# Logical proof of the equivalence of territory and area scoring

__Keywords__: Theory

## Theorem:

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).

## Proof of the theorem:

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 http://www.cs.cmu.edu/afs/cs.cmu.edu/Web/People/wjh/go/rules/AGA.commentary.html*

*See also Mathematics of Scoring for a more general treatment of the differences between territory and area scoring.*