1996-10-9 last update, 1996-7-7 first day
Robert Jasiek, jasiek@berlin.snafu.de
Copyright: All rights of the author are preserved according to the international law.

Examples 2

Introduction
Examples
Links

Introduction

This page continues Examples 1 and presents and comments on important, difficult, and extraordinary examples. See also The New Ko Rules. In general the comments refer to the new ko rules. In many examples analysis and allowed MSs under the Ing ko rules are different even if often the same conclusion is drawn; ko mouths belong to the same ko position, if their so called settlement is entangled. The examples are also interesting for study under super ko. Conventions are defined.

Examples

Ko coupling with two ko mouths and an adjacent ko mouth

There are two ko positions on the board: The ko position of the ko mouths b and c with a ko coupling and the ko position of the ko mouth a.
In the ko mouth a an infinite MS could be played. All its adjacent strings have two breaths. So the ko position a is neither a single DLife nor a single DDeath, but a FK. It is still a FK, if a B ko stone is in exactly one of the ko mouths b and c, since even with a B ko stone in a W could gain a second breath for the adjacent string.
Concerning the ko position b-c: Not all strings of the ko coupling have two breaths; it is not in a state of stable DLife. However, it is in a state of instable DLife, because B[c] would transform it into stable DLife: With a B ko stone in the ko mouth c all strings of the ko coupling have at least two breaths. And after a player would from the then existing configuration have played B[a], B[b], respectively W[c] and the breath number of the other player's string of the ko coupling have reduced to one, the other player immediately could regain a breath with B[ac]#, B[bc]#, respectively W[cb]#.
Now consider a B ko stone in exactly one of the ko mouths b or c: As described, the ko position b-c is now in a state of stable DLife. Either player with a ko capture in b or c would disturb b-c and could become disturber.
With a B ko stone in a and ko stones of different colour in b and c the ko position b-c is in a state of stable DDeath: Exactly two ko mouths of two board points each, the W string of the ko coupling has one breath, the B two breaths. W could disturb b-c and become disturber.
With a B ko stone in a and W ko stones in b and c the ko position b-c is in a state of instable DDeath: The W string of the ko coupling has exactly the breaths of b and c as its breaths.
As long as b-c is a ko position it is a DK and its set of board points remaines unchanged. This is relevant for the prohibition rule: It applies independently from a change between DLife and DDeath.
Since b-c can change between DLife and DDeath, the DLife will not have had the general nature of ko positions in a state of stable DLife: remaining on the board until the end of the game.
The interesting case is B to play: B[c].
Due to the basic ko rule W has no answer available and will pass or tenuki.
First let us assume a pass play: B[cpa] follows.
B[cpac] disturbs the DDeath. W becomes its disturber by rule of disturber.
B[cpacba] is to be expected. The ko stone MS in b-c has ended. Due to the disturber rule W has ceased to be disturber.
B[cpacbapp] follows, because B[cpacbapb]# would cause W to become again the disturber of b-c, of which the ko position is the same even if its ko type has changed, and this would happen from the same configuration of the stones on the union of all board points, that are not part of a ko position, and the same set of board points of the union of all ko positions, as it was at the moment of W first becoming the disturber of b-c, what is forbidden due to the prohibition rule.
B[cpacbappa]# is not forbidden by the prohibition rule, because during B's first play of a ko stone in a is the exception of the rule.
W again cannot disturb b-c due to rule of prohibition and must pass: B[cpacbappap]#.
B[cpacbappapcpppp] ends the game with W being removed.
With tenukis possible MSs are B[ctacbattabcatt..tt] or B[ctacbaptabcapt..pt]. With each tenuki the prohibition rule still allows a ko stone move at a and becoming disturber in b-c. Finally both player might pass instead of tenuki: [..pp]. Then the prohibition rule still allows B to play [..ppa]#, since B's preceding play of a ko stone in a falls under the exception, because it has been the first ko stone in the fighting ko during the period of repetition.
W, however, must pass now and looses its stones.
The Ing ko rules also conclude, that W will be removed. (See C14 in the official rules booklet.) They say, that it is only one ko position, a triple ko. It is a DDeath with W dead even with three W ko stones, because after B[b] or B[c] there is an asymmetry in the allowance of captures on b and c (Ing rules do not use a basic ko rule but a general variation terminology, that contains basic ko), a is external relative to the double ko part b-c with its asymmetry of a B eye breath favouring B, a is a general unreal breath due to B's outside unconditional life, and the whole ko position is a compound pattern of unreal breaths. The life and death of the W normal string is considered to be equivalent to the life and death of a W ko string in a. The analysis considers the features of the ko coupling without restricting a ko position to it. W is the player, who can disturb. In C14 B[acbapb]# would disturb for the second time from an unchanged rest of the board (invariation, which is not described clear as in the new ko prohibition rule) and W must pass instead. B[acbappapcpppp] follows and is identical to the new ko rule MS.
Unreasonableness of breath pattern analysis under Ing ko rules (1): Ko position is equivocal: Further ko mouths could be part of it. It is unclear, whether only board points of ko mouths belong to it. If the given ko position must never have further ko mouths, then the ko position would be ruled to be settled independently of the rest of the board; that would approach the practice of former Japanese rules.
Unreasonableness (2): The settlement of B's outside string must be presumed. A determination might be extremely difficult and depend on further ko mouths. So the settlement of the corner depends on the settlement of the outside and vice versa, thus the ko position is equivocal or in the case of doubt the whole board, what makes an analysis most complex also from a mathematical point of view.
Unreasonableness (3): "General unreal" depends on settlement.
Unreasonableness (4): If the ko position must not be called DLife at any time and a special rule is used, so that this ko position can be called DDeath, then innumerable exceptions will follow.
Unreasonableness (5): "unreal", "external", "eye breath", etc. are undefined.
Unreasonableness (6): If the eye breath would be a big region with B and W stones in it (and possibly ko mouths), it might be in fact no eye/territory breath, but unreal for B. This would be extremely hard to determine.
With the new ko rules no environment of ko positions exist and each ko position is maximally given by a ko coupling. So ko position is unequivocal and easy to determine in any case.

