Prisoner Counting For Stone Scoring

Citation (Copyright: Robert Jasiek) from a rec.games.go article:

Subject: Prisoner-counting for Stone-scoring
Date: Thu, 12 Feb 2004 09:21:05 +0100
From: Robert Jasiek <jasiek@snafu.de>
Organization: jasiek
Newsgroups: rec.games.go

Introduction:

Prisoner-counting for stone-scoring is by far the fastest counting method ever. This is possible because the scored intersections are already filled.

The counting method is of great relevance for historians because its existence means that a historical text that mentions "prisoners"[1] is not sufficient evidence that territory-scoring would have been meant. Besides prisoner-counting for stone-scoring provides a natural theory how Japanese style rules might have emerged.

Rules:

- removed stones are kept as prisoners
- for each pass the opponent receives 1 prisoner
- 2 successive passes end the game
- White makes the last pass; this is 1 extra pass if necessary
- A player's score is the number of his stones on the board.
- The result is determined by the following count:
- white prisoner stones minus black prisoner stones.

Abbreviations:

Sb := black stones on board at game end
Sw := white stones on board at game end
Rb := removed black stones during game
Rw := removed white stones during game
Pb := black passes
Pw := white passes
Mb := black moves
Mw := white moves
Xb := prisoners that are black stones
Xw := prisoners that are white stones

From the rules it follows that
(1) stone-score := Sb - Sw
(2) prisoner-count := Xw - Xb

Proposition:

(1) = (2).

(In words: "The prisoner-count determines the stone-score.")

Proof:

From the rules it follows that
(3) Xb = Rb + Pb
(4) Xw = Rw + Pw
(5) Mb = Sb + Rb + Pb
(6) Mw = Sw + Rw + Pw
(7) Mb = Mw             //   <= White passes last

(5)(6)(7) =>
(8) Sb + Rb + Pb = Sw + Rw + Pw

Transformation:

Sb - Sw                 //   (1) can be transformed due to (8)
= Rw + Pw - (Rb + Pb)   //   can be transformed due to (3)(4)
= Xw - Xb               //   (2)

QED

Notes:

The following aspects do not matter:
- group tax
- seki parity
- board parity

Compared to the area-vs-territory proof, there area-score and territory-score both have the summond (Eb - Ew), which counts the numbers of empty intersections scoring for Black/White. This includes the group tax, which has no score meaning if stone-scoring is regarded alone; it has only a strategic meaning.

In practice, shortly before the game end one player can still fill his territory so that he does not self-atari while his opponent already has to pass.

It becomes easy to resign late during the game.

Instead of formally speaking of passes one could also express matters more informally; i.e. that might have been done centuries ago.

I postulated prisoner-counting for stone-scoring last year but only now I found time to do the proof.

The proof is surprisingly simple. Therefore, considering the mathematical skill centuries ago, and regarding the conveniently fast counting, it is pretty much possible that in former times prisoner-counting for stone-scoring had already been used.

Application to examples is left as an excercise:)

With a group tax, the counting method can be abused for area-scoring if the players fill their territory until self-atari during the alternation. Defining "group tax" precisely is left as another "excercise"...

Of course, if White starts the game, then Black should pass last.

-- robert jasiek

DougRidgway Dan Gilder has used this ruleset and counting method in a beginner's leaflet. (dead link as of 2022-12-30)

[1]

Bill: What document or documents are you referring to, Robert? Thanks.

RobertJasiek: No particular ones. I have not organized what I have read somewhere about historical rules information in any usefully accessable form yet. So I just do recall that there are historical rules-related reports that mention prisoners. Some such reports were overinterpreted as being sufficient evidence by themselves for territory-scoring. Instead they might be evidence for stone-scoring or area-scoring... Such reports would typically be without even some game records, not to mention diagrams or score calculations showing counting.

Robert Pauli:

• Why "copyright"? Isn't everything on RGG free to copy ? Is this proof worth it ?
• The scoring rule should simply be
  • A player's score is the number of prisoners he made
• Proving this to be the same as the plainest scoring rule of all - a player's score is the number of his stones on the board - is too trivial to bother.

RobertJasiek:

Copyright

Because I want it to be copyrighted.

RGG

E.g., something does not become free after being illegally copied on RGG. I use this simple example just to show you that obviously something is not free just by appearing on RGG.

Proof worth it?

Yes. Not because of the "difficulty" of performing the proof but because of the great creativity required to formulate the proposition and prove it at all.

Score

Can you please state it more precisely? E.g., I am not sure of you still want to use pass stones at all.

Equality?

I was wondering about "prisoner-scoring", i.e. like prisoner-counting for stone-scoring but without the rule that White passes last. My first guess is that it might lead to pass fights. IOW, that one would not be equivalent.

Robert Pauli: The equality, of course, basically depends on an equal number of stones in both supplies (to put it one way). Details left to you . . .