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This diagram shows an occurrence of a double ko. (From Large Avalanche Turn Inward). The black group has one liberty (the marked eye) and two kos, giving him always two liberties. If White takes away a liberty by taking the ko at a, Black can take the ko at b. It is then up to White to find a ko threat. Black can answer the ko threat, and when White recaptures at b, Black takes at a. It is again White who has to play a ko threat, which Black can answer.
It should be clear that White cannot win in this situation - Black is alive in double ko. From the opposing view, White is dead in double ko.
This type of double ko provides the disadvantaged player (in this case, white) with an unlimited number of ko threats, which means he or she is likely to win any other kos that might arise on the board (provided they are smaller in size than the group living by double ko).
Note that in this kind of double ko life, it is necessary that some outside stones (in this case, the white stones at the top) die. It is also necessary that the inside group have an eye.
Here the double ko will normally be resolved into a single (two stage) ko, but the White stones that need to die are not part of any ko shape.
This is a double ko seki. If Black takes at a White takes at b, and vice versa. Then the first player to take cannot continue locally.
An interesting feature of double ko sekis is that they supply both players with an infinite number of unremovable ko threats. If another ko arises elsewhere on the board, the game will end in triple ko if neither player is willing to give something up. This can allow an otherwise dead group to stay on the board; see moonshine life.
For another example, see double ko seki.
Alternatively, if we remove the eye, white is also dead, but now with ko threats: taking one of the kos threatens to kill the black group. (Conversely, one could say that in this diagram, black is alive in double ko.)
Two dead kos
If the outside group is unconditionally alive, white is dead. There are no ko threats for anyone in this situation. Since a white ko capture is not sente, this is not a double ko.
Like in the previous diagram, the white strings are dead. It is not a double ko because a white ko capture is not sente.
This black group is involved in a number of kos, but they don't affect its status. Black cannot live by "winning" any or all of the kos. (If White wishes to take the stones off she simply connects at a. Then she can take the ko at b with atari, and the rest should be clear.)
Bill Fraser discovered this double ko. Note that the double ko does not affect the status of any group.
In this artificially constructed example, both the chain and the chain are connected to their respective outside stones via a common double ko. Neither of the marked chains can be captured, unless one side neglects to recapture `their' KO.
These problems involve double ko. This does not necessarily mean that the solution is double, but can also mean that some attempts fail because of double ko.
- Problem 3523
- Problem 3789
- Problem 4523
- Problem 4575
- Problem 5769
- Problem 8474
- Problem 9472
- Problem 11025