Logical proof of the equivalence of territory and area scoring
In a game, where
- there was no handicap,
- no player made any moves after his/her opponent passed
- and there are no sekis with eyes,
the following holds:
If White made the last move, territory scoring and area scoring give the same result. Otherwise there is a difference of one point (to Black's advantage under area scoring and to White's under territory scoring).
First assume that White has made the last move. We will use the following notation:
- Be: empty intersections surrounded by Black
- Bc: black stones captured by White
- Bs: black stones on the board
(and similar form We, Wc, Ws).
The result (positive if Black wins) under territory scoring will be Be+Wc-We-Bc, while under area scoring it will be Be+Bs-We-Ws
Now, the fact that White has made the last move, combined with the first two conditions, means that both players have made the same number of moves. Thus, Bs+Bc=Ws+Wc. Call this total T. Simple mathematics now gives:
territory result= Be+Wc-We-Bc = Be+(T-Ws)-We-(T-Bs) = Be-Ws-We+Bs = Be+Bs-(We+Ws) = area result
And thus, the two outcomes are the same. If Black made the last move, a similar calculation gives:
territory result = area result-1
See also Mathematics of Scoring for a more general treatment of the differences between territory and area scoring.