One of the more common mistakes I see in any endgame-related discussion is related to the value of a kō capture. The usual touted value of ½ points only applies to the very last kō of a game; and the more knowledgeable players who have heard of the ⅓-point theory often use it wrongly in relation to other endgame values. This essay seeks to remedy the above mistakes while deepening the reader’s understanding on how endgame calculations work.

Dia 1 shows a general kō fight position. As mentioned above already, a move at A is supposedly ⅓ points. But how do we know that?

Additionally, the position in Dia 1 is inherently beneficial to Black. Black can capture a stone at A, while White may only prevent her stone from being capture by playing at the same intersection. But how many points does Black expect under △?

Again, we can get trustworthy numbers by performing a shape duplication. See Dia 2, which holds three positions like in Dia 1.

In Dia 2, if White started playing the situation out, Dia 3 would follow. White connects at 1, after which Black captures a stone with 2. At △ and 3, then, there are two simultaneous kō fights going on, which means it is impossible for White to win both. White connects at 3, and Black connects with 4 at △. An even number of stones has been played, and Black has scored one point (the prisoner at △).

Let Dia 4: this time Black goes first. Black captures at 1, to which White responds by connecting a kō with 2. At △ and 4, there are again simultaneous kō fights, so Black connects with 3 at △. Finally, White connects the last kō with 4. Again, an even number of stones has been played, and Black scored one point.

From Dia 3 and Dia 4, it follows that in Dia 2, Black always expects to profit by exactly one point. Logically, it follows that in Dia 1, Black expects one-third of the profit in Dia 2: in other words, ⅓ points.

Therefore, in Dia 1, if White connects at A, White has reduced Black’s expected profit from ⅓ points to zero—white A is worth ⅓ points.

If instead, as in Dia 5, Black captures a white stone with 1, Black scores one point. However, the resulting board position has ⅓ points of expected territory for White: thus, Black moved his local profit from ⅓ points to 1 − ⅓ = ⅔ points, meaning black 1’s value is ⅔ − ⅓ = ⅓ points. Q.E.D.

A more interesting—and widely much less known—case is how much a move in a two-step kō fight is worth. Have a look at Dia 6!

This time, our problem is how much a move at A values, and how many points White expects at △.

If White connects at A, one point at △ becomes secure. If instead Black captures at A, we get to Dia 7. Although it may be immediately obvious, in Dia 7 we have a “zero” position: black A and white B (or white A and black B) are clearly of equal value, so in Dia 7 both players should expect no profit.

Curiously, it follows that Dia 6 is also a “zero” position—currently White expects no territory at △. If White connects at A, White gets one point of territory; and if Black captures at A, Black gets one prisoner (and the newly-created shape is “zero”).

What if, as in Dia 8, when Black captured at A, White in response connected at B?

Remember how in Dia 6, we had a “zero” position. In Dia 8, Black A scored one point as a white prisoner, while white B prevented another white stone from getting captured. The resulting position after white B is ⅓ points in White’s favour, so altogether in the A-B exchange, Black profited by 1 − ⅓ = ⅔ points. This result is inferior for White compared to Dia 7, because there, Black had to use his move turn to get the one-point profit; in Dia 8, Black got ⅔ points for free while retaining his move turn.

In summary, in a two-step kō, **each move fighting the kō is worth one point and backing out from the kō loses ⅔ points**.

Let us return to our homework from the last time, shown in Dia 9.

In the following image captions, “B” stands for a situation where Black has played one more stone. “W” respectively stands for White playing one more stone, and “WW” for White playing two more stones, etc. The number to the right of “B” or “W” means Black’s profit minus White’s profit (in other words, how many points more Black has than White).

Starting in Dia 10, if Black plays his local endgame move, we get 1-3 after which Black’s visible territory is nine points and White’s is four.

If instead White gets to play 1 as in Dia 11, we have an as-of-yet unclear endgame position that needs further calculations.

Continuing from Dia 11, either Black may get to play 2 as in Dia 12, or White further gets to reduce Black’s territory with 3 as in Dia 13. Dia 12 holds seven points for Black and five for White, while Dia 13 requires additional calculations.

Following from Dia 13, Black either answers with 4 as in Dia 14, after which later we may imagine Black 6 and White 7 getting played. In Dia 14, Black has seven points (six points of territory and one capture) while White has 5⅓ points (5 points of territory and ⅓ in the kō). Dia 15, again, needs more calculations.

If Black plays 6 as in Dia 16, we have 6⅓ for Black and five points for White. If instead White plays 7-9 in Dia 17, we have 5 for both Black and White.

Now to sum it all up! WWW in Dia 15 is the average of WW=1⅓ and WWWW=0, in other words ⅔. WW in Dia 13 is the average of W=1⅔ and WWW=⅔, in other words 1⅙.

But wait! In Dia 16, we have WW=1⅓, which is more than 1⅙. It follows that in Dia 15, Black should always respond to white 5 in order to maximise his points, and thus WW=1⅓.

W in Dia 11 is the average of 2 and WW=1⅓, in other words 1⅔. Finally, for the count in the problem position in Dia 9, we get the average of B=5 and W=1⅔, ie. 3⅓. In other words, in Dia 9, Black expects 3⅓ points more than White.

To get numbers for the individual territories, White either has 4 (as in Dia 10) or 5 (as in Dia 11) for an average of 4½, and Black has 3⅓ more than that, in other words 7⅚.

Black 1-3 in Dia 10, and White 1 in Dia 11, both change the count by 1⅔ points; in other words, an endgame move in the problem position in Dia 9 is worth 1⅔ points.

To reward the reader for fighting through the hard thinking exercise above, today there is no homework!

There seems to be some error with 3’s in the denominator not appearing correctly.

For a low level kuy (and bad in math math too)

This is way over my head but, maybe i should stick at playing only..

Good to have you back Antti!

This seems to be pretty hard stuff if we try it for the first time!

May i ask where you got this from? Is it a thing you came up by yourself or did you get it

from your trainings in japan though?

May I suggest a change in terminology for the two-step ko: http://senseis.xmp.net/?TwoStepKo

Here the position is seen in the diagrams, so everything is quite unambiguous even without the change, of course.

I find that it helps, in Diagrams 14 and later, to think of Black 4 and White 5 as miai, so that Diagram 13 and Diagram 16 are equivalent in terms of points.