Written by Wilton Kee on 30-Apr-2013.
1. Applicable to any types of boards with any sorts of connections between intersections for stone placement
2. Applicable to any number of players (even 3 or more)
3. All games end in finite number of plays, if the board is finite
4. All games end by consecutive pass plays of all active players, unless all players have been prohibited from playing
5. All games end with a definite score by scoring with a single board
1. Enclosure Rule is introduced to cater for 3+ player game. Full enclosure is required to prevent from being considered suicide. See Rule 9 (Enclosure Rule) and Rule 10 (Suicide Rule).
2. Rather than prohibiting certain stone play under a usual ko rule before the cycle is formed, Kee Ko Rule defines the ko player to penalise after the cycle is formed by prohibiting him from playing in the remainder of the game. This avoid the dilemma that either all possible plays are prohibited in certain situation or certain type of plays (e.g. pass plays) has to be exempted and a cycle is then formed without penalty. See Section F under "CONSIDERATION OF SOME KNOWN ANOMALIES".
3. The ko player is defined as the last player to play among players who made the least number of normal pass plays within the cycle.
4. Refreshing pass plays are introduced to preserve certain local outcome in a fully occupied board. For a player who is restricted by the usual ko rule (i.e. primary ko condition) to place a stone to alter a board position, he can now choose to make a refreshing pass play to relieve himself from the restriction and reject immediate scoring with such board position.
5. Along with the introduction of refreshing pass plays, secondary ko condition is introduced to restrict repetitive refreshing pass plays to ensure the game still ends with finite number of plays.
1. Flexibility in allowing different Enclosure Rule: The default definition of Enclosure Rule is not an obvious extension of a usual two-player game. One possible variant is to define an allied group as just a normal opponent stone unit without alliance with adjacent stone units of other opponents. This reduces the number of types of stone plays considered suicidal.
2. Flexibility in switching into allowing suicide: See Rule 10 (Suicide Rule)
3. Flexibility in switching into disallowing refreshing pass plays: This can be achieved by omitting Rule 5(c) and Rule 12(a)(ii) for refreshing pass play and Rule 12(b) for secondary ko condition
4. Compatible with both area scoring and territory scoring: Rule Section G is written for area scoring as default, but it can be changed to territory scoring as a rule variant
1. A board consists of a pre-defined number of intersections, with each intersection adjacent to some others in a pre-defined manner. In a usual game, a board is marked with 19 parallel vertical lines and 19 parallel horizontal lines, making 361 intersections.
2. The number of stones shall be sufficient to end a game. In a usual two-player game, lens-shaped black and white stones are used.
3. A usual game starts with an empty board, where all intersections on the board are unoccupied.
4. A usual game is played by two players. Each player possesses one colour of stones.
5. (IMPORTANT) In each turn, every player makes a play onces in a pre-defined order. A play is either one of the following:
(a) stone play: placing one stone on an unoccupied intersection of the board to occupy such intersection; or
(b) normal pass play: giving up the right of a stone play and choosing to do nothing on the board instead; or
(c) refreshing pass play: (to be omitted if refreshing pass plays are disallowed as a rule variant) also doing nothing on the board, but it has a different treatment than a normal pass play in Kee Ko Rule (Rule 12) and End of Game Rule (Rule 13).
In a usual two-player game, black player makes the first play, then white player, and then black player again and so on in alternation until the end of the game.
6. After a stone has been placed on an intersection of the board, it cannot be moved to any other intersection.
7. An unoccupied intersection which is adjacent to a stone on the board is called the liberty of such stone.
8. When a stone is adjacent to another stone of the same colour, these two stones are connected and belong to a single stone unit. The liberty of any one stone in the same stone unit is considered to be the liberty of the whole stone unit. Similarly, when an unoccupied intersection is adjacent to another unoccupied intersection, these two unoccupied intersections are connected and belong to a single unoccupied unit.
9. (IMPORTANT) Enclosure Rule: After a player makes a stone play, an allied group is defined as the union of stone units of all his opponents adjacent to each other. If a stone play is made such that there is an allied group in which none of the stone units have liberty, such stone play is enclosing.
is an enclosing stone play of Black.
is not enclosing because the white stone unit (with two stones) still have one liberty.
