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SecondCourseOnKo

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Three Step Ko
Path: SecondCourseOnKo   · Prev: TwoStepOneMoveApproachKo   · Next: TenThousandYearKo
    Keywords: Ko

Sometimes kos arise in which you have to take the ko more than once before winning it, when your opponent could win it immediately. Each time you have to take the ko is a step of the ko. These are called dan ko in Japanese.

Here is a simple three-step ko.

[Diagram]
3-step ko

If Black fills the ko the score is 3 points (for Black). If White takes the three steps of the ko with White 1, White 3, and White 5, and then fills the ko, the score is -4 points (i. e., 4 points for White). Each move in the ko gains 1 2/5 points. So the count in the original position is 1 3/5 points.

Very long step kos are possible, but you seldom see one more than four steps long.

-- BillSpight

Kiko: I don't understand the math here. Three points for Black vs. four points for White means playing the four moves (1, 3, 5, and connect) is worth 7 points. 7/4 = 1 3/4. How do you get 1 2/5?

Bill: If Black fills the ko (1 move), the result is +3. If White takes the ko three times (3 moves) and then fills the ko (1 move), the result is -4. 5 moves make a difference of 7 points. On average, each move is worth 7/5 points.

Karl Knechtel: Or you can do a full analysis of the move tree like I did at PickyEndgameMiaiValuesAttempt, and come to the same conclusion. You can do the same thing for a 1-step ko to get the 1/3 point value; that case is a lot simpler. Of course, counting moves between the results is a lot nicer mathematically, but I wonder if it would be any easier to program ;)

The complicated calculation is analogous to summing an infinite series. It reminds me of the story of the "fly and bicycles" problem being posed to Von Neumann...

At any rate, it appears that in general the plays in an n step ko have a miai value of (3n-2)/(n+2).



See my objections at two step ko -- Andre Engels



Path: SecondCourseOnKo   · Prev: TwoStepOneMoveApproachKo   · Next: TenThousandYearKo
This is a copy of the living page "Three Step Ko" at Sensei's Library.
(OC) 2003 the Authors, published under the OpenContent License V1.0.