Cycle tampering
Let’s look at a few smaller variations of the wonderful beast shared by @Glunkolin above. The basic building block here is a seki where one side can repeatedly sacrifice two stones:
Without superko, black could infinitely repeat the sequence [A3 A1 A2 Pass]. This does not pose a problem for Japanese rules, because white gains one prisoner each cycle. However, in area scoring rules, we really need some rule such as positional superko to prevent this repetition.
There are multiple variations on generalized ko-rules, but the two most common are:
- Positional superko (PSK) forbids any move that would recreate an earlier board position.
- Situational superko (SSK) forbids any move that would recreate an earlier board position with the same player to move.
This means that in the sequence [A3 A1 A2] above, the final black move would not be legal under PSK (because the position is repeated), but it would be legal under SSK (because in the repeated position, the player to move has changed from black to white).
Note though that passing is always legal - superko restrictions only apply to board plays. So under SSK the sequence [A3 A2 A1 Pass] is totally fine, even though white’s final pass recreates the initial position with the same player to move. However, black would not then be able to continue the sequence with a new play at A3.
My personal preference is for PSK - it’s slightly cleaner to state, and it seems easier to keep track of. All analysis in this thread assumes PSK unless otherwise explicitly stated.
Normally, superko means that a position like the above would stay as seki. But in exceptional cases, strange things can happen:
At first glance, both white groups look alive. But black has this sneaky sequence:
- 3 cannot be at F5 because it would repeat the initial position.
- 5 cannot be at F5 because then white kills with D1.
After 7, all white moves are self-atari, so she passes. Then black plays 9, probably accompanied by an evil laugh:
White is forbidden from capturing the two stones, because that would repeat the position after move 6 in the sequence! As a result, all the white stones will die.
One way to make sense of what just happened is that normally, PSK prevents black from doing anything nasty in the corner. But by playing some forcing moves in the middle of the “cycle”, black turned PSK to his favor: suddenly white is the one who is forbidden from repeating, even though black was the one that initiated the cycle. Quite an evil trick indeed!
Seeing this threat in advance, white would do better by playing 4 at F5, sacrificing the bottom left group to save the upper group.
This particular example seems to be due to John Tromp himself, more discussion about it here.
Two crucial and somewhat “unrealistic” elements of the position above are:
- The ko threat at B2, which white could easily remove if she had the chance.
- The lack of non-passing moves for white after move 7.
Here is an alternate version with an unremovable ko threat and some black territory:
- Black can at any time exchange A5-A7 as a ko threat, but white cannot move first in the seki without killing herself.
- However, the black territory around G4 means that black can’t start the J7 sequence immediately, because white has several non-passing moves available.
- Correct play is for black to first play inside his own territory until it consists entirely of 1-point eyes (white cannot prevent this), and only after that start the J7 sequence.
- White will die somewhere, but she can choose quite freely where she will die. She may ignore the ko threat on the left and die there, or she may play an eye-filling move on the bottom instead of passing, or she may go along with the entire sequence and allow black to kill the top right.
What happens if we give white a third eye?
Now killing is no longer possible for black, white has a move available at G1, and there’s no way for black to take that away.
With all this in my mind, you may be able to solve the following problem!
How should black play from here, and what is the result when both players play optimally?
As usual: Area scoring, no komi, positional superko.
Click the diagram above if you want to play out sequences! This is especially helpful because OGS will prevent you from making moves that violate PSK. You may also download the sgf and open it in CGoban or some other program which may be more helpful with scoring than OGS.