Triple ko stones (C3-C4 in Ing's SST Laws of Wei-ch'i / 1991)

The question is not, if the normal strings can live. If one side only succeeds in bringing its normal string to live, it loses. Go is a game, in which one wins by achieving more points. Also ko strings count. In this case, it is important, which side could make life for its triple ko stones.
The given MS, that is cyclical, is [1 2 .. 10] with 10 one point below 2. All captured stones are ko stones. 2-10-4 resp. 7-5-9 are triple ko stones.
Three ko mouths occur: 1-8, 3-6, and 2-7-10-5-4-9. No ko coupling exists. Thus there are three ko positions.
On the triple ko stone ko position no ko stone MS of infinite length can be played. Thus it is a single DLife. During the cyclical MS it is never adjacent to an empty board point. Thus it remains a ko position the whole time.
The ko mouth 1-8 is only a ko position, if the board point of 6 is not empty. During [3 4 5] it is no ko position. Else on the ko position 1-8 an infinite MS [1 8]* can be played; it is no single DLife. In the case of a W ko stone in the ko mouth all adjacent B strings already have two breaths. In the case of a B ko stone even the string of 6 can gain a second breath by connection. So neither the ko position is a single DDeath. Thus it is a FK.
The same applies for 3-6, which is no ko position during [8 9 10] and else a FK.
Note, that during the following MS only moves on the board points 1 .. 10 are played, so that it is a ko stone move-sequence. This is important for an application of the prohibition rule.
Before a move the ko positions currently on the board are 1-8 and 2-7-10-5-4-9.
[1] : Still only the rules of basic ko and fighting need to be applied. However, for following applications of the prohibition rule this will be the first ko stone move in a fighting ko during the starting ko stone move-sequence. Now all three ko mouths are ko positions.
[1 2] : In go language one would call 2 a ko threat of the ko 1-8. By definition of disturbing 2 disturbs the DLife. W becomes its disturber.
[1 2 3] : 1-8 ceases to be a ko position. The set of board points of all ko positions is given by the ko mouths 3-6 and 2-7-10-5-4-9. The current MS starting with 2 is not any longer a ko stone MS in the DLife, so it has no disturber any more.
[1 2 3 4 5 6] : 1-8 is revived as a ko position. There are now three ko positions.
[1 .. 7] : B becomes the disturber of 2-7-10-5-4-9.
[1 .. 8] : 3-6 ceases to be a ko position. B ceases to be disturber.
[1 .. 10 1] : 3-6 is a ko position again. B played a ko stone in 1-8 earlier, but that play as the exception is ignored for the prohibition rule. So [1 .. 10 1]# is not prohibited, even though it is a ko stone move in the same fighting ko.
[1 .. 10 1 2]# is prohibited by prohibition rule: W would again become disturber while the set of board points of all ko positions is the same. Also when W played [1 2]# all three ko positions were present. Thus W must tenuki now, before he can play at 2, 4, or 8. 4 may be a different disturbing move, but important is only, that by 4 as well as by 2 W would become disturber of 2-7-10-5-4-9.
W will connect at 3, B at 8, and B wins the game.

