Re: glossary: fickle and firm
Dan Hoey
> After some discussion, Jeff and I agree that the definition is
> correct provided 0^0 and 1^1 are added to the list of
> nim-values. Here's the (hopefully final) version of my definition:
>
> "Game *G* is tame" means nim-values(*G*) are one of* *0^1, 1^0, 0^0,
> 1^1, 2^2, 3^3, 4^4, 5^5, etc... and for every *H* in *G*', either
> *H* is tame or all of the following are true:
Okay, so the four conditions are trying to rephrase the extension of
"tame" from ONaG to WW.
> 1. Grundy+(*H*) > Grundy+(*G*) or *G* has a tame child *K* such
> that Grundy+(*H*) = Grundy+(*K*).
I understand this as a way of saing "H does not affect Grundy+(G)".
> 2. Grundy-(*H*) > Grundy-(*G*) or *G* has a tame child *K* such
> that Grundy-(*H*) = Grundy-(*K*).
Ditto for Grundy-(G).
> 3. There exists a J in H' such that Grundy+(*J*) = Grundy+(*G*)
> 4. There exists a J in H' such that Grundy-(*J*) = Grundy-(*G*)
These are the requirements for reverting through the wild options.
However, it is required that J be tame in both cases.
For instance, G=*[((2//1)(2/1))/0] == {{{{{{2}},1},{{2},1}}},0} has
nim-values 1^0, the same as *[0]={}. So H=*[((2//1)(2/1))/] does not
affect Grundy+(G). Furthermore, the single option of H,
J=*[(2//1)(2/1)], has nim-values 1^0, the same as G. But the
nim-values of G+*[2] are 3^4, so G can't be tame.
I found this by playing around with wild games like *[2//1] and *[2/1]
until I found a wild J with nim-values that agree with a tame game. I
suspect this behavior is typical for a wild animal that wears a tame
mask, perhaps with a few exceptions.
Dan
Josh Purinton
> > 3. There exists a J in H' such that Grundy+(*J*) = Grundy+(*G*)
> > 4. There exists a J in H' such that Grundy-(*J*) = Grundy-(*G*)
>
> These are the requirements for reverting through the wild options.
> However, it is required that J be tame in both cases.
Oops, thank you for the correction and the example. Did you have any
time to form an impression of my "Tame Game" conjecture?
"A game G is tame (in the WW sense) iff it is indistinguishable from a
sum of nim heaps in any game in which the non-G components are all nim
heaps. More precisely, G is tame iff there exists a sum of nim heaps H
such that for any sum of nim heaps N, o+(G+N) = o+(H+N) and o-(G+N) =
o-(H+N). "
--
Josh Purinton <josh.p...@gmail.com>
Dan Hoey
> conjecture?
> "A game G is tame (in the WW sense) iff it is indistinguishable from
> a sum of nim heaps in any game in which the non-G components are all
> nim heaps. More precisely, G is tame iff there exists a sum of nim
> heaps H such that for any sum of nim heaps N, o+(G+N) = o+(H+N) and
> o-(G+N) = o-(H+N). "
Aaron usually has a pretty good handle on these things, so I generally
borrow my opinions from him when it comes to things I can't imagine
how to prove. So far, the best candidates I've found for
counterexamples are *[(2/0)210] and *[(2/1)210], both of which act
like *[3] in the sums I've tested.
*[3] is discriminated from *[(2/1)210] by *[2/20] and from *[(2/0)210]
by *[((2/1)10)2/10].
Dan