mitsun wrote: Oops, I wrote this up a few days ago, in response to the previous problem, but forgot to post it. Covering old ground now, but with a slightly different counting method, and perhaps a bit less math. (I also neglected the possible ko in the lower left.)
The lower left has a gote play worth 7 points (difference counting). The center has a gote play worth 5 points. The lower right has a gote play worth 2 points. The upper left has a gote play worth 1 point.
Quite so. 
Quote: The upper right is a bit more complicated. There is a play for B that gains 3 points immediately and creates an 8 point follow-up. In games like this where the follow-up is large compared to other moves on the board, this play can be evaluated as 3 point sente. In other situations, it might be evaluated as gote 3+(1/2)*8, where the (1/2) is some sort of probability or statistical average over board positions. W might evaluate this as (2)*3 point reverse sente, but again the (2) is only a general estimate.
It is true that on some boards by correct play this corner will be played as gote by Black. However, it is classified as a sente, and is counted as three points of territory for Black. The reverse sente gains three points.
Quote: If B starts with the UR play, perhaps realizing that it is sente, then W responds as expected. The remaining gote plays are then taken in order of size. If I counted correctly, B is ahead by only 6 points on the board. This agrees with the count {+3,+7,-5,+2,-1} = +6.
If B starts with the LL gote, W should take the large center gote. B can again take the UR play in sente. The remaining gote plays are then taken in order of size. B is again ahead by only 6 points on the board. This agrees with the count {+7,-5,+3,+2,-1} = +6. This is the same as the previous play, just changing the sequence a little.
If B starts with the LL gote, it is a mistake for W to take the UR play (perhaps reasoning that it is reverse sente). The remaining gote plays are then taken in order of size. This time B is ahead by 7 points on the board. In some rulesets, I guess W has to pass at the end, giving B one more point. This agrees with the count {+7,-3,+5,-2,+1} = +8.
Difference counting (what I am calling swing counting) may be useful for comparisons, but it does not tell us what each play gains. Somehow this is seldom taught, leading to much confusion.
In the original position Black is ahead by 4.5 points. Then with correct play Black gains 1.5 points for a total of 6 points. With komi White wins by 0.5.
Variation 1: Black starts with the sente in the top right. Sente gains nothing, so the board count remains 4.5. Then Black gains 3.5 in the bottom left, for a count of 8. Then White gains 2.5 in the center right, for a count of 5.5. Then Black gains 1 point on the bottom side, for a count of 6.5. Then White gains 0.5 point on the left side, for a board score of 6. As advertised.
Variation 2: Black starts with the gote in the bottom left, gaining 3.5 points for a count of 8. then White takes the reverse sente in the top right, gaining 3 points for a count of 5. Then Black gains 2.5 in the center right, for a count of 7.5. Then White gains 1 point on the bottom side, for a count of 6.5. The Black gains 0.5 point on the left side, for a board score of 7. As advertised.
To show that the original board count is 4.5, the board score should be 3 if we let White play first, taking the gote while avoiding the ko, and letting Black take the sente in the top right.
- Click Here To Show Diagram Code
[go]$$Wc White first
$$ ---------------------------
$$ | . . . . . X . O O 5 4 O . |
$$ | X X . . . X . O X O X X X |
$$ | O X X . . X X X X O O X O |
$$ | O O X , X . X O O O X X . |
$$ | . O O X . X O . . O O X . |
$$ | . . . . X O O . O O X X . |
$$ | . . O . X O X . O X X . . |
$$ | . . O X X O O O . 6 O X X |
$$ | . . O X . X O . X . O O X |
$$ | . O O O X . X O O O . . O |
$$ | . O X X X X . X O . O O . |
$$ | O X 1 2 . . . X X O . . . |
$$ | . 3 . . . . . . . . . . . |
$$ ---------------------------[/go]
- Click Here To Show Diagram Code
[go]$$Wcm7 White first
$$ ---------------------------
$$ | . . . . . X . O O O X O . |
$$ | X X . . . X . O X O X X X |
$$ | O X X . . X X X X O O X O |
$$ | O O X , X . X O O O X X . |
$$ | . O O X . X O . . O O X . |
$$ | . 7 6 8 X O O . O O X X . |
$$ | . . O . X O X . O X X . . |
$$ | . . O X X O O O . X O X X |
$$ | . . O X . X O . X . O O X |
$$ | . O O O X . X O O O . . O |
$$ | . O X X X X . X O . O O . |
$$ | O . O X . . . X X O . . . |
$$ | . O 5 4 . . . 2 1 3 . . . |
$$ ---------------------------[/go]
Board score:
Black has 22 points in the group spanning the board from top to bottom, plus 10 points on the right, for a total of 32.
White has 13 points on the left side plus 6 points in the center and 10 points in the bottom right corner, for a total of 29.
Net score: +3. As advertised.
Edit: If White is still komonster, which I believe is the case, White can make the ko on the bottom side by playing  at 10, and later forcing Black to take the half point on the left side before gaining one point by filling the ko. That way White gets 2 points on the bottom side, for a net score of +2. The gains and losses still add up:
4.5 - 3.5 + 2.5 - 1 + 0.5 - 1 = 2
Edit: Actually, to be accurate, it costs White 2/3 point (0.67) to set up the ko. Sente may gain nothing, but it can lose something. Later filling the ko gains 1.67. My mistake above. Here is the corrected accounting:
4.5 - 3.5 + 2.5 - 1 + 0.67 + 0.5 - 1.67 = 2
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elsewhere
takes back at 
takes the 
stone is en prise. Now we know already that this position is gote with a play gaining 1 point. Let’s continue play at temperature 1.
elsewhere
moves to the position in the previous note. We already know that 
takes back at 1, 
elsewhere
are both reverses called hane-and-connect. Each reverse gains 1 point.
is a 5 move reverse that gains 1.5 points.
at 2.
captures the 
is ambiguous, but we count it like a sente. With the captured stone White makes 4 points locally. The swing is 7 points. Each play gains 3.5 points.
is a slight inaccuracy. The solid connection is technically better. White still wins, though. 
take ko.
takes ko 
fills ko at 25. 
takes ko at 23. 
is the winning ko threat.
??
It looks like it threatens to make another point by a White play at e. However that is never going to happen because e is a ko threat for Black. So all it does is give White one more ko threat at c. But it gives Black 