Subject: Lasker-Maas rules for the game of Go (1995.March first draft)
From: rem@BTR.Com (Robert Elton Maas, B.S. in mathematics, Putnam top 5)
"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 removing 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.
Subject: Lasker-Maas rules for the game of Go (1995.August incomplete draft)
From: rem@BTR.Com (Robert Elton Maas, B.S. in mathematics, Putnam top 5)
The following is not complete, but merely the parts of my proposed rules that differ from the standard rules that are the same just about everywhere (Japan, China, AGA, Europe, New Zealand, but not Ing which are so weird they might not be Go at all). These rules are a slight modification by me from the rules in the Appendix on Go in Edward Lasker's "Modern Chess Strategy". (Oops, I wrote this all up from scratch, forgetting where I put another draft I wrote this past March, then before posting this I discovered the older file, so I'll post both at the same time now, then merge them sometime later.)
- Board and stones general description: (no change)
- Placement of handicap stones: A matter for tournament rules or personal agreement, not the rules themselves, but once they are placed and the game started they are considered the same as stones previously placed during the game in regard to liberties life/death etc.
- Black first unless special agreement such as fixed handicap, alternating: (no change)
Intersection = where the lines cross or meet (including edges and corners).
Group = maximal orthogonally-connected set of stones of the same color (a single stone not connected to any others counts as a group).
Group of the played stone = the group containing the stone that was just placed = the stone that was just placed, and any other stones which are now part of the group containing that stone (i.e. all the stones of groups of the same color having at least one stone adjacent to the stone that was just played).
Breathing space (for a group) = empty spot orthogonally adjacent to any stone of the group.
- Capturing: If any group of one color has no breathing space (because it had only one breathing space but that space has now been occupied by a stone of the other color that was just placed), ALL the stones of that group are immediately (before it is the next person's turn to pass or place a stone) removed from the board (i.e. the group is captured). If several groups of the same color satisfy this no-breathing-spaces condition, ALL of them are immediately captured (all the stones of all the captured groups are immediately removed).
- Legal moves: (Japanese/American rules, no suicide, but with SuperKo mostly per Lasker's original statement):
Whenever it is one player's turn to play next, that player may pass, or may place one of his stones on one of the intersections of the board subject to the following restrictions:
The intersection must have just-previously been empty (no stone on it).
Either the group of the placed stone must have at least one breathing space, or at least one group of the other color must be captured (per the earlier condition, no breathing space) thus creating breathing space for the group of the placed stone.
The resultant position (after placing the stone and removing the stones of any thereby captured groups of the other color) must NOT repeat the exact whole-board position at any previous time (at the end of a play) with the same person to play.
- Prisoners and supply of stones: During the first stage of the game, there is an unlimited supply of stones of both colors to use, one at a time, for placing on the board. Captured stones (stones removed from the board when their group is captured) are put in a special storage area that is empty at the start of the game (except that by special arrangement a number of prisoners of one color or the other may be put there to compensite for differing skill of the players and/or to compensite for the advantage of the first move). During the second stage of the game, stones are taken not from the unlimited supply but from the storage area for prisoners. If during the second stage the prisoners storage area becomes empty of one or the other color of stones, an equal number of stones of each color is put into the prisoners storage area to allow play to continue from that source of stones.
- Ending regular play: When the two players pass consecutively, the first stage of the game is ended and the second start begins. When the two players pass consecutively AFTER the first two passes, the second stage is ended and the counting begins.
- Final life/death and counting: All stones on the board at the end of the second stage are ruled alive for the purpose of counting. Any maximal orthogonally-connected sets of empty intersections which touch (have at least one empty intersection touching) only one color of stone, not the other, count as territory for the player who was placing stones of that color. Each stone still in the prisoner storage area counts against the player who was placing stones of that color. (An adjustment to this score may be agreed-upon before the game began, for example a half point in White's favor to avoid draws.)
- Early resolution: At any time the two players may agree to fill neutral points in parallel, or that certain groups are to be removed immediately without actually capturing them, and to end the game with these agreements before carrying out the full two stages specified above. (In timed games, some agreement must be specified whose clock will run under what circumstances when players are negotiation towards such an early resolution of the game.)
- Equivalence: Any method of play mathematically equivalent to the above can be substitued, such as a computer program that doesn't actually place stones on a board but merely updates its memory and its display of its memory, and keeps track of the number of prisoners and stones played during the two stages separately and combines the sub-totals in an appropriate way at the end of the game.
- Comment: The score here is equivalent to Japanese score, but without the fiat rules about what's alive and what's dead. Playing stage two from the prisoner's storage is equivalent to deciding what's alive and what's dead by the Chinese or Lasker "kill it, or else it's alive" rule, without screwing up the essentially Japanese score. Therefore score is in 1-point increments instead of the 2-point increments of the Chinese subtract-whole-territory counting method. As a result komi can be meaningfully adjusted in 1-point increments to have a 50% chance of better matching the diffence in skill and advantage of first move.
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