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[go]$$W initial position
$$ ----------------
$$ | . X . . . X O .
$$ | X O O . X O O .
$$ | X . O . X O . .
$$ | X X . X X O . .
$$ | . X X X X O . .
$$ | . . . . . . . .[/go]
This is quite an interesting position.
Black to play connects for 13 pts. of territory.
- Click Here To Show Diagram Code
[go]$$W Ko
$$ ----------------
$$ | . X . 2 1 X O .
$$ | X O O 3 X O O .
$$ | X . O . X O . .
$$ | X X . X X O . .
$$ | . X X X X O . .
$$ | . . . . . . . .[/go]
This ko is an obvious possibility when White plays first. If Black takes and wins the ko the result is again +13. (All territory values are from Black's perspective.)
- Click Here To Show Diagram Code
[go]$$W White wins the ko
$$ ----------------
$$ | . B W C O C O .
$$ | X O O O X O O .
$$ | X . O . X O . .
$$ | X X . X X O . .
$$ | . X X X X O . .
$$ | . . . . . . . .[/go]
If White wins the ko White gets 1 pt. for each captured stone plus ⅓ pt. for the
stone in the remaining ko, for -2⅓. Each play in the ko gains on average 5 1/9 pts. ((13 + 2⅓)/3), and the position after
in the previous diagram has an average territorial count of 2 7/9.
Now let's look at the position just before White makes the ko.
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[go]$$W 1 Black prisoner
$$ ----------------
$$ | . X . X O . O .
$$ | X O O . X O O .
$$ | X . O . X O . .
$$ | X X . X X O . .
$$ | . X X X X O . .
$$ | . . . . . . . .[/go]
Suppose that Black is komaster. Then, OC, Black can take and win the ko for 12 pts.
- Click Here To Show Diagram Code
[go]$$W 1 Black prisoner, White sente
$$ ----------------
$$ | . X . X O 1 O .
$$ | X O O 2 X O O .
$$ | X . O . X O . .
$$ | X X . X X O . .
$$ | . X X X X O . .
$$ | . . . . . . . .[/go]
White to play can fill the ko with sente, for 10 pts. If Black takes and wins the ko she gains only 2 pts. in 2 moves, for an average value of 1 pt. per move.
However, White has another arrow in her quiver.
- Click Here To Show Diagram Code
[go]$$W 1 Black prisoner, White makes ko
$$ ----------------
$$ | . X . X O 2 O .
$$ | X O O 1 X O O .
$$ | X . O . X O . .
$$ | X X . X X O . .
$$ | . X X X X O . .
$$ | . . . . . . . .[/go]
If White makes the ko and Black takes, the resulting position has an average value of 13 - 5 1/9 = 7 7/9 pts., which is better for White than 10 pts.
This means that when the largest plays on the rest of the board gain less than 3 pts. on average, White should normally take sente, but when the largest plays elsewhere gain between 3 and 5 1/9 pts. on average, White should normally make the ko and exchange a play somewhere else for losing the ko. This may seem counterintuitive, but in that case White should make the ko in order to lose it.
Let's back up one move.
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[go]$$W 1 Black prisoner, White plays first
$$ ----------------
$$ | . X . 1 O . O .
$$ | X O O . X O O .
$$ | X . O . X O . .
$$ | X X . X X O . .
$$ | . X X X X O . .
$$ | . . . . . . . .[/go]
In this position White to play can prevent the ko for a local score of -1. If Black plays at 1 and White makes the ko (in order to lose it) Black will get 13 pts. in 2 net moves. Then each move is worth on average 4⅔ pts. ((13 + 1)/3). When the ambient temperature is less than 4⅔ and more than 3, and Black plays first, that is the expected play.
Now let's back up to the original position.
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[go]$$W White first
$$ ----------------
$$ | . X . 2 1 B O .
$$ | X O O 3 X O O .
$$ | X . O . X O . .
$$ | X X . X X O . .
$$ | . X X X X O . .
$$ | . . . . . . . .[/go]
Then when the ambient temperature is between 4⅔ and 3 we expect this line of play when White plays first, which is the main line.
takes the ko,
plays elsewhere, and
wins the ko. (OC, there may be a ko fight to force Black to fill.
) When the temperature is exactly 4⅔ each play gains on average 4⅔, and the territorial count is 8⅓.
When the temperature is 3 or less, we expect the following line of play.
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[go]$$W White first
$$ ----------------
$$ | . X . 2 1 B O .
$$ | X O O 4 X O O .
$$ | X . O . X O . .
$$ | X X . X X O . .
$$ | . X X X X O . .
$$ | . . . . . . . .[/go]
fills the ko
The local score is 10. The change in the territorial value from 8⅓ to 10 represents an average loss of 1⅔ pts. to White, which explains why White does not wait, but makes the ko in order to lose it.
Next I'll look at the White komaster situation.