drmwc wrote:
Consider the case of 6.5 komi
Except for missing or wrong assumptions, let us study the mistakes you make.
For the discussed game, the komi is right though.
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Suppose black wins by 0.5 points,
This is one of the cases to be studied.
Mistake 1: You do not discuss the other most interesting case that White wins by 0.5.
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and white's territory was T,
It is possible to call it T.
Mistake 2: You do not specify whether there may still be dead stones in White's territory or whether you they have already become prisoners.
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with white having taken P prisoners and black 0 prisoners. (One can assume 0 black prisoners without loss of generality: black and white return an equal number prisoners until the black total is zero, so P may be negative.)
This construction is possible but I am not delighted as it creates a necessity for additional explanation of how to deal with this "without loss of generality" in practice.
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Then black must have T+P+7 points.
Note: Your 7 seems to be the sum of 6.5 komi wrongly accounted for Black instead of White plus 0.5 points of Black's winning margin. Compare Mistake 5.
Mistake 3: If White, from White's perspective, has T points (of empty intersections?) of territory, then, in general, Black, from Black's perspective, need not also have T points of territory. Instead, White can have Tw and Black can have Tb points. Furthermore, note that Tw or Tb might be odd numbers so that a sum or difference of Tw and Tb need not be an even number.
Mistake 4: Previously, you have set P as White's prisoners, from White's perspective, and, WLOG, 0 as Black's prisoners. Therefore, you may not now change Black's prisoners from 0 to P, from Black's perspective.
Mistake 5: The komi 6.5 favours White. If you account the komi for Black, you may therefore not add 6.5, but you must subtract 6.5.
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So the total territory is 2*T+7+P.
Note: You do not mean territory but the territory score.
Mistake 6: You apply previous values despite your previous mistakes.
Mistake 7: When considering both players' territories, we must assume either player's value perspective and form a difference of his territory minus the opponent's territory, such as Tw - Tb from White's perspective. However, your 2T = T + T forms a sum instead of a difference. Therefore, the total territory score does not contain a 2T. It would only make sense to use a sum when considering both players' total number of territory intersectiions. This you do not do; you refer to the territory score, i.e., an expression that might contain a difference such as Tw - Tb.
Mistake 8: In genral, 2T is wrong because the players need not have the same amount of territory or same numbers of territory intersections.
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Now consider the number of stones on the board after the last dame point is filled,
Note: that the last play occupies the last dame is a consequence of your too restricted related assumption.
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assuming no seki.
For the basic case, this assumption may be made. However, more cases need to be considered because seki is not that rare.
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Suppose there are B black stones.
Note: you mean black stones on the board. Of course, the value may be called B.
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Then there are B+P white stones if white played the last dame; or B+P-1 if black played it.
Mistake 9: The live (non-seki) white stones on the board do not magically disappear. They are also not accounted in P because live stones on the board do not become prisoners. The without loss of generality construction does not let them disappear, either, because it only accounts prisoners but does not account live stones. You do also need the number, say, W of life white stones on the board, or consider the difference B - W or, opposite perspective, W - B or define B (then better called S) to be such a difference.
Note: Due to your incomplete or wrong accounting of live stones on the board, I cannot judge yet how many, if any, additional mistakes your stone numbers of the last play cases might contain.
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Under the no seki assumption, the total number of stones on the board plus territory must equal 361.
Note: This presumes correcting your assumptions, such as all teire already being filled.
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Hence the total territory plus number of stones is 2*T+7+2*P+2*B if white played last,
Notes: Due to your mistakes, I cannot be sure what you mean by "total territory" here. I also cannot be sure whether the addition of 2P and the addition of 2B are the right operations or either should be subtraction. Besides 2P or 2B might have to be subtituted by differences, see above.
Mistake 10: You started with White's prisoners P, then made the mistake to derive Black's prisoners P. Your new mistake is to use this sum P + P = 2P in your term.
Mistake 11: Likewise, your new mistake is to use the sum B + B = 2B in your term, firstly because the players might have different numbers of live stones on the board and secondly because it must be clarified whether a difference W - B or B - W needs to be used here.
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which is odd.
Mistake 12: Given your earlier mistakes, this conclusion is premature.
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So this could equal 361, and is possible.
Note: Since now you compare your term to 361 (the number of board intersections), it remains necessary to convey the meaning of "the total territory plus number of stones", which also includes the '7'.
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If black played last, the total is 2*T+6+2*P+2*B, which is even, which is impossible as 361 is odd.
Mistake 13: as before.
Note: as before, but now you account 6 = 7 - 1, where the -1 deserves explanation.