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 in 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 [ext] 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.


Logical proof of the equivalence of territory and area scoring last edited by MtnViewMark on December 28, 2008 - 21:22
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