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Numbers
Path: CGTPath   · Prev: ChilledGo   · Next: AnotherNumber
    Keywords: EndGame

Many go positions chill to numbers in combinatorial game theory (CGT). Conventionally, we just call them numbers. If there are no kos involved, it's never optimal to play in a number instead of a hotter play, and it is never wrong to play in the hotter of two numbers. With no kos involved, go numbers always have a power of 2 in the denominator. The ones with larger denominators are hotter; play in them has a miai value of 1 - 1/D, where D is the denominator.

Here are some examples:

Nearly every yose book has a diagram like this. (Note on diagrams. These are not full board diagrams. The outer stones that frame the position are assumed to be alive.)

[Diagram]
Numbers

  • The corridor a is worth -1/2 (1/2 point for White).
  • b is worth -1 1/4.
  • c is worth -2 1/8, d is worth -3 1/16, and e is worth -4 1/32.

The whole thing, their sum, is worth -10 31/32.

In the play of numbers (without ko) the player with the move (sente) can "round the number" to the nearest integer in his direction. Here Black to play can play to -10, White can play to -11.

[Diagram]
Black plays first

That's it. ;-)

Well, the rest is miai.
B1 had a miai value of 31/32.
-10 31/32 + 31/32 = -10.
Since that is an integer, neither player needs to play, and the result will be the same (with correct play), regardless of who plays first. Of course, it will be played out before counting, but that is a formality.

[Diagram]
-10, Black plays first

[Diagram]
-10, White plays first

All same same.


[Diagram]
-10 31/32, White plays first

This is worth -11.
W1 was worth 31/32.
-10 31/32 - 31/32 = -11 15/16.
B2 was worth 15/16.
-11 15/16 + 15/16 = -11.
The rest is miai.


[Diagram]
Black mistake

B1 is worth 15/16.
W2 is worth 31/32.
This position is the same as the previous one, worth -11.

Black's mistake of 1/32 point ended up costing a full point, because it allowed White to round down to -11. Scary!


Empty corridors are prototypical numbers, but there are many others. For instance,

[Diagram]
Another number

This is also a number, as you may verify. If that's not clear, see another number.



[Diagram]
Yet another number

See yet another number.



[Diagram]
One more number

See one more number.



-- BillSpight


Migeru: Still trying to translate surreal numbers into go positions. In ZeroInCGTTerms I studied { | } = 0. Now I can construct {0| }=1, { |0}=-1 and {0|0} (not a number).

[Diagram]
{0| } = 1

Black can play to { | } = 0, but white has no legal moves. The fact that Black has one more legal move than white translates into a score of 1. This is very interesting, because it indicates that the original game of go was exactly a combinatorial game of the kind "the last player to legally move wins"[1] (see tedomari).

{ |0} = -1 is the color-reversed version of {0| } = 1.



[1] Bill: No pass go is not the same as the original game surmised by Zhang Hu, which is a kind of territory scoring with a group tax.



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This is a copy of the living page "Numbers" at Sensei's Library.
(OC) 2004 the Authors, published under the OpenContent License V1.0.