Here denotes the stone of the third player (named "Red") following White to play. Same definition for any diagrams thereafter.
is not enclosing because even though the white stone unit (with two stones) has no liberty, its allied group contains the red stone which still has liberty.
is enclosing because it takes away the remaining liberty of all stone units of the adjacent allied group.
is not enclosing even if it takes away the remaining liberty of one stone unit of both opponents, because the red stone unit is adjacent to the white stone unit on the bottom which still has liberty and belong to the same allied group.
is enclosing because it encloses one allied group (the white stone unit on the left) even if it does not enclose another allied group (the red and white stone units on the bottom).
is enclosing as long as it encloses one allied group, even if it does not have liberty and does not enclose another allied group.
10. (IMPORTANT) Suicide Rule: If a player makes a stone play which is not enclosing and it takes away the liberty of its stone unit, such stone play is suicidal.
(default rule) A suicidal stone play is prohibited.
(if suicide is allowed as a rule variant) If a stone play is suicidal, the stone unit of such stone play is immediately removed from the board.
Red's stone play on the trangle mark is suicidal even if it surrounds most of both allied groups because it encloses neither of them.
11. (IMPORTANT) Capture Rule: If a player makes a stone play which is not suicidal and it takes away the remaining liberty of any opponent stone units, such opponent stone units are immediately removed from the board.
Red's stone play on the trangle mark is suicidal, so it cannot capture the two adjacent white stones and all black stones will eventually be captured by White. If it were not considered as suicidal, it could capture the two adjacent white stones and Black could then play on to make himself alive there.
would capture the two white stones.
would capture both the two white stones and the one red stone.
would capture the two white stones and the one red stone.
would capture the one white stone on the left and the one red stone.
In general, whether a stone play is enclosing is important only if it does not have liberty because it would then affect whether it is suicidal. If it is suicidal, it cannot capture any opponent stones. As long as it is not suicidal, whether it is enclosing or not does not affect its ability to capture.
12. (IMPORTANT) Kee Ko Rule: The ko player is prohibited from playing in the remainder of the game once either one of the following two conditions is satisfied:
(a) Primary ko condition: A player has made a stone play or a normal pass play which results in a board position identical to one which has ever been encountered by his succeeding player since the latest of:
(i) the moment just after the start of the game if there has never been any player being prohibited from playing, or the moment just after the last player being prohibited from playing if otherwise
(ii) (to be omitted if refreshing pass plays are disallowed as a rule variant) the moment just after the latest refreshing pass play (if any)
and there has been at least one stone play since such encounter.
(b) Secondary ko condition: (to be omitted if refreshing pass plays are disallowed as a rule variant) A player has made a refreshing pass play on a board position identical to one which was encountered by his succeeding player at either one of the following two moments:
(i) the moment just after the start of the game if there has never been any player being prohibited from playing, or the moment just after the last player being prohibited from playing if otherwise
(ii) the moment just after any previous refreshing pass play made after the moment defined in (i) (if any)
The ko player is defined as the last player to play among players who made the least number of normal pass plays since such encounter.
13. (IMPORTANT) End of Game Rule: The game ends with scoring once either one of the following conditions is satisfied:
(a) All players who have not been prohibited from playing have consecutively made a normal pass play on a board position since the moment just after the start of the game if there has never been any player being prohibited from playing, or since the moment just after the last player being prohibited from playing if otherwise.
(b) All players have already been prohibited from playing.
14. The game also ends when the ranking can be agreed by all players.
15. If the ranking cannot be agreed by all players at the end of the game, the ranking would be determined by scoring based on the board position at the end of the game:
(a) Stones are removed from the board if all players agree.
(b) With respect to each player, each intersection occupied by a stone of such player on the board, as well as each unoccupied intersection which belongs to an unoccupied unit fully surrounded by stones of such player, is counted towards one point of such player.
(c) An unoccupied intersection which belong to an unoccupied unit surrounded by stones of more than one players on the board are not counted towards any point of any player.
(d) The ranking is determined according to the total points possessed by each player. The higher is the total points obtained by a player, the higher is the ranking.
See Hanezeki.
Observation: Anomalies exist in rule sets which attempts to rule out stone units with only one liberty in scoring.
