# Message #1828

From: Andrey <andreyastrelin@yahoo.com>

Subject: Re: [MC4D] God’s number for 3^N

Date: Thu, 07 Jul 2011 06:11:13 -0000

I just understood that we want not maximal, but minimal values of coefficients to use the formula. They are for N=5: 0.70295 for FTM and 0.22201 for QFTM.

Andrey

— In 4D_Cubing@yahoogroups.com, "Andrey" <andreyastrelin@…> wrote:

>

> Andrew,

> You are right - I forgot to mention a couple of things:

> - formula works only for N>=5 (where we haven’t additional invariants for corners orientations)

> - and even in that cases it’s only asymptotic: (2/9+o(1))*N*3^N and (3/4+o(1))*3^N. I’m not sure in constant 3/4: for N=170 computations give something like 0.739.. and this coefficient slowly decreases. Maximal coefficient for QFTM is 0.222674992 for N=71, and for FTM - 0.7642 for N=17.

>

> Andrey

>

>

> — In 4D_Cubing@yahoogroups.com, "Andrew Gould" <agould@> wrote:

> >

> > A couple problems I’ve noted for FTM:

> >

> > When N = 3: your lower bound gives 20.25 when we know it’s actually 20.

> >

> > When N = 3: your possible twists is 24 when I count 18.

> >

> > When N = 4: your possible twists is 192 when I count 184.

> >

> >

> >

> > I’m wondering if your FTM lower bound equation is a bit high.mainly because

> > it beat my lower bound :-)

> >

> > 3^4: your FTM lower bound is 60.75, mine is 56.

> >

> >

> >

> > –

> >

> > Andy

> >

> >

> >

> >

> >

> > From: 4D_Cubing@yahoogroups.com [mailto:4D_Cubing@yahoogroups.com] On Behalf

> > Of Andrey

> > Sent: Wednesday, July 06, 2011 11:27

> > To: 4D_Cubing@yahoogroups.com

> > Subject: [MC4D] God’s number for 3^N

> >

> >

> >

> >

> >

> > My estimates show that lower counting limit L for God’s number for 3^N is

> > 2/9*N*3^N for QFTM (as implemented in MC5D and MC7D - with 2*N*(N-1)*(N-2)

> > possible twists) and 3/4*3^N for FTM (where any twist of face is counted as

> > 1, so we have N!*2^(N-1) possible twists). Actual God’s number is probably

> > between L and 2*L.

> > By the way, if we take puzzle 2*1^N (with only one twisting face), its God’s

> > number in QFTM is N. But counting limit gives something like

> > N*(log(2*N)/(2*log(N)) that is N/2*(1+o(N)). So lower limit is almost the

> > half of the actual number.

> >

> > Andrey

> >

>