Half-eye

A half-eye is an unfinished eye: a potential eye that can be completed or destroyed depending on who plays first. Typically, the moves at a half-eye are locally small for both players, and are only played in order to make a group live or die, or to keep it on the run.

It is so called because two half-eyes are effectively a whole eye, as they are miai.

A three-quarter-eye – where forming one eye threatens to form a second – is sometimes also referred to as a half-eye; both are instances of fractional eyes, whose values need to combine to ``geq 2`` for a live group.

Table of contents Basic example How half-eyes combine One eye plus a half eye is unsettled Two half-eyes make a whole eye Four half-eyes make life Common half-eye shapes Half-eyes at the edge of the board Half-eyes in Combinatorial Game Theory See also Notes and references Discussion

Table of diagrams A half-eye for Black A half-eye for Black A half-eye for Black Black group with a half-eye One eye, two half-eyes Four half-eyes Half-eye Half-eye Half-eye Half-eye due to shortage of liberties Corner half-eye Edge capture half-eye Farmer’s hat half-eye Edge five-space half-eye

Basic example

A half-eye for Black  

The marked point is a half-eye for Black ...

A half-eye for Black  

... because by playing first, it becomes an eye; but ...

A half-eye for Black  

... White can remove the eye by playing first.

How half-eyes combine

One eye plus a half eye is unsettled

Black group with a half-eye  

This unsettled black group has a half-eye at a.

The key point is of course b, because Black needs to play there to complete his second eye.

[edit]

Two half-eyes make a whole eye

One eye, two half-eyes  

One eye and two half-eyes make a living group.

[edit]

Four half-eyes make life

Four half-eyes  

Half-eyes can be used to count up to two eyes.

In this example, White has four half-eyes at a, b, c and d, which is equivalent to 1/2 + 1/2 + 1/2 + 1/2 = 2 eyes, so White lives.

Common half-eye shapes

Once one understands the idea of half-eyes, it is useful to be able to recognise them in common situations. In particular, this helps one solve some life and death problems.

(See Landman^[2] for many examples of fractional eye shapes, including many corridors.)

Half-eyes at the edge of the board

Assume in the following examples that Black connects through to the rest of his group. All White’s stones are assumed alive.

Half-eye  

Half-eye  

Half-eye  

Half-eye due to shortage of liberties  

White can remove the eye with the throw-in at a (if Black captures afterwards then White ataris). The marked stones are important here.

This is quite a common shape. See for instance the diagram "Variation 1 : mistake at 7" in Gokyo Shumyo, Section 1, Problem 14 / Solution.

Corner half-eye  

Edge capture half-eye  

Farmer’s hat half-eye  

(See also the article Farmer’s hat half-eye)

Edge five-space half-eye  

This shape is related to the shape "Half-eye due to shortage of liberties" above.

Black has several moves to make an eye here; but in all cases White can answer so that Black's eye is gote (i.e., so that Black ends in gote when making the single eye, with no possible follow-up to make a second eye).

White’s only move to remove the eye is the hane at a. There are a couple of variations involved.

See Edge five-space half-eye? for more detail.

To do: add more similar cases.

Half-eyes in Combinatorial Game Theory

The methods of CGT may be used to treat a half-eye as a sort of combinatorial game with the result:

• 1 eye for Black, if Black goes first.
• 0 eyes for Black, if White goes first.

This could be written as ``{1_e|0_e}``, with the subscript ``""_e`` to show that ``1`` and ``0`` stand for the number of eyes and do not have their usual meaning in CGT^[1] (nor do they stand for the resulting score).

This approach applies to other fractional eyes as well, and has been worked out by Howard Landman^[2]. This yields fractional values which must add up to at least ``2`` for a group to live, with one exception: one might expect ``{2_e|0_e}`` to be equivalent to ``1`` eye, but ``{2_e|0_e}+{2_e|0_e}`` and ``1_e+1_e`` live^[3], while ``{2_e|0_e}+1_e`` is unsettled.

To do: make an article on eyespace values? explaining the main points of Landman’s paper as a more theoretical counterpart to fractional eyes.

See also

• Fractional eye — Eyespace regions that can be considered to be ⅓ eye, ⅔ eye, ¾ eye, 1¼ eye and 1½ eye. Also refers to Landman’s paper^[2].
• Eyes collection — A collection of articles on all aspects of eyes.
• Farmer's hat half-eye — A particular half-eye shape at the edge.

Notes and references

[1] In CGT, ``0`` and ``1`` are respectively defined as the games ``{|}`` and ``{0|}`` (i.e. ``{{|}|}``).

[2] Eyespace Values in Go by Howard Landman: http://www.msri.org/publications/books/Book29/files/landman.pdf

[3] We assume here that the incomplete eyes occur within the same group – if they can be disconnected

Discussion

So the result of this game is

• 2 eyes for Black, if Black goes first.
• 1 eye for Black, if White goes first.

(This is denoted { 2 | 1 } in Combinatorial Game Theory, I believe)

-- Jan de Wit

Bill Spight: See “Eyespace Values in Go” by Howard Landman: http://www.msri.org/publications/books/Book29/files/landman.pdf

Jan de Wit: Another nice reference is Martin Mueller’s Ph.D. thesis “Computer Go as a Sum of Local Games: An Application of Combinatorial Game Theory” which can be found at ftp://ftp.inf.ethz.ch/pub/publications/dissertations/th11006.ps.gz. This also has the most accessible introduction to Combinatorial Game Theory which I’ve found so far.

Further discussion moved to Combinatorial Game Theory.