Molasses ko

B[cadbb] does not lead to a cyclical MS. c-d is no ko position.
a respectively b is temporarily no ko position.
The ko mouth a consists of exactly two board points. The B ko stone is adjacent to a string with two breaths. If the ko mouth had a W ko stone, it would be adjacent to at least one string, that would not get another breath with one move other than capturing the ko stone. Thus the ko position of the ko mouth a is a single DDeath.
After W[a] the ko position of the ko mouth a is still DDeath. The analysis does not give a different result. Instead of W[a] W[p] would also have provided all W strings, that are adjacent to the ko mouth, with at least two breaths. Thus W[a] does not disturb.
The more interesting case is B[c] : It is played from a position with a being ko position, a B stone in b and only W stones in c-d.
After B[c] the W string adjacent to the ko mouth a needs to capture the ko stone to gain a second breath. In the case of a W ko stone in the ko mouth the single B stone b could not gain a second breath with one move other than capturing a. So now the ko mouth a is no longer DDeath. It is a FK.
B[ca] : This is an exception of the prohibition rule for a position with a being a ko position, a B stone on b and c, and W stones on c and d.
B[cad] : This is played from a position with a being a ko position, a B stone on b and c, and W stones on c and d. a ceases to be a ko position. Now b is a single DDeath.
B[cadb] : This is played from a position with only b being a ko position, B stones on c and d, and a W stone on a. W disturbs the ko position of the ko mouth b. W becomes disturber of the DK. B must not recapture immediately due to rule of basic ko.
B[cadbpc] : This is played from a position with only b being a ko position, B stones on c and d, and a W stone on a. Now b is a FK.
B[cadbpcb] : This is an exception of the prohibition rule for a position with b being a ko position, B stones on c and d, and W stones on a and c. d ceases to be a ko position. a is a DDeath now.
B[cadbpcbd] : This is played from a position with only b being a ko position, B stones on c and d, and W stones on a and c. b ceases to be a ko position. a becomes a single DDeath.
B[cadbpcbda] : This is played from a position with only a being a ko position, a B stone on b, and W stones on c and d. B disturbs the DDeath a. He is disturber of a.
B[cadbpcbdapc]# is prohibited by prohibition rule, because as with B[c] a would have been a ko position, at b would have been a B stone, at c and d would have been W stones.
B[cadbpcbdapd] : This is played from a position with only a being a ko position, a B stone on b, and W stones on c and d. a is a FK now.
B[cadbpcbdapda] : This is an exception of the prohibition rule for a position with a being a ko position, B stones on b and d, and W stones on c and d.
B[cadbpcbdapdac] : This is played from a position with only a being a ko position, B stones on b and d, and W stones on c and d. a ceases to be a ko position; b becomes a DDeath.
B[cadbpcbdapdacb]# is prohibited by prohibition rule, because as with B[cadb]# b would have been a ko position, at a would have been a W stone, at c and d would have been B stones.
B[cadbpcbdapdacpa] : B wins all.
Molasses ko is an extreme borderline case between DDeath and FK and no ko position. This is one of the reasons for its difficulty under most rule sets.
Under Ing ko rules, because not both players can successively pass and leave the position as is during B[cadbpcbdap], the position (all involved points and the two surrounding strings) is unsettled, i. e. FK. Already after B[ca] the W strings a and c-d are hot and cannot be captured immediately.

