The proverb 'Eyes win Semeais' means that in a semeai (capturing race), if one group has an eye and the other has not (in Japanese the term 'me-ari-me-nashi' is used for this situation), the player with the eye has an advantage. An example of this proverb at work can be found in this diagram: at first sight White appears to have 6 liberties and Black has 5, so it seems that White would win the semeai even if Black went first. The truth is rather the opposite: Black will win, even if White goes first. The reason for that is Black's eye at a. Because playing here would be suicide for White, it is necessarily the last liberty White will fill. White will have to fill the shared liberties (5 and b) first.
To count the liberties correctly in a situation like this, use the following rule:
There is a second way in which eyes influence semeais, less well-known (and in general also less important). This diagram is an example. Black and White both have four dame.
Black has an eye and White does not, but at first sight that does not seem to matter, because there are no shared liberties. Nevertheless, White is not able to win the capturing race, as this diagram shows.
The reason for this is the approach move that White has to make, with . At first sight, Black has a similar approach move problem at a. However, this problem does not worry Black, who can simply solve it by making b the last liberty that Black fills up. White, on the other hand, is forced to play the eye as the last move - so White has to make the approach move first. Stating this as a rule:
-- Andre Engels (2D Europe)
Richard Hunter deprecates this proverb at Counting Liberties and Winning Capturing Races.
See also counting liberties and the pages in the Eyes Collection.