QUESTION 7
How to round fractions of local endgames with star, up, down?
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[go]$$B Example 1: count 1 1/2*
$$ . . . . . . . .
$$ . . . X X X X .
$$ O O O . . . X .
$$ . X . . X X X .
$$ . X X X X . . .
$$ . . . . . . . .[/go]
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[go]$$B Black follower B = 2 1/2
$$ . . . . . . . .
$$ . . . X X X X .
$$ O O O 1 . . X .
$$ . X . . X X X .
$$ . X X X X . . .
$$ . . . . . . . .[/go]
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[go]$$W White follower W = 1/2
$$ . . . . . . . .
$$ . . . X X X X .
$$ O O O 1 . . X .
$$ . X . . X X X .
$$ . X X X X . . .
$$ . . . . . . . .[/go]
The black follower has the count B = 2 1/2. The white follower has the count W = 1/2. Therefore, the combinatorial game in the initial position is {B|W} = {2 1/2|1/2} = 1 1/2 + {1|-1}. This chills to 1 1/2 + {0|0} = 1 1/2*.
1 1/2 belongs to this count of the local endgame, which is not an empty corridor but also carries a *.
For an empty corridor, we would round 1 1/2* like 1 1/2: if Black starts, round up to 2; if White starts, round down to 1.
This rounding also has a meaning here: if only this local endgame is on the board, the following sequences occur (for the sake of simplicity, I ignore the dame) with the results B' = 2 or W' = 1, as predicted by the aforementioned rounding:
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[go]$$B Black starts, result B' = 2
$$ . . . . . . . .
$$ . . . X X X X .
$$ O O O 1 . . X .
$$ . X 2 . X X X .
$$ . X X X X . . .
$$ . . . . . . . .[/go]
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[go]$$W White starts, result W' = 1
$$ . . . . . . . .
$$ . . . X X X X .
$$ O O O 1 2 . X .
$$ . X . . X X X .
$$ . X X X X . . .
$$ . . . . . . . .[/go]
However, if we only consider the first move, we get the intermediate positions with the count B = 2 1/2 or W = 1/2.
Recall the count 1 1/2* of the initial position. If we treat 1 1/2 and * separately, we can consider rounding of only the remainder * of the local combinatorial game. This is rounded up to 1 if Black starts or rounded down to -1 if White starts. Therefore, the starting Black achieves 1 1/2 + 1 = 2 1/2 or the starting White achieves 1 1/2 - 1 = 1/2. These are the counts of the intermediate positions.
Which rounding makes sense? Why? Are the relations between rounding and counts only accidental or which of the mentioned rounding techniques have a general scope of application?
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[go]$$B Example 2: count -1 1/2^
$$ . . . . . . . .
$$ . . O O O . . .
$$ . O O . O O . .
$$ . O . . . O O .
$$ . O . X . . O .
$$ . O O X O O O .
$$ . . . X . . . .
$$ . . . . . . . .[/go]
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[go]$$B Black follower B = -1/2
$$ . . . . . . . .
$$ . . O O O . . .
$$ . O O . O O . .
$$ . O . 1 . O O .
$$ . O . X e . O .
$$ . O O X O O O .
$$ . . . X . . . .
$$ . . . . . . . .[/go]
e = -1/2. f = -1*. Therefore the black follower has the count B = -1/2 and the white follower has the count W = e + f + (-1) = -2 1/2*.
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[go]$$W White follower W = -2 1/2*
$$ . . . . . . . .
$$ . . O O O . . .
$$ . O O . O O . .
$$ . O . 1 . O O .
$$ . O e X f . O .
$$ . O O X O O O .
$$ . . . X . . . .
$$ . . . . . . . .[/go]
The local endgame in the initial position of Example 2 is {B|W} = {-1/2|-2 1/2*} = -1 1/2 + {1|-1*} = -1 1/2 + {1 | -1 {0|0}}, which chills to -1 1/2 + {0 | 0 {0|0}} = -1 1/2 + {0 |{0|0}} = -1 1/2 + {0||0|0} = -1 1/2 + {0|*} = -1 1/2^. The calculation has already not been so easy and the UP is surprising in a room surrounded mostly by white stones. The real difficulty, however, is the question of correct rounding.
For an empty corridor, we would round -1 1/2^ like 1 1/2: if Black starts, round up to -1; if White starts, round down to -2.
This rounding also has a meaning here: if only this local endgame is on the board, the following sequences occur (for the sake of simplicity, I ignore the dames) with the results B' = -1 or W' = -2, as predicted by the aforementioned rounding:
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[go]$$B Black starts, result B' = -1
$$ . . . . . . . .
$$ . . O O O . . .
$$ . O O . O O . .
$$ . O . 1 . O O .
$$ . O . X 2 . O .
$$ . O O X O O O .
$$ . . . X . . . .
$$ . . . . . . . .[/go]
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[go]$$W White starts, result W' = -2
$$ . . . . . . . .
$$ . . O O O . . .
$$ . O O . O O . .
$$ . O . 1 . O O .
$$ . O 3 X 2 . O .
$$ . O O X O O O .
$$ . . . X . . . .
$$ . . . . . . . .[/go]
However, if we only consider the first move, we get the intermediate positions with the count B = -1/2 or W = -2 1/2*.
Recall the count -1 1/2^ of the initial position. If we treat -1 1/2 and ^ separately, we can consider rounding of only the remainder ^ of the local combinatorial game. This is rounded up to 1 if Black starts or rounded down to 0 if White starts. Therefore, the starting Black achieves -1 1/2 + 1 = -1/2 or the starting White achieves -1 1/2 + 0 = -1 1/2. For Black's start, this is the count B = -1/2 of the intermediate position. For White's start, the -1 1/2 is NOT the count W = -2 1/2* of the intermediate position.
Since the rounding behaviour with UP (and DOWN) differs from the behaviour with STAR, I have to ask my questions again for Example 2, but this time for UP and DOWN:
Which rounding makes sense? Why? Are the relations between rounding and counts only accidental or which of the mentioned rounding techniques have a general scope of application?
And you thought that rounding was easy... For infinitesimals, it is a matter of well-definition and correct relation to semantics.