Smallest Group with Two Eyes

Difficulty: Beginner Keywords: Life & Death

Table of contents
Introduction

Smallest groups in the corner

Smallest groups on the side

Smallest groups in the center

All Smallest Groups in One Sight

The no. of stones of smallest groups should be...

The smallest group enclosing a living center group

Related Links
Table of diagrams
First corner group
Second corner group
Third corner group
Fourth corner group
First side group
Second side group
Third side group
Fourth side group
First center group
Second center group
Third center group
Fourth center group
Smallest groups with [two eyes]
The smallest group enclosing a living center group
The smallest group enclosing a living center group (2nd possibility)

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Introduction

This is a list of the smallest groups with two eyes. They do not need a single further stone to defend them, no matter how many stones may be played to attack them.

Note by aLegendWai: Some have arguments relating to the smallest groups with two eyes.
- These special (though maybe arguable, you decide!) smallest groups are somewhat alive by means of special conditions like seki (mutual life) {some of them occur in rare situations (or theoretically)}.
- The 12 smallest groups (See ^[1] for all 12 groups) in the following are alive as normal.

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Smallest groups in the corner

In the corner, six stones are the minimum needed to complete a group with two eyes.

This is similar to the second corner group. The eyes are the same, the stones are slightly different.

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Smallest groups on the side

On the side, the smallest groups with two eyes have eight stones.

This looks like the second side group. Different stones, but the same eyes.

Again the same as the second group

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Smallest groups in the center

In the center, 10 stones are required to make a group with two eyes.

These all have the same internal eye space: 10 stones, but only one eye shape.

[1]

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All Smallest Groups in One Sight

Here's a board containing all of the above groups:

On a side note, this diagram also illustrates a drawback of living small: very few points are gained, but relatively many stones are used up.

Here, the comparison isn't quite fair because, in this artificial setup, the move count is not balanced: 119 white vs. 96 black stones. Nevertheless, White's wider formation (241 zi) is far more efficient than Black's cramped groups (120 zi). White owns two thirds of the whole board, whilst Black's area is barely half that size.

Bonobo: I have made an image of this, hope it’s useful for newbies, clubs etc., free to use/share (and please let me know if such images are too big for SL):

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The no. of stones of smallest groups should be...

For the final result, please read The no. of stones of smallest groups should be....

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The smallest group enclosing a living center group

rsun: As a further extension to the problem, the smallest group containing a living center group of the opposite color has 19 stones.

Notice that the White group has 10 stones and also forms the smallest center group. Variants of the smallest center group (see above) also lead to a Black group of 19 stones. However, this family of solutions is not unique!

As illustrated in the next solution, White has 12 stones, which is not the smallest center group, but Black still has 19 stones.

White need not be the smallest center group for Black to be the smallest group that contains it. (This is not a mathematical proof so please correct me if you find a better solution. Seki positions are not considered.)

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