Lasker-Maas rules for the game of Go (1995.March first draft)

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Robert Elton Maas, B.S. in mathematics, Putnam top 5

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Aug 9, 1995, 3:00:00 AM8/9/95
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"Lasker-Maas" rules for the game of "Go" -- First draft, 1995.Mar.19

(These are basically modeled after the rules in the appendix on Go in
the book "Modern Chess Strategy" by Edward Lasker, modified slightly by
Robert Elton Maas to be more precise, especially regarding how to end
the game and decide final life&death and count the score fairly.)

General: Go is played on a square matrix (usually represented by grid
of crossing lines where the intersections of the lines are the points
of play, usually physically slightly elongated from a square grid in
the direction that makes it appear square when viewed in perspective
from players seated across it, but exactly square when drawn on a
computer screen or printed representation or public display board).

General: One player places black stones on the points on the board
while the other places white stones.

General: The board starts empty, then players alternate placing a stone
of their color on some unoccupied point of the board, but may pass at
any time in the game rather than place a stone. There is an infinite
supply of black and white stones available for play (although in
practice only a few hundred are needed to play on a full-size i.e.
19-by-19 board). There is also a place to put "captured" stones, which
is initially empty (usually this is actually two separate places, one
for each color stones).

However, by agreement before the game begins, a "handicap" may be used
to give black a large advantage, either by allowing black a
pre-determined number of consecutive moves on the initial board before
allowing White to begin placing stones alternately, or by setting a
standard pattern of black stones on the board before White begins
placing stones alternately. Also to fine-tune this "handicap" a small
number of black stones can be placed in the captured-stone place before
the game begins (called "komi"). (Handicap is usually used to balance
the game so that black has a chance to win when white is much stronger.
Komi is usually used to compensate for black's first-move advantage
when the two players are of nearly identical level of skill.)

Definition: At any time, two stones of the same color which are
orthogonally adjacent (along rows or columns of matrix, i.e. along the
grid lines of physical representation of board, not diagonally) are
"directly connected". If there is a chain of such direct connections S1
S2 ... Sn such that each Si is directly connected to S(i+1), then S1 is
"connected" to Sn. Otherwise no stones are connected. (Lemma: Being
"connected" is a transitive relation). (Lemma: If two stones are
connected, they are of the same color.)

Definition: A "group" is a maximal set of mutually connected stones.
(I.e. a set of stones each connected to each other, and such that no
other stone is connected to any of them.) (Lemma: Any two distinct
groups have no stones in common. Thus each stone is a member of exactly
one group.)

Definition: For any stone, a "direct breathing space" for that stone is
an unoccupied space orthogonally adjacent to that stone. For any group,
a "breathing space" for that group is any direct breathing space of any
stone of the group.

Rule 1: A stone may be placed on ANY unoccupied point on the board,
except as restricted by rule 2 or rule 4.

Rule 2: A stone may not be placed such that in the resultant position
the group of which that stone is a member doesn't have any breathing space.

Rule 3: When a stone is placed, if it occupies the last breathing space
of one or more opponent's (oppositely-colored) groups, i.e. after
placing the stone the opponent's group(s) have no breathing space
whatsoever, such opponent's groups are "captured", i.e. removed from
the board immediately and placed in the prisoner's place. Even if the
stone that was placed was part of a group that didn't have any
breathing spaces initially, after removing the opponent's group(s) the
group now does have breathing space, so the play is allowed.

Rule 4: If, after placing a stone and removed any opponent's groups
that are accordingly captured according to rule 3, the position of the
whole board is exactly the same as it was any previous time of the same
game, that move is not allowed.

General: After placing any handicap or komi, then playing alternately
for a while using stones from the infinite supply, when both players
pass consecutively (without any intervening non-pass plays), the first
phase of the game is over. Play then continues just as before except
that stones to be placed on the board are taken not from the infinite
supply but from the prisoner's place. (Note: Rule 4 applies just as if
the two passes had never occurred.) If either color of prisoners
becomes empty, a pair of stones one of each color is moved from the
infinite supply to the prisoners' place (so that neither player will be
prevented from playing due to lack of prisoner stones). When both
players pass consecutively again (which must be two NEW passes, not one
of the first two passes plus an immediate third, but may be the third
and fourth consecutive pass immediately after the first two), the
second phase is finished.

