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Mar 24, 2008, 9:26:51 AM3/24/08

to Josh Purinton, sprouts...@googlegroups.com, danny purvis, Aaron Siegel, Thane Plambeck

"Josh Purinton" <josh.p...@gmail.com> writes:

> 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

Mar 24, 2008, 12:33:13 PM3/24/08

to Dan Hoey, sprouts...@googlegroups.com, danny purvis, Aaron Siegel, Thane Plambeck

On Mon, Mar 24, 2008 at 9:26 AM, Dan Hoey <Ho...@aic.nrl.navy.mil> wrote:

> > 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.

> > 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>

Mar 24, 2008, 1:12:30 PM3/24/08

to Josh Purinton, sprouts...@googlegroups.com, danny purvis, Aaron Siegel, Thane Plambeck

> Did you have any time to form an impression of my "Tame Game"

> conjecture?

> 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

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