Relevant treatment in Kee Rules: Every stone on the final board position is counted towards one point in scoring with no exceptions.
See Unremovable Ko.
Observation: Anomalies exist in rule sets which attempt to avoid scoring board positions with an unfilled single ko.
Relevant treatment in Kee Rules: All board positions can be used for scoring as long as all players not prohibited from playing have consecutively made a pass play.
See Unremovable Ko. Bulky five is usually dead, but White can create a ko threat inside the bulky five to gain either from it or from the irremovable ko elsewhere. However, an irremovable ko threat elsewhere is not sufficient, as an irremovable ko threat would be removed after Black's response to the threat. Black then has nothing to lose by placing a stone at (a) after .
See Bent four in the corner is dead. White can create a ko inside the bent four to gain either from it or from the irremovable ko elsewhere. Actually even just irremovable ko threat elsewhere is sufficient because White's capture of after the irremovable ko threat is still a ko. An example of irremovable ko threat which is not an irremovable ko is a typical seki.
See Unremovable Ko.
Observation: Anomalies exist in rule sets which attempt to score all positions locally (e.g. Japanese rules).
Relevant treatment in Kee Rules: Kee Rules rely on actual alternate plays of players globally to determine life and death.
See Rules Beast 1. Ko threat at lower-left corner can be triggered by after White's recapture.
Observation: Anomalies exist in rule sets which does not consider the number of normal pass plays within the cycle into consideration.
Relevant treatment in Kee Rules: The player who is prohibited from playing in the remainder of the game is the ko player, who is defined as the last player to play among players who made the least number of normal pass plays within the cycle.
Observation: Anomalies exist in rule sets which does not release the ko restriction before a game is scored.
Relevant treatment in Kee Rules: A player can make a refreshing pass play to be relieved from the restriction and reject immediate scoring with such board position. More elaboration in Section A under "EXAMPLES OF APPLICATION".
Black 9 is a pass play but it repeats the board position produced by Black 3. Its validity is questionable because it completes a cycle and White made a pass play earlier as well (White 6) within the cycle.
Observation: Anomalies exist in rule sets which always treat a pass play as a valid play. However, it is impractical to prohibit a pass play because a player may end up having no valid plays available.
Relevant treatment in Kee Rules: Rather than prohibiting certain stone plays but unconditionally allowing a pass play, Kee Rules determines the ko player to penalise after a full cycle is formed. The player making a pass play in a cycle may still be prohibited from playing in the remainder of the game if he is the last player to play among players who made the least number of normal pass plays within the cycle.
Question: Is White able to capture the black stones with only one liberty and one simple ko if there are no sensible unoccupied intersections to place stones?
Suppose Black captures the ko by . If White recaptures the ko immediately, he would be prohibited from playing in the remainder of the game under Rule 12(a) (the primary ko condition). So White would not immediately recapture the ko. Also, since White is not satisfied with the current board position, he can choose to make a refreshing pass play. After that, Black cannot fill the ko due to lack of liberty, so
can only be another pass play. It does not matter whether it is normal or refreshing.
After a refreshing pass play, White is now allowed to recapture the ko by . Again Black would not recapture the ko immediately under Rule 12(a). So
can only be a pass play. It does not matter whether it is normal or refreshing.
White can then capture all the black stones by .
Conclusion: White is able to win the game by capturing the black stones with only one liberty and one simple ko.
Question: Is Black able to capture the white stones with only one liberty plus one simple ko in a similar "fully occupied" board even if there is a third player (named "Red") in the game?
Unlike the previous example, White would not be immediately prohibited from playing in the remainder of the game if he recaptures the ko immediately by . However, if White passes, Red cannot capture the black stone under Suicide Rule (Rule 10) because it is not enclosing.
After White's recapture, if Red makes a normal pass play, it would trigger a cycle and White would be prohibited from playing under Rule 12(a) (primary ko condition) because Red made a normal pass play within the cycle but White as the last player who has not.
If White recaptures the ko again by , Red cannot make a normal pass play for the same reason. If Red makes a refreshing pass play this time again, White would be prohibited from playing under Rule 12(b) (secondary ko condition).
