No Eye versus No Eye Capturing Race

When both parties involved in the capturing race (semeai) have no eye, the fight is calculated as follows.

1. Fewer than Two Inside Liberties

If there is one or no inside liberty, there is no possibility of seki: one of the groups will die. It is the one with fewer outside liberties. If both groups have an equal number of outside liberties, the one who has sente wins the fight.

In a formula:

OL1 > OL2 => 2 dies, else sente kills

Example 1

No eye - no eye - no inside liberties  

There are 4 outside liberties for White (marked a) and 4 for Black (marked b). There is no inside liberty.

The player who has sente kills the other.

Example 2

No eye - no eye - 1 inside liberty  

There are 4 outside liberties for White (marked a) and 4 for Black (marked b). The circled point is the only inside liberty.

The player who has sente kills the other.

Demonstration  

Black goes first to give atari at 7 and capture at 9. Of course, White should realize this and keep 2, 4, 6 and 8 in reserve as ko-threats.

Example 3

No eye - no eye - 1 inside liberty  

There are 5 outside liberties for White (marked a): the utmost left is an approach move and as such is also counted as a liberty. Still 4 outside liberties for Black (marked b). The circled point is the only inside liberty.

Black is dead.

Demonstration  

So, Black should keep 1 to 9 in reserve as ko-threats.

2. Two or More Inside Liberties

If there are two or more inside liberties, seki is possible. The group with more outside liberties is alive. The threshold for the other group to live in seki is:

OL1 + 1 = OL2 + IL [1]

If the numbers are equal^[2], then sente decides whether the defender can make seki or the attacker can kill. If one number is larger^[3], the status is settled as seki or death. The oddity in this formula is that the attacker counts one of the inside liberties for himself.

GigaClon I don't quite understand this section could some one post a diagram?

[2]tderz: Above formula ^[1] can be rewritten as OL1 - OL2 = IL -1.
Ok, easier defined: if the difference in outside liberties^[4] between the two groups ( Δ = OL1 - OL2) minus the Fighting shared Inside Liberties (FIL, FIL>= 2) is zero, OL1 - OL2 - FIL = 0 then sente decides the outcome.

Read also CapturingRaceFormula (and CapturingRace) , where it has been described earlier:

''The side that needs to put the other into atari first must take responsibility for filling the shared liberties, and is called the attacker. The attacker will need an advantage in exclusive liberties to succeed; the difference in exclusive liberties is called Δ. The number of shared liberties he or she must fill before the other side is in atari defines F (FIL). To win, the attacker must have

Δ >= F

where Δ = F is the critical (unsettled) case where the attacker must move.
Which side is the attacker is determined first by comparing eyes: the side with the weaker eye must fill the shared liberties. Any eye beats no eye, all small eyes are the same, big eye beats small eye, and bigger eye beats big eye. If the eyes are the same, the side with more exclusive liberties must attack.
F is determined by the eye of the defender. F = the number of shared liberties S if the defender has an eye, or F = S - 1 if no eye.''

The groups in the following diagrams do not have eyes => F = S - 1 (S = IL), because White = the attacker must have one liberty (to avoid self-atari) while putting the defender in atari
F = IL - 1

IL=2 ; OL1, OL2 = 0, delta = 0, which is too small to win  

dia. [5]: The values are IL=2 ; Δ = OL1, OL2 = 0 => OL1 - OL2 - IL = -2 => the situation is settled as seki
Δ = 0 F = S - 1 = +1
Δ = 0 < F = +1, hence White cannot win.
Black has the same number of liberties and cannot win either,
which therefore results in seki

[edit]

Δ < F means that that there are too few liberties (e.g. OLw, OLb) to win the fight!

IL=2 ; OL1 = 1 (at o), OL2 = 0, delta = 1, delta = F  

dia. ^[6]: Δ = OL1 - OL2 = 1 - 0 = 1,
Δ - IL = 1 - 2 = -1

Δw = +1 seen from White F = S - 1 = 1

[edit]

=> Δ - F = 0 => __whoever will start (have sente) will decide the outcome => the situation is unsettled

• - White could kill ^[8] by approaching at any of the IL i and capture in time or
• - Black could achieve seki (return to dia ^[5] by playing o

[edit]

Δ = F means that that there just enough liberties - for either side - to win the fight! (capture / seki)

IL=2 ; OL1 = 2(o), OL2 = 0, delta = 2,  

dia. ^[7]: Δ = 2 ; Δ > F means the attacker (White) has more than enough excess liberties to win the fight. Here Black would waste his sente by adding o, it would not alter the outcome.
White can

• tenuki once and end up in the unsettled situation of dia. ^[6] or
• tenuki twice, resulting in seki of dia. ^[5]

delta - IL = 0, sente decides  

dia. ^[8]:

tderz: Question w.r.t these edit hotspots on the right side of a Senseis page: their position is decided by what?

Are they created automatically?

unkx80: Yes, at the start of every section/subsection, marked by lines starting with a "!".

Thx for the info.

See also the pages in the Eyes Collection.

[4] inclusive all approach moves, hence not only physical ones.