1996-10-05 last update, 1996-10-05 first day, Robert
Copyright: All rights of the author are preserved according to the
This page gives examples with comments referring to super
ko. If not explicitly differently stated, comments are valid for positional
super ko and for situational super ko. The Tromp-Taylor rules with either
super ko rule shall be the presupposed rule set. Besides SK for super ko
further Conventions are defined. If a whole
string should have marks on it, just ignore them; they are immaterial.
- fight, W+-[att..]
- In any basic ko after one player has captured in it, the other player
must not recapture immediately, because the resulting whole board position
would be the same as before the capture of the one player. (Note: Also
the same player moves, what is important under situational SK.)
- For example, W[attatt] is possible.
- coexistence, B+[pp],
- The position arises shortly before the game end. Let us assume B to
play. B[p] or B[a] are conceivable.
- B[pp]# ends the game with both sides remaining on the board.
- B[pbapp]# would end the game.
- B[pbapb]# forces W to get another liberty, but B[pbapba]# is prohibited,
because it would repeat the starting position. W must pass and looses all.
So B[pb]# has been a mistake and B[pp]# would have been correct.
- Under positional SK B[abpab]# is forbidden, because it would repeat
the whole board. B must pass and would loose all. Thus B[a] is a mistake.
Under positional SK only B[pp] is possible. This means coexistence.
- Under situational SK B[abpab]# is allowed, since it is W's turn then.
After B[abpabp] B must pass, because B[abpabpa]# would repeat the situation
after B[a]. So B[abpabpp]# ends the game with coexistence after some superfluous
moves; B[pp] would have been enough to end the game with the same outcome.
- Now let us assume that the double ko exists together with another basic
ko c with a W ko stone in it on the board, Both players might try to use
the double ko for ko threats. SK does not allow B[cbacba]#. If W does not
want to sacrifice the double ko, he should have better not played B[cb]#.
This means that ko threats for c should be searched elsewhere only. The
double ko behaves neutral as to c.
- W[attatt..] is a possible ko fight.
- W[acbacttbacbatt..] is another possibility of equal character.
- The players may choose between move-sequences of 1 or 5 moves in the
triple ko, but they need to search their threats alternatively in each
- B[cttctt..] is a possible ko fight.
- B[cbacbttacbactt..] is another possibility of similar character.
- Play closely resembles triple ko.
- W can be removed before the end of the game: B starts a sequence of
5 moves, then no W ko threat - even on a bigger board - would alter the
character of the position and finally W is left with nothing else than
a pass play.
- Under positional SK any try by W to capture B, e. g. W[cbp], can be
countered, e. g. W[cbpcb]# is prohibited.
- Under situational SK any try by W to capture B, e. g. W[cbp], can be
countered, e. g. W[cbpcbacba]# is prohibited. When analysing this MS you
will notice the difficulty of keeping track of a situation including the
right to move in comparison with the position only. More variety as to
possible MSs is provided, but the outcome is the same.
- In a similar position with the ko mouth a being external, but still
adjacent to the big W string the analyses is similar.
- If W tries to prevent removal, B[abcabc]# is prohibited. W cannot avoid
- Assume W to have had the last move for a capture at a.
- B gets no more chance to capture at a. So he also cannot play at b.
B will not play at c or d.
- If B would not already have the marked stone on the board, he could
vary the board position and remove W: W[aMpapbpp].
- If W should have brilliantly played yose, so that B had no chance to
remove at a and b, W deserves life in the corner.
Sending-2-returning-1 and bent-4
- B[pp] ends the game with coexistence.
- Under positional SK B[aca]# or B[bcb]# are disallowed. So just B[pp]
might be played.
- Under situational SK B[aca]# or B[bcb]# are allowed. E.g. B[acapp]
ends the game with coexistence. B[acapb]# is unwise, since B[acapbcb]#
would repeat a situation and be prohibited.
- B[cpp] ends the game with coexistence.
- B[cpabac] gives a bent-4 ko. With SK B[cpabacbp]# allows B to capture
- B should try to kill something. B[abcdbadc]# is prohibited. B then
gets the right side and W the left.
- In history strong players interpreted positions by seeing "eyes"
in them. This resulted in rules with attempts of definitions for shape.
SK does not have problems of locality.
At b shall be a B stone, at d shall be a W stone
- The traditional name for the position is eternal life. Without restriction
as to repetition B[acbd]* would be possible.
- A proper MS with ko threats is B[acbttdacttbdattcbdtt..].
- Molasses ko seems to be the most interesting ko configuration. The
reason for the name will be obvious from the following MSs: In general
every fifth move of a game before its last stage is a tenuki.
- Til today Molasses ko is the only known example with different outcomes
under positional SK respectively situational SK.
- The special feature of Molasses ko is that not both players want to
pass successively, because the other player's strings can be threatened.
However, after four moves a pass play is required. Hence a halfcycle within
the ko configuration consists of an odd move number.
- Under positional SK B[cadbecbdapcadb]# is prohibited. W looses all,
because B has got the last tenuki. The previous fight on the rest of the
board was about getting this last tenuki. Skill was needed to get more
tenukis. Molasses ko under positional SK produces similar strategy to play
under the primitive ko rule.
- Under situational SK B[cadbpcbda]# is not prohibited, but B[cadbpcbdaecadbpcbda]#
neither. In general Molasses ko remains in coexistence til the end of a
game. The game will end with B[cadbecbdapp].
- For Molasses ko neither a removal of one colour nor a coexistence is
surprising; a removal can considered to be caused by the continuous availability
of a threatening move, a coexistence can considered to be caused by a balance
of liberties. One variety of SK is not chosen to fit a special desire for
shape interpretation; special examples with unfamiliar behaviour ought
not to cause preference for a rule set. Otherwise one will have to change
the rules each time when new spectacular examples are discovered.
- W[c] gives a sufficient number of liberties for the time being.
- A complete move-tree-analyses for quadruple ko is difficult under SK.
This is true for humans, not for computers. If you should be able to solve
the position, you might as well try the general n-tuple ko...
- Single pass plays make the analysis more difficult.
- A typical length of MSs with maximal length is about exponential in
the number of involved kos, i. e. 2^n for n kos.
- If both players should be able to read out the whole move-tree and
conclude that all MSs would result in coexistence, they might as well pass
as soon as possible. If their reading is incomplete and they want to kill,
then an extremely long MS follows.
- Maybe the outcome of a n-tuple ko can be given by an explicit mathematical
- Concerning future research: One would like to see a proof that for
any position respectively situation during a part of a cyclical MS a reasonably
defined outcome of all involved stones could be explicitly given. Then
for any kind of repetitive shape reading would be superfluous. (This is
wishful thinking. Ing ko rules have tought us just how arduous a general
shape analysis would be.)
Ring of 4 basic kos
Two double ko stones
- With ten involved points the position is rather intriguing.
At b shall be a B stone, at d shall be a W stone
- This is not too difficult.