Double Ko
Double ko refers to any situation in which a group has two kos, and these determine its status. Groups can live (including life via seki) in double ko, but they can also die in double ko.
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Examples of double ko
1. Life in double ko
This diagram shows an occurrence of a double ko. (From the Large Avalanche Turn Inward joseki). The black group has one liberty at 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 b, Black takes 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.
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.
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).
A smaller ko occurred in the top right corner. White uses as a ko threat and takes the top right ko at .
If Black uses a ko threat elsewhere () and takes the ko, White uses the other ko capture, , as a ko threat and retakes the top right ko. Black has wasted a ko threat for nothing.
2. Seki in double ko
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.
3. Death in double ko
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.)
4. Double ko connections
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.
5. Two kos in different locations
Sometimes, two separate kos occur simultaneously. In such cases, neither side can win both because the opponent can use a ko capture as a ko threat for the other ko. For an example, see BeginnerExercise136.
Similar shapes which are not double ko
If the surrounding Black group is unconditionally alive, White corner 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.
As 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. Black has only false eyes and simply dead.
Problems involving double 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
Double ko under Ing rules
In a double ko life situation such as the first diagram on this page, Ing rules may only allow the disadvantaged side to use it as a ko threat twice. See Ing Ko Rules and Disturbing Ko.