Message #1826

From: Andrey <andreyastrelin@yahoo.com>
Subject: Re: [MC4D] God’s number for 3^N
Date: Thu, 07 Jul 2011 02:32:59 -0000

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
>