Triple ko and eternal life; At b shall be a B stone, at d shall be a W stone

The triple ko with a ko coupling and the eternal life are both FKs.
B[a] is a ko stone in a fighting ko. This is the exception of the prohibition rule, but this will not become of importance. Due to rule of fighting W cannot answer at c.
B[ae] : Due to rule of fighting B must not answer at e or g.
After B[ae] B will capture 4 W stones of the eternal life and W wins the triple ko. B should have better chosen B[ga].
W[eaf] can be expected with W to play.
The rule of fighting has made the prohibition rule superfluous in this example.

Triple ko (one single ko stone and two double ko stones); at a shall be a B stone, at c shall be a W stone; B to play

B[edbacbdace] classifies all 5 involved short strings as ko strings. a-b, c-d, e are three ko mouths with the same ko coupling and thus form one ko position. Not both strings of the ko coupling have two breaths; it is no stable DLife. Neither b nor e is a move, that gives both strings of the ko coupling two breaths each; it is no instable DLife. So it is no DLife. The ko position does not have exactly two ko mouths; it is no DDeath. Thus it is a FK.
After B[bd], when W at a would reduce the breaths of the B string of the ko coupling to one and B could regain a second breath at e, the exception of the stable disturbing life definition comes into use: This ominous example has block-of-4 ko mouths. The W move at a would be adjacent to a ko string of the opposing colour, not adjacent to a W ko stone, and reduce the breath number of the B ko coupling string to one. The ko position still will be a FK.
The same exception applies, if no W stone would be at c. So whatever ko stone move-sequence will be played, the ko position is a FK.
Each played ko stone must be followed by a ko threat.
The Ing ko rules virtually restrict reasonable play in this special ko position to [ettett], but allow some intervening moves in the big ko mouths. Thereby all aji is eliminated by a win of the ko position.

A ko coupling and an adjacent ko mouth

Two ko positions are on the board. b-c with a ko coupling is a stable DLife independently of which ko stone is in a. With two B ko stones in b and c b-c is a instable DLife. With a B ko stone in a still not all adjacent W strings have only one breath. So a is a FK.
W's only chance is to win the FK and connect at a. Then all strings remain in coexistence.
With a B stone at a the ko mouth d is a single DDeath. B can win the ko at a by capturing d or connecting at a. He can then before the end of the game dissolve the ko position d and proceed by connecting at c (or b respectively). This would leave b (or c) as a single DDeath. A try to capture the W string of the ko coupling by disturbing at b instead would not succeed due to the prohibition rule, which would then allow B to disturb the ko position only once.
B[a] plays a ko stone in fighting ko as an exception to the prohibition rule. d is now a ko position.
B[ac] : W is disturber of b-c, while three ko positions are on the board.
B[acba] : W ends as disturber. d is no ko position any longer. W's move is also an exception of the prohibition rule, since it is played from three ko positions being on the board.
B[acbac] : B is disturber of b-c, while a and b-c are the ko positions on the board.
B[acbacba] : B ceases to be disturber and plays again a ko stone in the fighting ko a, while a and b-c are the ko positions on the board, what he is allowed to do due to the exception of the prohibition rule. Now again three ko positions are on the board.
W is prohibited to disturb b-c again with the three ko positions being on the board , so B[acbacbapa] and B wins all.
A W stone at e is useless for W as discussed in another example.
The nature of the prohibition rule is, that the first player to play a ko stone in a fighting ko of a string ko (more than one ko position played through together) has an advantage: In general the other player cannot use ko threats in other ko positions for the FK.
Under the Ing ko rules I suspect the following interpretation: It is not decided, who wins the upper left ko. So it is FK. The settlement is entangled, so only one ko position is on the board.

