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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.)
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.
That's it. ;-)
Well, the rest is miai.
All same same.
This is 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,
This is also a number, as you may verify. If that's not clear, see another number.
See yet another 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).
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.
RafaelCaetano: Migeru, do you think multiplication is necessary? Unless there's an application, I'd rather leave it unmentioned. Too complicated. Why aren't you happy with {0|} meaning 1 "free move" by Black? :-) [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. Path: CGTPath · Prev: ChilledGo · Next: AnotherNumber This is a copy of the living page "Numbers" at Sensei's Library. ![]() |