Combinatorial Game Theory
who wins, and by how much
Generator

One finished board, two scores

One finished board, two scores
One finished board, two scores. A Go endgame as its accounting: some plays worth points and some neutral points somebody has to fill. Under territory scoring a side counts the empty points it surrounds, so filling a neutral point is worth nothing. Under area scoring it counts its stones as well, so filling one is worth a point. The two conventions therefore differ by the number of stones each side places, which after the neutral points are filled is decided by their parity — and neither player did anything differently.

A Go endgame as its accounting: some plays worth points and some neutral points somebody has to fill. Under territory scoring a side counts the empty points it surrounds, so filling a neutral point is worth nothing. Under area scoring it counts its stones as well, so filling one is worth a point. The two conventions therefore differ by the number of stones each side places, which after the neutral points are filled is decided by their parity — and neither player did anything differently.

1 essay calls go-scoring. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every one of these figures is a drawing of the same construction, so a correction to it corrects them all.

Where the two conventions come apart, counted. Every small Go endgame solved under both scoring conventions. The scores agree exactly when the number of neutral points is even and never when it is odd, which is the parity of the stones each side ends up placing. A counted fraction name different winners. And a smaller fraction are played differently, which is the half of the finding a rules argument does not predict: a neutral point is a one-point play under one convention and worth nothing under the other, so the two rule sets disagree about the order of the endgame and not only about its total. Out in the world

Two ways to count a finished board

Territory scoring and area scoring are both in daily use and they are not variants of one rule. Over seventy-nine small endgames they agree exactly on the thirty-nine with an even number of neutral points and on none of the forty with an odd number — sixteen name a different winner, and twelve are played differently, which is not something a convention is supposed to do.

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