Cycle Law

[Robert Jasiek]

Proposition:

In a situational cycle the difference of passes equals
the negative difference of removed stones.

Proof:

Let Mb be the number of black moves during the cycle.
Let Mw be the number of white moves during the cycle.
Let Bb be the number of black plays during the cycle.
Let Bw be the number of white plays during the cycle.
Let Pb be the number of black passes during the cycle.
Let Pw be the number of white passes during the cycle.
Let Sb be the number of black stones on the board before the cycle.
Let Sw be the number of white stones on the board before the cycle.
Let Tb be the number of black stones on the board after the cycle.
Let Tw be the number of white stones on the board after the cycle.
Let Rb be the number of black stones removed during the cycle.
Let Rw be the number of white stones removed during the cycle.
Let D := Pb - Pw.

The cycle is situational => Mb = Mw.
It is a cycle => Sb = Tb AND Sw = Tw.
By definition of move, play, pass => Mb = Bb + Pb AND Mw = Bw + Pw.
By the above => Bb + Pb = Bw + Pw <=> Pb - Pw = Bw - Bb <=> D = Bw - Bb.
By definition of move and of removals =>
Sb + Bb = Tb + Rb AND Sw + Bw = Tw + Rw.
By the above => Bb = Rb AND Bw = Rw.
By the above => D = Rw - Rb.
QED.

Corollary 1:

Pb = Pw <=> Rb = Rw.

Note:

Especially Pb = Pw = 0 => Rb = Rw.

Corollary 2:

If the cycle is positional but not situational,
then D is adjusted by +-1.

Corollary 3:

For an equivalence proof of area and territory scoring
the following are requirements:

a) Mb = Mw.
b) The difference of pass stones equals D.

--
robert jasiek
http://www.snafu.de/~jasiek/rules.html