Conclusion: Red is not able to help White and Black is able to win the game by capturing all the white stones.
It can be clearly seen that Red is still not able to help White because the stone play of both of them to recapture is not enclosing and is considered suicidal under Suicide Rule (Rule 10). Simple ko is at most for two players to capture back and forth under the default combination of Enclosure Rule, Suicide Rule and Capture Rule, even in a 3+ player game.
has no immediate consequence, but as
is made as a normal pass play, Black is prohibited from playing in the remainder of the game because it triggers a cycle and White has a normal pass play within the cycle but Black has not. White wins by playing the remainder of the game by himself alone.
triggers a cycle, but Black is prohibited from playing in the remainder of the game because White has a normal pass play within the cycle but Black has not. White wins by playing the remainder of the game by himself alone.
Black 3 has to be a refreshing pass play to avoid scoring with such board position. Then a pass of White does not end a game even if White 4 is a normal pass play.
Black is allowed to capture at after Black 3 as a refreshing pass play.
Similarly, White 6 has to be a refreshing pass play to avoid scoring with such board position. Then a pass of Black does not end a game even if Black 7 is a normal pass play.
White is allowed to capture at after White 6 as a refreshing pass play.
Now Black 9 can neither be a normal pass play or a refreshing pass play. If it is a normal pass play, then a normal pass play of White 10 would end the game and White wins by 2 points (or 1 point using stone counting method). If it is a refreshing pass play, it would repeat Black 3 as a refreshing pass play and trigger secondary ko condition such that Black would be prohibited from playing in the remainder of the game because the number of normal pass play of both players are the same within the cycle.
Conclusion: White wins all 2 points (or 1 point using stone counting method). Therefore Black 1 should pass.
Question: Starting from an empty board, can Black do better than a draw?
Flow of plays with White's optimal plays (assuming no refreshing plays at the moment):
Assuming White would optimally respond in the way described below and no refreshing pass plays are made, below is the only possible flow of plays. We will discuss what if there are refreshing pass plays in later paragraphs.
White has no choice but to occupy the opposite corner. Note that there are no two different boards produced by Black which are followed by the same board produced by White here.
Suppose Black is not satisfied with a draw. Black would then occupy one more intersection. Each board in Situation W-1 corresponds to two possibilities of Black play.
White has no choice but to capture the two Black stones. If only the 1st, 2nd, 5th and 6th board position on the left are concerned (see Situation W-4), note that there are no two different boards produced by Black which are followed by the same board produced by White here.
Black can then occupy either one of the two possible unoccupied intersections. Each board in Situation W-2 corresponds to two possibilities of Black play.
Instead of capturing the Black stone, here White chooses to make a normal pass play. Note that there are no two different boards produced by Black which is followed by the same board produced by White here.
Black would then capture the two White stones. Similar to Situation W-2, there are actually only 4 possible combinations for Situation B-4. The left is already the simplified list.
Here White chooses to occupy either the lower-right corner or the upper-left corner. White has to control his response on Situation B-4 to give the four board positions shown on the left to win the game. Note that there are no two different boards produced by Black which is followed by the same board produced by White here.
Now Black has two sets of choices, i.e. either making a normal pass play to return to Situation B-2 (but only 1st, 2nd, 5th and 6th there), or placing a stone to produce Situation B-5. Note that this is the only situation created by White within the flow which Black is satisfied (better than a draw) and that is why here we consider the possibility of Black to make a normal pass play.
Note that by Situation W-4, Black can only produce either the first or the third on the left. However, the other two is relevant in case Black makes a refreshing pass play (to be discussed later).
White has no choice but to capture the three Black stones. Note that there are no two different boards produced by Black which is followed by the same board produced by White here.
Black has no choice but to occupy the opposite corner.
Here White chooses to make a normal pass play to return to Situation W-1.
The arguments why White would win if Black is not satisfied with a draw (assuming no refreshing pass plays at the moment):
1. There are two possible cycles here, one bigger from Situation W-1 to Situation B-6, and one smaller from Situation B-2 to Situation W-4.
2. If the game runs through the bigger cycle, White would win because he has two normal pass plays within the cycle but Black has none.
3. If the game runs through the smaller cycle, White would also win because:
(i) The cycle begins from a board produced by Black (Situation B-2).
