tapir / Minimal Ko

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old rgg post by bill, just wikified

Second smallest ko post by Bill (rgg)

Dear Simon,

The second smallest ko (by territory scoring, AFAIK): (Well, actually, the third smallest ko, because there is a ko with a miai value of 1/5, that I discovered a couple of years ago.)

[Diagram]

outer stones alive, white komaster

If Black plays first, she fills at 1, with a result of 1 point.

Suppose that White takes the ko, Black plays elsewhere, and White fills the ko. That leaves this position, after 2 net local plays by White.

[Diagram]

outer stones alive, white komaster, 1 captured black stone

If Black plays first, she connects at 1, for a result of 0. If White plays first, he throws in at 1, Black takes at 2, then White plays a threat and Black responds, then White takes the ko back. Next Black fills at 6, and White fills at 7, yielding this position:

[Diagram]

outer stones alive, white komaster, 2 captured black stones, 1 white stone

Play continues. B 8 takes the ko, White plays a ko threat, Black responds, White takes the ko back, Black plays elsewhere, and White fills the ko.

In 2 net White local plays from B, White reaches a score of -1. In 1 net local play Black reaches a score of 0. When White is komaster, this throw-in ko at B has a miai value of 1/3 and a count of -1/3.

From A White takes 2 net local plays to reach B, with a count of -1/3. From A Black takes 1 play to reach a count of 1. So the miai value of the ko at A is 4/9, and the count is 5/9.

(When Black is komaster, White will not play the throw-in at B, and the count at B is 0. So then the miai value of the ko at A is 1/3, and the count is 2/3. The difference between Black's being komaster and White's being komaster is 1/9 point.)

Best,

Bill

Explanation

Bill: If I may. :) This is from [ext] this note. It is a ko that is hotter than a 1/3 point ko. :) I later discovered a ko position with a miai value of 1/4 point. It is currently on Half point ko /discussion.

Tapir: That was kind of what I was wondering about: How can this be the second smallest ko? (I don't really understand the 1/9 difference. This supposes there are outside moves of average value, isn't it?) Either way, is there any difficulty in constructing a 1/4, 1/5, 1/6, 1/7, 1/n point ko by just making one side having to play some approach moves? Or... yes, then the count changes as well.

Bill: You did not copy the whole original, which was clearer, I think. Anyway, suppose that Black plays first.

[Diagram]

Black fills

This is worth 1 pt. (for Black).


[Diagram]

W1 takes and W3 fills


From that position, suppose that Black protects.

[Diagram]

Black protects

This is worth 0. Black has 1 pt. of territory and White has 1 prisoner.

[Diagram]

White throws in and wins ko fight

This is worth -1. That means that the position before the throw-in was worth -1/3, and each play in the ko was worth 1/3.


Backing up to the previous ko. each play in the ko was worth 4/9. (Three moves between positions worth +1 and -1/3.)

So, after White takes the first ko, the position is worth 1/9. But after Black throws in and White connects four stones, it is worth 0. Therefore the throw-in loses 1/9 point on average.

Tapir:

[Diagram]

Even smaller? (When Black is ko master.)


Bill: When Black is komaster each move gains 1.5.

Tapir: Thank you. Want to assure you that I am learning quite a bit in the process. :)

What if?

[Diagram]

outer stones alive, white komaster

Even a ko master doesn't win two kos at once.

Bill: B2 loses 1/9 point on average, when White is komaster.

Tapir: When there are moves elsewhere, isn't it?

[Diagram]

outer stones alive, white komaster, 1 black stone captured

[Diagram]

outer stones alive, white komaster, 2 black stones / 1 white stone captured

As we are talking about the minimal ko, we can assume when this is played no other point are left on the board. This result is clearly better for Black than the one used to calculate the value. (Of course the assumption of tenuki when there is still a point a stake in the fighting of a minimal ko, is a little strange from start on. When fighting over the last point in the minimal ko, there should be no other point on the board otherwise fighting this minimal ko would be wrong to begin with.)

Bill: This ko is hotter than a 1/3 point ko. It's miai value is 4/9 when White is komaster.

[Diagram]

Original example realistic play, Black doesn't make a ko, same result (assumption no points elsewhere, when fighting the minimal ko)

Bill: B2 also loses 1/9 point.


What is the minimal ko?

The half point ko has a net count of 1 - as only 1 point is at stake. While this is the smallest stake, the local tally can be different and thus the miai value of a ko can in principle be even smaller than the 1/3 of the "half point ko". While adding an unlimited number of approach moves changes the local tally, it isn't easy to construct approach moves in a way that the net count stays at 1.

[Diagram]

This one is smaller. (Found by Bill Spight.)

The net count is 1, while approach moves are present. What is at stake isn't the stone in the ko, but the point at circle. Black at a when ko master, can make an approach ko it wins, but - and that is the trick of this problem - White can't make the ko, but only atari at b in sente.

If you, only compare Black winning the ko with White winning the ko, the miai value is much bigger here (White would resolve the ko by capturing stones.) So, what makes the diagram above so minimal is that only Black can set up the ko (and only will do so if ko master), while the best White can do is safeguard a single point in sente by playing at b.

[Diagram]

Not minimal anymore.


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