Message #1824

From: secnodmouse <paul.timmons@btinternet.com>
Subject: Re: God’s number for 3^N (and what about 2^N?)
Date: Wed, 06 Jul 2011 20:46:14 -0000

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

Without having analysed your results fully (as I am not qualified to comment not having the same empirical savvy) I would think your lower bound limit is much closer to the truth than my "shooting from the hip" suggestion of 3^N in an earlier thread. As a matter of great interest to me had you any thoughts on the lower bounds for the 2^N case? Surely (and sorry for calling you Shirley -;)) - the effort should be concentrated on the 2^N case as this is likely to be more more tractable? This proved to be the case in a neat presentation for the 2x2x2 cube group (whereas the 3x3x3 is extremely unwieldy):

http://cubezzz.dyndns.org/drupal/?q=node/view/177

P.S. Sorry looks like you’ll have to cut and paste this link - don’t have the time to work out how to make it auto clickable.