(ii) Both players have the same number of normal pass plays within the cycle.
(iii) If White's response to Situation B-4 are controlled to give only Situation W-4 as shown, there are no two different boards produced by Black which is followed by the same board produced by White. This means if White has repeated the situation then Black would have repeated one step earlier.
Conclusion: With the pre-defined optimal plays by White as set out above, these arguments ensure White would win the game if there are no refreshing pass plays.
What if Black has made a refreshing pass play in between?
First of all, notice that Argument #2 above is valid regardless of where the cycle starts. Argument #3 above is also valid as long as the cycle starts from a situation produced by Black. Below is the optimal response of White upon a refreshing play by Black on each situation.
If it is made immediately after Situation W-1:
If it is made immediately after Situation W-2:
If it is made immediately after Situation W-3:
If it is made immediately after Situation W-4:
If it is made immediately after Situation W-5:
The arguments why White would win if Black is not satisfied with a draw (even if there are refreshing pass plays):
1. Using the above response to Black's refreshing pass plays, Black would either immediately lose (if refreshing pass play at Situation W-5) or always be the player who first repeats the situation within the original flow after the last refreshing pass play. When the situation repeats, he would be the ko player who is prohibited from playing because Black would never have more normal pass plays than White in any cycles formed within the original flow of plays.
2. This means Black has to make further refreshing pass play to avoid triggering primary ko condition when repeating the situation. Sooner or later, Black would make a refreshing pass play on the same board, definitely sooner than White. When the refreshing pass play repeats, He would still be the ko player who is prohibited from playing because Black would never have more normal pass plays than White in any cycles formed within the original flow of plays.
Conclusion: With the pre-defined optimal plays by White as set out above, these arguments ensure White would win the game even if there are refreshing pass plays. White wins all 4 points (or 3 points using stone counting method) as Black will sooner or later be prohibited from playing if he is not satisfied with a draw. Therefore Black 3 should pass.
Question: Starting from an empty board, can Black do better than a draw?
Flow of plays with White's optimal plays (assuming no refreshing plays at the moment):
Suppose Black is not satisfied with a draw. Black would then occupy one more intersection. Each board in Situation W-1 corresponds to two possibilities of Black play.
As Black is not satisfied with a draw, Black would then occupy one more intersection, either to form a line or a triangle. Each board in Situation W-2 corresponds to two possibilities of Black play.
There are actually only 4 possible combinations for Situation W-5. The above is the simplified list.
There are actually only 2 possible combinations for Situation W-6. The above is the simplified list.
Now Black has no choice but to capture the five white stones to return to Situation B-1.
The arguments why White would win if Black is not satisfied with a draw:
1. There is only one possible cycle here, namely from Situation B-1 to Situation W-6.
2. All the 12 board situations from Situation B-1 to Situation W-6 are different, except Situation B-4 and Situation W-4 where there is a normal pass play by White.
3. The cycle contains one normal pass plays by White but none by Black. This means White would win without refreshing pass plays.
4. If there are refreshing pass plays by Black, White can choose to make a normal pass play, which returns to the same situation within the original flow of plays. White would still win by repeating the situation because he has one normal pass play within the cycle but Black does not.
Conclusion: With the pre-defined optimal plays by White as set out above, these arguments ensure White would win the game no matter there are refreshing pass plays or not. White wins all 6 points (or 5 points using stone counting method) as Black will sooner or later be prohibited from playing if he is not satisfied with a draw. Therefore Black 5 should pass.
Question: Does the first player (Black) have any sente advantage to break a double ko seki and win under Kee Rules?
Suppose Black with sente is not satisfied with this being a seki. After and
, Black 3 has to be a refreshing pass play (assuming all intersections outside the region are occupied). White 4 can choose to be a normal pass play.
After and
, Black cannot make any further refreshing play because it repeats the refreshing play Black 3 and Black has the same number of normal pass play as White within the cycle.
Conclusion: Black 11 triggers secondary ko condition and Black would be prohibited from playing in the remainder of the game. So Black 11 can only be a normal pass play and the game ends by scoring with the position as if it is a seki. The first player (Black) does not have any sente advantage to break a double ko seki and win under Kee Rules.