Prisoner-counting for Stone-scoring
Robert Jasiek
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" 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