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5.2 Counting by Territory and Prisoners in Area Rules III

We have seen that the game has a boundary at which the outcome is not affected if one player is allowed to play twice consecutively. Let's consider whether a game played by area rules III can be counted by territory and prisoners.

Let n1 be the number of black stones played on the board up to the first pass (including handicap stones in a handicap game), and let n2 be the number of white stones. After this pass, play continues until the end of the game. Let A1 and A2 be the numbers of black and white stones on the board at the end of the game, and let B1 and B2 be the numbers of black and white stones removed by capture. Let C1 and C2 be the numbers of points surrounded by Black and White, respectively.

Let N1 be the total number of stones played by Black, including handicap stones, and N2 be the total number played by White. Then,

N1 = A1 + B1 and N2 = A2 + B2.

Black's and White's scores t1 and t2 are calculated as follows:*

t1 = A1 + C1 - (n1 - n2)/2
t2 = A2 + C2 + (n1 - n2)/2

The difference T between the two scores is:

T = t1 - t2 = (A1 - A2) + (C1 - C2) - (n1 - n2)

But (A1 - A2) = (N1 - B1) - (N2 - B2) = -(B1 - B2) + (N1 - N2), so

T = (C1 - C2) - (B1 - B2) + (N1 - n1) - (N2 - n2)

If the scores s1 and s2 are counted by territory and prisoners as follows, the difference will agree with territory rules III:

s1 = C1 - B1 + (N1 - n1)
s2 = C2 - B2 + (N2 - n2)

Think about what these equations mean. C1 is territory, B1 is prisoners, and (N1 - n1) is the number of black stones played after the first pass. The score in area rules III is being redefined as follows:

Score = surrounded points - prisoners + (number of stones played after first pass until end of game)

In general, after the first pass in area rules III, the player with fewer remaining moves will be able to play some extra moves in order to match the number played by his opponent. We can make this a requirement: the two players must play equal numbers of moves after the last competitive move; that is,

N1 - n1 = N2 - n2

Adding this requirement does not alter the outcome, and the two players' scores become:

s1 = C1 - B1
s2 = C2 - B2

In practice mistakes can easily be made in counting the number of stones played by the two players after the last competitive move, so playing the same number of stones after this point is a practical procedure.

We must remember, however, that the players cannot always play equal numbers of stones after the last competitive move. There are cases in which one player cannot play anywhere on the board. In these cases, the player should be allowed to play a move that immediately becomes a prisoner. That is, the player should be allowed to pass and give one stone to his opponent as a prisoner.

The rule can therefore require equal numbers of moves after the last competitive move, with the provision that one stone be given to the opponent as a prisoner for each pass after the last competitive move. There will rarely be cases in which one side has nowhere to play, so usually the players can actually play equal numbers of stones.

We have arrived at an important conclusion: go scored under area rules III as the sum of stones and territory can also be counted using territory and prisoners. This shows that the game is exactly the same, whether the object is to play more stones or to take more territory and prisoners. From the standpoint of traditional territory counting, it also means that if the number of moves played after the so-called end of the game is added to each player's score, the addition of this rule enables unusual positions to be completely resolved through actual play without causing any loss of points.

Another point to note here is that when the last competitive move is made correctly under area rules III, the number of remaining neutral points (other than one-sided neutral points) should be even. Accordingly, there is a difference from traditional territory rules, under which the last competitive move can occur even if the number of neutral points is odd. To make the neutral points truly neutral in territory rules, it will be necessary to add further changes to area rules III.

*
Implicit in these formulas is the provision that in a handicap game played under area rules, the handicap stones are not counted as part of Black's area. This is the standard Chinese practice.
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