Bless your next 356 days

In which a recurrent myth is shattered.Before you read on, think about the following questions:

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1) What is the value of a move at A?
2) How many points should you allot to Black during positional judgement?

If you already know the aswers, please scroll until the end.

When counting the value of an endgame move that is gote for both players, we calculate the difference between Black playing and White playing. Obviously, if White connects at A, neither player has any points. But what if…

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…Black takes?

Then it gets more complicated. Fortunately White’s points do not change, so we only need to count Black’s points. (And substract White’s eventual prisoner from Black’s gain.)

From here on, there are two paths:

Either Black connects again…

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…and makes 5 points (2 points+3 prisoners)…

…or White takes back the one stone.

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Now Black has 3 prisoners and White has 1.

But we can’t count the points yet as the shape is not settled.

The path splits up again:

White goes in…

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Black is left with the 2-prisoner advantage.

Black may also block.

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Now Black has 3 1/3 points.

Where did the 1/3 point come from?

Wonder no more, for the notion of a “half-point ko” is a lie! Those innocuous kos are measured in thirds.

Of course, the actual outcome in a game is restricted to natural numbers. Having that ko-stone in atari either gives Black 1 or 0 points. But only in 1/3 of cases does Black get that point, and in 2/3 Black will lose the ko. The difference to a 50-50 concept is minuscule, and is only apparent when added up.

In the following outrageous shape, Black is holding onto three of those kos.

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And Black has exactly 1 point here.

It doesn’t matter who starts playing.

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Black starts and gets one prisoner.

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White starts and Black still gets one prisoner.

Since Black gets 1 point per three kos (and not 1.5), putting a ko-stone in atari gives Black 1/3 point. Thus follows the chart below. Each path per fork has an equal likelyhood of happening which is represented by percentages in the chart:

 

3656d

This should sum up to 3 5/6. If you play this move instead of a 4-point move, you will lose a point in one out of six games.

In order to answer the second question, another halving is involved.

Much like we count 1 point for Black during positional judgement in this situation below…

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The move is worth 2 points, but on average Black owns 1 point.

…Black is holding onto 1 11/12 points in this shape.

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Which looks like a condom.

If you think this is absurd, have a look at what the EGF Academy’s top student wrote, in all seriousness, in the Academy’s endgame move value quiz:

3569a

That said, I wish everyone who cares a pleasant 2018. If my counting is right, my blessing lasts until Christmas.

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