A ko coupling and an adjacent ko mouth

Two ko positions are on the board: a and with a ko coupling b-c. a is a single DDeath. b-c starts as a stable DLife.
With W[a] W is disturber of a. b-c is now in a state of stable DDeath.
Due to the rule of basic ko B cannot recapture a. So he disturbs b-c with W[ab]#. W ceases to be disturber of a.
W[abca] follows. B ceases to be disturber of b-c. b-c is again a stable DLife.
W cannot disturb a again. He might disturb b-c once, but then he can disturb nowhere. So he may as well pass now: W[abcapdpbpcp] and then B removes the last W string, too.
To solve the fate of all strings it has been necessery to start with the unbalanced ko position a and then to proceed to the other ko position. The start has come from a DDeath with W having the so called dead strings and being the only player able to disturb and thus getting a disadvantage in the following fight. So it could have been predicted, that W has been bound to loose.
Under the Ing ko rules I suspect the following interpretation: It is decided, that the upper left ko can only be won by B. So it is a DDeath. The settlement is entangled, so only one ko position is on the board. B wins all.

Moonshine life

b-c is a stable DLife. a is a single DDeath.
If before the end of the game B wants to prove W's death in the upper left corner, he must play at a.
W can resist by disturbing b-c once and only once: B[abcabcapd] can be played according to rules of basic ko and prohibition. Now with a B stone at d and a ko position less on the board the players could defiantly become disturber in b-c once again each before a final game end with the double ko remaining on the board.

A main cause for Ing ko creation

Here are two stable DDeaths. In both ko positions only B can be the disturber. After B[ab] B must not play in that ko position any more due to prohibition by basic ko rule. After B[abcd] B also must not play in the other ko any more. Now the prohibition rule forbids him to play a ko stone in either ko.

A problem of definition

When analysing ko positions under the Ing ko rules one must not believe one's eyes: Here is a position where black has a balanced pair of one external and one internal shared beath, what is equivalent to one eye breath. c-d are paired ko breaths. White has also an eye breath. a and b are a miai eye for black. It is disturbing life.
Under the new ko rules simply the definition of stable DLife applies.

Quadrupel ko

The ko position is an instable DLife. W[c] ensures a stable position. W[c] is a prologue, what creates the first stable position.
W[cd] gives the first disturbing move. B is the first disturber of the DK. With W[cdbcdbc] the ko position has been repeated for the first time since the position just before the occurance of the disturber. Now the disturber rule prohibits B to disturb again for the time being. B may, however, answer to a disurbing action initiated by W.
At some time the prohibition rule might eliminate the ban on either player as to becoming disturber in the ko position.

Sending-2-returning-1 and bent-4

The two single stones are ko stones. a-a-c is a ko mouth and a ko position, b-b-c is a ko mouth and a ko position as well. This is a case of overlapping ko mouths and ko positions. Both ko positions are single DLifes. a respectively b disturbs.
W will be removed, because W cannot win a bent-4 ko. B can remove W by B[apb].
If W resists: B[acapcpa] and B wins the bent-4.

Sending-2-returning-1

B cannot play c and a to play the bent-4 ko, because W has an irremovable large ko threat at d. The last point on the board, that is worth something, is c. If, however, B should want to get more, he might try a or b. Due to area counting this costs nothing.
The upper right corner is a "seki", i. e. coexistence. a-a-c and b-b-c are two single DLifes.
B might start the move-sequence B[aca]. With adding a stone to a ko stone in B[a] B is the disturber of a-a-c.
B may disturb both ko positions only once. Then he should take the last endgame point c.

A strange DDeath

The string of c-d is no ko string: Even W[cbdaacb] does not lead to a repetition of the board position. So the MS is not cyclical. a and b have a ko coupling, so a-b is a ko position.
a-b is in a state of instable DDeath and after B[a] it is in a state of stable DDeath.
W[c] leaves the ko position in instable DDeath. B answers with W[ca]# to give stable DDeath, in which only W could disturb.
W[cad] : This is again stable DDeath. But now it is B, who could disturb!
B is indeed forced to disturb: W[cadb]#.
W[cadba]# and B will be captured. So B should have started with B[a], if he got a chance.
W[cadbpcadb]# : Now it is W who disturbs.
W[cadbpcadba] : W will become captured, so he should not have passed.
The player to take first one breath of the opponent's string of the ko coupling wins the ko position, though it is a DDeath!

