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zero in CGT terms
Path: CGTPath · Prev: DifferenceGame · Next: Reversible How do you recognise go positions that are equal to 0, in the sense of CGT? Since the idea of number appropriate in CGT corresponds to territory in go, zero games should comprise positions recognisable as 'no net territory for anyone'. Here is a 0 in go:
One way to write it is { 6 | -10 || 8 | -8 } If Black plays first White can reply and get 10 points (White scores are negative); if White plays first Black can reply and get 8 points. Neither player can afford to play. Since neither player will play, we can simplify this to { | }, where neither player has any option. { | } = 0 by definition. Here is another 0, this time a clear case of miai:
We can write it { 2 | 0 || 0 | -2 }. In this case neither player can gain by playing here.
Note on the diagrams: When we're getting picky, it's important to indicate where the plays are. So instead of the usual convention of surrounding positions by open points, I (Bill Spight) have surrounded them by empty space or by nothing. The stones that frame the position cannot be captured. See assumptions of yose problems. The most basic game in Conway's theory is the "empty game" { | } where neither player can legally move. An example in go is
Any position consisting entirely of (any number of) minimally alive groups (of either colour) is an example of { | }. I suspect that this is the reason why mathematical go is equivalent to go scoring with a group tax.
It is easy to see that { | } = 0. If you want, you can check that 0 + G = G + 0 = G for any game G, using the definition of addition. - Migeru Original page by Bill Spight, moved and edited by Charles Matthews. Path: CGTPath · Prev: DifferenceGame · Next: Reversible This is a copy of the living page "zero in CGT terms" at Sensei's Library. ![]() |