On an N x N board, suppose that prior to the preliminary end of the game
(the first two consecutive passes in territory rules I, equivalent to
the first pass in area rules III) Black has played M1 moves and White
has played M2 moves. We will continue to use 1 to designate Black and 2
to designate White. Let Li (i = 1, 2) be the number of stones the board.
Then Black has lost M1 - L1 prisoners and White has lost M2 - L2. Let
Ti be the number of stones played after the preliminary end, and Pi be
the number passed as prisoners. Let Qi be the number of stones on the
board at the end of the game, and let Si be the amount of territory
The number of prisoners captured after the preliminary end is Li + Ti -
Qi. The total number of prisoners is therefore Mi + Ti - Qi + Pi.
||Under area rules III, the scores are:
|Black Q1 + S1 - (M1 - M2)/2
White Q2 + S2 + (M1 - M2)/2
The final term (M1 - M2)/2 is the half point added and subtracted when
Black makes the last competitive move, which occurs when M1 - M2 = 1.
The difference D(area III) between Black's score and White's is:
||D(area III) = black - white = (S1 - S2) + (Q1 - Q2) - (M1 - M2)
Under territory rules I, the scores (territory minus prisoners) are:
||Black S1 - M1 - T1 + Q1 - P1
White S2 - M2 - T2 + Q2 - P2
D(ter. I) = (S1 - S2) + (Q1 - Q2) - (M1 - M2) - (T1 + P1 - T2 - P2)
By the rule of equal number of moves after the preliminary end of the game,
||T1 + P1 = T2 + P2
T1 + P1 - T2 - P2 = 0
D(ter. I) = (S1 - S2) + (Q1 - Q2) - (M1 - M2)
D(ter. I) = D(area III)
So we have proved that area rules III and territory rules I are in
complete agreement, even though one of them counts stones plus territory
while the other counts territory minus prisoners. This comes from the
addition of the last competitive move condition to area rules III.
Another consequence of this proof is that since area rules II do not
have the half-point adjustment, when Black makes the last competitive
move, area rules II are one point different from area rules III and
territory rules I. Note that this does not depend on the size N of the
N x N go board. There is a misconception that the one-point difference
arises only when N is an odd number, but we have seen that this is not
Table 2 gives the numerical values for the Go-Miyamoto game.