Numbers

Path: <= CGT path =>
    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


See also Numbers Example 1: [thumbnail diagram]

More on the /Discussion page.


Path: <= CGT path =>
Numbers last edited by Bill on November 9, 2007 - 04:39
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