Unremovable ko for both sides
Consider this position:
Black cannot fill the ko at A3 (then white could create a dead eyeshape with C1). So eventually white can capture on A3, but white should not fill the ko afterwards! This is because as long as the threat of a direct ko exists, black needs to keep the ko threat on E8, leaving the two white stones on top alive for now. If white fills the ko, black no longer needs that insurance, and could safely capture 2 stones to gain 5 points.
So this is a case where neither player wants to fill the ko, but of course both sides would prefer to have it captured in their favor when entering scoring. Thus if the opponent just took the ko, you would rather play inside some territory (either your own or the opponents’) instead of passing. The one who runs out of such moves first will be forced to enter scoring without the ko, so if the result hinges on the ko the game has turned into a form of no-pass go, where optimal play can involve playing into the opponents territory to gain extra moves.
In this minimal example there are no territory-filling moves available, but that is what would happen if we embed this on a larger board with territories for both sides: the winner would be determined not only by the sizes of territories, but also by the shapes of those territories. (No-pass go deserves its own post in this thread in the future)
The position is taken from the Sensei’s article Unremovable ko for both sides, which states that it was discovered by user 序列號 in 2008. The idea of a ko that neither side can “finish” is fascinating, and it’s an interesting beast to consider under various rulesets. However, the above example is quite complicated and comes in multiple parts: the ko itself, and then two types of unremovable ko threats elsewhere on the board. The sizes of all the groups involved must be precisely balanced to make the ko truly unremovable.
Recently, a new example was discovered by Francisco Criado and René Martínez:
Go read the lifein19x19 post from Criado for a very nice explanation of how this one works! In my opinion it is much simpler than the earlier one, and has some desirable properties such as not relying on unremovable ko threats.
This one we can easily put in a global context where we can see the no-pass go play out:
Assume it is black to play. I believe correct play is as follows:
- Black pass.
- White takes ko.
- Black fills an eye.
- White pass.
- Black takes ko.
- White fills an eye.
…etc
In this case, black has 5 eye-filling moves available while white has only 4. Thus black will get to take the ko last, and claim B1+C1 as points.
If we took away one of black’s eyes in the starting position, it would instead be white that gets to capture last and claim B1+C1. In other words, simply adding a black stone on A7 changes the area score 4 points in white’s favor!
I made sure to fill the territories with 1-point eyes to remove the possibility of playing inside the opponent’s territory instead of your own. This lets us easily analyze the situation by just counting who has more eye-filling moves available. With bigger eyes, it becomes non-trivial to find the correct endgame sequence. Let’s look at examples of that in a future post - for now I just wanted to share this unremovable ko discovery.