Counting: At the end of the second phase, empty points are treated as
if they were a third color of stones for the purpose of defining
"connected" and "group". Any group of empty points such that one of the
points is orthogonally adjacent to any white stone, and one of the
points (same or other) is orthogonally adjacent to any black stone, is
not counted. Otherwise, any group of empty points is counted as
"territory" for whichever color stones some of its points touch, the
number of points counted being equal to the number of empty points in
the group. For each color, the number of stones remaining in the
prisoner place is subtracted from that color's score. The color with
the largest score is the winner. In case of tie, white wins. (To
compute a numerical margin of victory in such a way that ties are
avoided, a half point is always added to white's score, then the scores
for the two colors are subtracted.)

Comment: The first phase is "Japanese" style play, where placing stones
in one's own territory reduces it, while the second phase is "Chinese"
style play, where placing stones in one's own territory does not reduce
it and is necessary to remove dead enemy stones from the board in order
to claim the territory they occupied and the adjacent formerly empty
territory. Score is "Japanese" style, whereby plays inside your own
territory reduce the number of empty points thus reduce your score, and
may be "counted" at the end of the first phase of the game. The second
phase is merely the method of resolving questions of life and death
that the players couldn't simply agree upon at the end of the first
phase. Because plays during the second phase are from the prisoner
place instead of from the infinite supply, such resolution doesn't
change the Japanese-style score, because each such move both fills a
spot (reducing territory or increasing number of dead stones eventually
in prisoner place) and consumes a prisoner (thus decreases the number
of stones in prisoner place), which effects cancel.

Efficiency: At any point in the game the two players can agree to the
result of the game and cease play beyond that point. (For example, they
may agree certain groups are dead and the rest are alive, and that no
further moves inside one's own territory need be made, and allow a
computer to remove those agreed-dead stones and immediately count the
score.) (If under time controls, the time-control rules need to provide
some way to negotiate such agreement without causing either player to
forfeit the game due to running out of time during such negotiation.)

Pecularities: Traditional Japanese rules have a lot of pecular
situations that have special rules that go against these more clean
rules, such as bent four in corner, triple ko, etc. Traditional Chinese
rules discard the low-order bit of the score-difference, thus not
allowing as fine control of komi as Japanese and Lasker-Maas rules. The
rules by Ing have a weird definition of two kind of "ko"s that don't
make any sense to me. Even the new Japanese official rules have several
strange cases that are difficult to understand or which are outright
"wrong". By comparison, I know of only one very strange consequence of
the Lasker-Maas rules: If in approaching the end of the first stage,
one player has more surplus ko threats than the total number of dame
pairs, that player can gain an extra point by not filling the last
one-point ko. He takes the ko, and the oppoinent uses a dame as a ko
threat, player answers with another dame instead of filling the ko, and
opponent takes the ko. This player then makes a ko threat and retakes
the ko. This continues until all dame are exhausted, then the opponent
must pass rather than play a dame, and this player immediately passes
also to end the first phase without filling the ko. During stage two,
the other player still can't retake the ko, so fills one of his own
points of territory to capture some dead stones, and this player
immediately fills the ko. (Can any reader think of any other pecularity
of these rules, except on very tiny boards, i.e. smaller than three in
either dimension?)

Observation: If the rule were three passes instead of two to end the
first phase, or if the rule were "white must pass last", etc., it
wouldn't change anything, not even the unfilled ko pecularity, because
in computing the final score it doesn't matter who plays next after the
two or three passes: If the person to fill the ko is next, he'll fill
it immediately because it won't cost him a point to fill it during
phase two. If the other person is next, he can't legally fill it, and
he has no ko threats, so the other gets to fill it anyway. This
eliminates "pass tricks" whereby one person passes to avoid filling a
point of territory, and leaving one of his groups in atari, but then
relies on getting first move in phase two to resolve the atari by
making that necessary connection. Except for the unfilled-ko trick,
there is never any advantage to leaving 'necessary' connections undone
at the end of the phase one, and in that one case it doesn't matter who
gets first move in phase two. (Can anyone refute that? Can anyone prove
it?)

Meta: I've avoided Go jargon such as "ko" and "dame" in the actual
rules above (so the rules are self-contained), but used them liberally
in the side remarks such as parenthetical remarks and observations. I
hope that's fine with everyone.

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