Capturing race formula
A capturing race between two groups with only outside liberties, eye liberties, and shared liberties can be evaluated with a simple formula. 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. 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 and the shared liberties.
- If the defender has an eye, F = S, where S is the number of shared liberties.
- If the defender doesn't have an eye but here are shared liberties, F = S - 1.
- If there are no shared liberties, F = S = 0.
If the attacker fails, two outcomes are possible: seki and death. If seki is possible, it will happen. Seki is possible only if the eye status of the two groups is the same and there are enough shared liberties: two for eyeless groups, one for groups with eyes. Otherwise, the attacker loses everything.
The first known appearance of this formula in Western print is in a 1982 DGZ article by Karl-Friedrich Lenz or K.-F. Lenz for short. He wrote several good pieces on Go & copyright and studying techniques on the internet. Please check out http://k.lenz.name/d/v/Elementar.pdf, Lernstrategie Jura .
The Second book of Go by Richard Bozulich contains two chapters by Richard Hunter on this subject, which later resulted in a book Counting liberties and winning capturing races.