Eye versus Eye Capturing Race
|Table of contents||Table of diagrams
1. Small Eye versus Small Eye
- the group with more outside liberties plus eye liberties is alive (group 1). If the number is equal for both, the one who has sente is alive.
- the other group (group 2) adds the inside liberties and compares:
- OL1 + EL1 vs OL2 + EL2 + IL
Black has a small eye in the corner (big eyes can be small in the corner), containing two liberties. White too has two eye liberties. There are no outside liberties. The one to play is alive. The other counts the one inside liberty. The count becomes 3 against two. Both are alive in seki.
Black: OL + EL = 4. White: OL + EL = 2. Black is alive. IL = 2. Black to play kills. White to play makes seki, by playing on an outside liberty or eye liberty.
White: The approach move a gives White 3 OL + EL. Black has 5 OL + EL. Black lives. White counts the inside liberty but still has only 4 liberties. White is dead.
If one of the groups has a big eye, the situation changes dramatically. The big eyed group counts all inside liberties. The small eyed group doesn't.
- OL1 + EL1 + IL vs OL2 + EL2
Black has a big eye with 5 liberties and one outside liberty. White has a small eye with 1 liberty and 6 outside liberties. Black counts the two inside liberties and gets 8 against 7. Against all appearances, White is dead. Let's have a look.
If White goes first, after filling the outside liberties and the big eye, White runs out of options. Filling the inside liberties leads to death (White a, Black b, White c, Black d).
Let's calculate this "difficult" situation.
Black: 1 OL + 2 EL + 4 IL = 7
White: 5 OL + 2 EL = 7
The one who has sente kills the other. This is quite surprising, because if we naively count adjacent empty spaces, Black has 6 and White 10 liberties !
Big eyes hide extra liberties behind the apparent points in the eyespace. These extra liberties cannot be removed at the same time as the apparent ones. Therefore, a big eye of bigger size than another big eye keeps more liberties in reserve. This is the reason why there is a fundamental difference between fights between groups with same sized big eyes and fights between groups with different sized big eyes.
3.1. The Big Eyes are of the Same Size
If the big eyes are of the same size, the reasoning of small eye versus small eye fights applies. The one with more OL + EL is alive (group 1). The other is alive in seki if the added IL lift Black's liberties over group 1's OL + EL.
The eye is a four space big eye for both. Black is alive. White counts the inside liberty and White's life depends on sente :
Black has 3 OL + EL. White has 5 OL +EL and is alive. Black counts the inside liberty but remains at 4. Black is dead.
Black OL + EL = 4. White OL + EL =3. Black lives. White counts two IL lives in seki.
3.1. The Big Eyes are of Different Size
If the big eyes are of different size, the group with the bigger eye (which does NOT mean more eye liberties !) enjoys the existence of more hidden liberties. The owner will count the inside liberties and the reasoning of big eye versus small eye fights applies.
In this example, even though Black's eyespace is almost filled, White is dead. Black OL + EL + IL = 6. White OL + EL = 5. Let's verify.
White goes first and has no choice but filling the IL, but let's remember that the IL are Black's liberties.
We see that the fight is reduced to one between same size big eyed groups. Black is alive. There are no inside liberties, so White will die.
Without a deep understanding of big eyes it demands superhuman effort to understand what's going on in this fight. It looks as if White has an enormous amount of liberties. Yet, White is dead.
Black has the bigger eye, containing 12 - 2 = 10 eye liberties. Black counts the 4 inside liberties for a grand total of 14 liberties.
White has 8 - 1 = 7 eye liberties plus 6 outside liberties for a total of 13. White dies.
Imagine the last diagram to pop up in a fast game (except for the artificial position). You know about capturing races and the role of big eyes and after a quick count you decide to leave it. Your opponent, who doesn't know, starts filling liberties while you go along, only to be astonished losing the race by one move.