A strange DLife

Also the triple stones are ko stones: W[eadbfeadbf] is a cyclical MS.
The ko position has a ko coupling and consists of four ko mouths. It is in a state of stable DLife.
W[e] is forced. The reason, why it does not make sense to label this to be disturbing, is W[epdfda] resulting in another stable DLife. (Even though W[epd]# disturbs, due to the normal stone W[epdf]# W ceases to be disturber and the prohibition rule does not consider W's prior disturber existence for future restriction.)
With W[e] and the ko position remaining unchanged until the end of the game the difference of stone numbers on the ko position after filling in will be W+4.
With W[ea]# B becomes the disturber.
After W[eap]# the W normal string cannot be captured. However, W[eapd] is an endgame tesuji sequence leading to an equal number of unshared breaths and two further B normal stones on the board. The counting for the former ko position will be B+4.
W might try to improve his endgame with W[eadf], but still filling in will result in B+4.
If B should miss W[eadf]#, then W[eadbfe] : Still all four ko mouths exist.
W[eadbfep] : B cannot capture W. B is prohibited to disturb again due to prohibition rule.
For a W try to further improve on his result note, that this position will be the first to be repeated since the occurance of B as current disturber.
W[eadbfebcabca]# : B himself repeats a board position and the disturber rule now does not allow him to disturb as long as he will remain disturber.
W[eadbfebcabcab]# : W continues the ko stone MS in the DK after disallowance for B to disturb in the DK. W also disturbs. So due to rule of disturber B ceases to be disturber and now W becomes the disturber.
W[eadbfebcabcabcabca]# : This repeats the board position for the first time since the position just before the occurance of W as the disturber. So now disallowance applies for W. Further play will be restricted for both at least temporarily according to rule of prohibition. The filling in counting is again B+4.
Of course, after B's failure the correct endgame for W would have been W[eadbfed] with W+4.

A stable DLife

The ko position b-c has a ko coupling and is in a state of stable DLife.
With a W stone at a the ko position is in a state of instable DDeath.

A stable DLife

The ko position a-b has a ko coupling and is in a state of stable DLife. A breaths reduction by B[a] or B[b] could be compensated by W[..c]#.
If the ko position was not stable DLife, then it would be instable DDeath.
B[acab] brings all strings of the former ko coupling to life and gives one ko mouth to each player. So the philosophical observation of the nature of the ko position judges, that it is reasonable to call the ko position stable DLife rather than assume all W stones to be dead in DDeath.

A rare ko coupling

The ko position with a ko coupling starts from stable DLife.
B[b] : B disturbs. The ko position is a FK then.
B[ba] : The ko position is instable DLife then. So the rule of fighting will not apply for the next move.
A B move left of b then gives stable DLife.
A W move left of a then maintains stable DLife an disturbs.
B at b then is prohibited by the prohibition rule.
Whatever, the ko position will end as a so called seki.

A one-sided stable DLife

Under the new ko rules the ko position is in a state of stable DLife, but all W stones will be removed due to B's so called better eye shape.
With their unlimited knowledge about shapes the Ing ko rules will know about eye breaths and wisely call W dead in DDeath. That is part of the religion of the Ing ko rules: Even if humans cannot define eye breath, the rules know the truth. Still some doubters say, that they are not rules at all... (They ought to believe the truth.)

Ring of 4 single ko positions

This example offers the most serious difference between the new ko rules and the Ing ko rules.
Under the new ko rules no algorithm for global considerations is needed and four FKs, that are basic kos, are on the board. Due to the prohibition rule after B[abcdbadca] W must not play a ko stone and B wins all.
Under the Ing ko rules only one ko position is on the board, which is balanced and therefore a DLife. (Well, surely an algorithm exists for detection of the special case of rings of 2n single kos with a total of 4n breaths, of which 2n are eye breaths. But what will happen with removable surrounding strings?) B[abcdpcbadpppp] is possible.

Exercises

The infamous three ko positions

a and d are in moments, when a ko stone could be played in them, FKs.
The ko type of b-c changes with more relevance.
Possible pass plays or tenukis make an analysis a little bit more complex.
Will after W[b] the main part of W or B be removed or will all end as a coexistence?