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 socalled 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 onesided 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.
