in sente.If white continues to follow based on point-order...]
(;W[da]C[*** See variations.];B[ca];W[ai];B[fg];W[fh];B[bi];W[ci];B[ai]C[Black wins.])
(;W[ai]C[*** Now what?];B[ca]
(;W[da]C[If white plays D9, it reverts to the other branch.];B[fg];W[fh];B[bi])
(;W[fg]C[Even if white gets both F3 and B1, the profit from D9 is too large, so black still wins.]
;B[da];W[bi]))
(;W[fg]C[*** Now what?];B[ca];W[da]C[White must capture, because A1 can gain a maximum of 3 points.]
;B[ai]C[Black wins.]))
(;B[tt]C[Assuming white to play, following the point value move order...]
(;W[da]C[*** See variation.];B[ca];W[ai];B[fg];W[fh];B[ff];W[bi]C[White wins.])
(;W[ff]C[If white blocks black's sente play...];B[ca];W[da];B[ai]C[Jigo.])
(;W[ai]C[*** Now what?];B[ca]C[If white gets both F4 and B1, it is 4 points. So the maximum white can get by not playing D9 is 4 points.

Therefore, D9 is at least as good as optimal play.]
;W[da];B[fg];W[fh];B[ff];W[bi]C[White wins.])))

But there is a payoff in the failure diagrams.

Do you mean that there is value in providing failure diagrams?

Oh, there is. But I have something specific in mind.

If captures at A-01, Black can make jigo.

[mitsun]

For W to play, it looks like a win is possible.

http://www.lifein19x19.com/forum/download/file.php?id=5380

[Bill Spight]

@ mitsun

Your main line is jigo.

[mitsun]

Oops, I was so sure the hane would be correct in some variation, I fooled myself and counted wrong. The right answer is simpler.

W captures two stones on top, B must also capture two stones, then W takes first in the lower left corner. This leaves two continuations for B, but in each case W gets the last move and wins by 1 point.

[lightvector]

I love Bill's simple endgame puzzles . Every time I learn something about how to conceptualize the fight for the last point. Maybe some day eventually knowing how to fight for the last point will win me a game (hasn't happened yet in any memorable way, sadly).

Here's my analysis.

$$Bc
$$ +-------------------+
$$ | O O a b X X O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |
$$ | X . X X O . O X X |
$$ | . . X O , O X . . |
$$ | . X X O O . X X X |
$$ | X X O . O d O X O |
$$ | X O O . O . O X O |
$$ | c X X O . O X X . |
$$ +-------------------+

Click Here To Show Diagram Code

[go]$$Bc
$$ +-------------------+
$$ | O O a b X X O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |
$$ | X . X X O . O X X |
$$ | . . X O , O X . . |
$$ | . X X O O . X X X |
$$ | X X O . O d O X O |
$$ | X O O . O . O X O |
$$ | c X X O . O X X . |
$$ +-------------------+[/go]

Firstly, on the bottom, C and D are easy. They are both the same size of play, but are different in followups.

C is a 1 point ambiguous that leaves behind a 1 point gote if white plays.
D is a 1 point gote directly.

Both players want to come out with sente from the top to play the bottom because doing so wins tedomari and an extra (half-)point.

If black gets the first play on the bottom, black takes D and gets tedomari by recapturing when white takes C.
If white gets the first play on the bottom, white plays C and gets tedomari because the recapture is miai with D.

Now, about A/B. This is a strange beast. Black has the choice between the following sequences:

Black shuts the position down in sente with a local score of 0.

$$Bc
$$ +-------------------+
$$ | O O 1 2 X X O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |

Click Here To Show Diagram Code

[go]$$Bc
$$ +-------------------+
$$ | O O 1 2 X X O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |[/go]

Black shuts the position down in gote with a local score of +1 for black.

$$Bc
$$ +-------------------+
$$ | O O 2 1 X X O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |

Click Here To Show Diagram Code

[go]$$Bc
$$ +-------------------+
$$ | O O 2 1 X X O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |[/go]

$$Bc
$$ +-------------------+
$$ | . . 5 3 4 . O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |

Click Here To Show Diagram Code

[go]$$Bc
$$ +-------------------+
$$ | . . 5 3 4 . O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |[/go]

White has the choice of these two options in reverse as well. So whoever has sente can spend it to gain 1 point here, so the miai value of A/B is *also* 1 point! But it's weird. It's as if whoever's next to play can choose for there to be an extra 1 point move on the board but then must immediately take that move, or else can shut the position down in sente and choose for nobody to be able to get a 1-point move here.

Given that both players want tedomari, whoever's first to play in this problem here should obviously shut the position down in sente so that they can get tedomari on C and D as discussed before, thereby winning by 1 point.


To answer Bill's question about mistake variations, if white to play in this problem takes C first, then recapturing C becomes miai with D, so in order to gain tedomari, black *does* want there be an extra 1-point move that he can take for tedomari, so he spends sente by playing B, achieving a draw. Like so:

$$Bc
$$ +-------------------+
$$ | O O 2 1 X X O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |
$$ | X . X X O . O X X |
$$ | . . X O , O X . . |
$$ | . X X O O . X X X |
$$ | X X O . O d O X O |
$$ | X O O . O . O X O |
$$ | W . . O . O X X . |
$$ +-------------------+

Click Here To Show Diagram Code

[go]$$Bc
$$ +-------------------+
$$ | O O 2 1 X X O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |
$$ | X . X X O . O X X |
$$ | . . X O , O X . . |
$$ | . X X O O . X X X |
$$ | X X O . O d O X O |
$$ | X O O . O . O X O |
$$ | W . . O . O X X . |
$$ +-------------------+[/go]

$$Bc
$$ +-------------------+
$$ | . . 5 3 4 . O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |
$$ | X . X X O . O X X |
$$ | . . X O , O X . . |
$$ | . X X O O 9 X X X |
$$ | X X O . O 7 O X O |
$$ | X O O . O 8 O X O |
$$ | W 6 . O . O X X . |
$$ +-------------------+

Click Here To Show Diagram Code

[go]$$Bc
$$ +-------------------+
$$ | . . 5 3 4 . O . . |
$$ | X X X O O O . O O |
$$ | . . X O X . O O X |
$$ | X . X X O . O X X |
$$ | . . X O , O X . . |
$$ | . X X O O 9 X X X |
$$ | X X O . O 7 O X O |
$$ | X O O . O 8 O X O |
$$ | W 6 . O . O X X . |
$$ +-------------------+[/go]

Question to Bill:

As long as there there is a sufficiently rich collection of plays on the board elsewhere with miai values between 1 and 2 points, assuming no ko, it seems to me that under correct play, we can assume the sente variation will always be chosen and the gote variation will never be chosen. Whichever player prefers there *not* to be an extra 1-point move can simply shut it down in sente once the temperature of the board falls in this range. Is this right?