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Method of Multiples
Path: Endgame   · Prev: Count   · Next: MiaiValue
  Difficulty: Beginner   Keywords: EndGame

Under Construction

The method of multiples, while seldom a practical way of determining the count of a position, provides a good explanation of it, I think. :-)

If there are no kos or future kos in a position, and some combination of multiples of that position has a definite score, the count of that position is that score divided by the number of multiples. (The combination has a definite score if the net score after correct play is the same, regardless of who plays first. IOW, the combination is miai.)

Example 1:

[Diagram]
2 copies

(Per convention, stones next to empty space are immortal.)

The score of this position is +4 (for Black). If White plays first the result is the same (e. g., W1 - B2). There are 2 copies of the original position, so the count of each is 4/2 = 2.

Note that B1 or W2 is gote.

Example 2:

[Diagram]
4 copies, Black first

[Diagram]
4 copies, White first

Regardless of who plays first, the score is +5. Since there are 4 copies, each has a count of 1 1/4.[1][2]

Note that each move is gote. Also that the number of copies is a power of 2.

Example 3:

[Diagram]
8 copies, Black first

[Diagram]
8 copies, Black first (ii)

[Diagram]
8 copies, Black first (iii)

[Diagram]
8 copies, Black first (iv)

The final result is +11.


[Diagram]
8 copies, White first

[Diagram]
8 copies, White first (ii)

[Diagram]
8 copies, White first (iii)

[Diagram]
8 copies, White first (iv)

The score is +11 regardless of who plays first.

There are 8 copies, so each one has a count of 1 3/8.

Note that the first play is gote, that the number of copies is a power of 2, and that the second player plays last.

(You can also see what I meant about being impractical. ;-))

-- Bill Spight


[1] Note on fractional counts: Some people object to fractional counts because the final scores of go games are integers. There are two answers to that. First is that, if the game is not over, a fractional count can serve as an estimate of the local score. The second is that, as with these examples, you can have a whole board position that is miai, and therefore with a definite score, that includes regions with fractional counts.


[2]

[Diagram]

Hyppy: Just throwing a twig in the fan here on the example above.

Bill: Yes, to use the method of multiples you have to find correct play. :-) B2 is the losing play. Now Black can get only 4 points.



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This is a copy of the living page "Method of Multiples" at Sensei's Library.
(OC) 2003 the Authors, published under the OpenContent License V1.0.