Almost Almost Fill
KarlKnechtel: To "almost-almost fill" (and a better term really is needed) an eyeshape is to fill all but two liberties inside.
Consider a group with a single eyeshape (the "large group"), which is almost-almost-filled with some enemy stones (the "small group"). Assume the large group cannot escape or connect out anywhere. What happens next depends on two basic variables:
- The number of outside liberties of the large group. Possibilities:
- None (case L0)
- One (case L1)
- More than that. (case L2)
- The shape of the small group, and the shape formed by adding a stone to the small group in either position. Possibilities:
- The shape is a killing shape, and at least one of the extensions makes a killing shape. (case S0)
- The shape is a killing shape, but neither extension makes a killing shape (case S1)
- The shape is not a killing shape, but at least one of the extensions will make a killing shape (case S2)
- The shape is not a killing shape, and neither is either extension (case S3)
- Whose turn it is.
The results are:
- With owner of large group moving first
L0 L1 L2 S0 R1 R1 R1 S1 R3 R3 R3 S2 R1 R2 R2 S3 R3 R2 R2
- With owner of small group moving first
L0 L1 L2 S0 R1 R1 R1 S1 R3 R3 R3 S2 R1 R1 R1 S3 R3 R3 R2
Here, the results are:
- R1: Large group dead, small group alive.
- R2: Large group alive, small group dead.
- R3: Both groups alive in seki.
Then again, all of this ignores the possibility of various ko situations. ^^;
- Either a or b by White threatens to kill.
- Either a or b by Black threatens to make Black alive with two eyes (not in seki any more) and kill the white stones.
Black's threat is worth 6 points: 4 points for eye space, plus 2 for white prisoners, minus the 0 existing value of the seki. White's threat is worth 34 points: 16 for black prisoners, plus 18 for resulting territory. (Not 36 points, as one might normally count, since the circled points aren't Black's territory to begin with.)
Proof that White threatens to kill: playing at both a and b results in this situation. We have S0 (the white stones make a squared four and a play at either marked point makes a bulky five) and thus R1 - Black dies.
When Black responds to White's threat, we have case S1 - the white stones make a triangle (a killing shape) but any extension makes a bent four (in case a) or a twisted four (in case b) - not killing shapes. S1 implies R3 - Black restores the seki. The indicated black plays are the only sufficient answer to the white threat.
When White responds to Black's threat, we again have S1, with identical analysis. Thus White restores the seki. The indicated white plays are the only sufficient answer to the black threat.
The original diagram is indeed seki; with alternating play, it reduces to an obvious seki no matter who starts. However, the position gives either player a ko threat. It is certainly in Black's interest to use this threat if a ko comes up, since doing so also denies a large ko threat to White. White should only use the threat if the ko is important enough, but it's still preferable to using some other 34-point ko threat. (I think.) Because of the huge difference in the threat value, it may be in Black's interest to play the threat immediately, so that White does not get the big threat later.
Of course, White does have a big threat regardless, since any seki offers an unremovable ko threat. However, while the first threat costs nothing and removes a small threat from Black, the second threat costs 8 points outright if Black answers. See Losing Ko Threat.
First of all, I think the definition of "Almost Almost Filled" (AAF) needs to be modified slightly. An AAF group is one where all inner liberties are shared by both Black and White (invader and invadee). For example:
Of course, you'd probably never see this in an actual game, but it helps illustrate the point. The above example has four liberties, all shared by the inner black and white groups. In such an AAF case, you now have three possible options (Damezumari not taken into account).
- White forces Black into a non-killing shape.
- Black forms a killing shape.
- White cannot force Black into a non-killing shape and Black cannot create a killing shape with only one liberty (almost filled).
Of course the names for these are:
The above example illustrates the seki point well. Black has a killing shape, Greek Cross, but the shape is still AAF. Black can create another killing shape, Hana Roku, at any of the circled points. If black does this, however, there are still 2 liberties left and therefore still AAF.
Assume, then, that Black makes the Hana Roku and we're left with a second AAF example.
Now, any move Black makes will be AAF, but will not be a killing shape. Any move White makes will be AAF and a killing shape. Therefore this results in seki. Neither Black nor White wishes to make the next move. It should be fairly easy to visualize White having the first move . . .
Again this results in seki (in fact, Black does not need to respond in this case).
Therefore we can prove (?) that any shape that's AAF and contains a killing shape is in seki. The only time this is not the case is when Black can move inside, reduce inside liberties to one, and still have a killing shape as below.