# Message #2118

From: Roice Nelson <roice3@gmail.com>

Subject: Re: [MC4D] Calculating the number of permutation of 2by2by2by2by2 (2^5)

Date: Mon, 07 May 2012 13:19:31 -0500

Hi David,

Thanks for helping me realize the state space reduction in the 3D case is *not

quite* 48, and similarly for other dimensions. I missed this

previously, but should have considered it when quoting the cube lovers post

about there being 12 equivalent 1 quarter turn moves (rather than 48). It

feels like a somewhat subtle point. It also makes calculating the

*exact*number of states which are "the same" in this discussion more

difficult!

seeya,

Roice

On Sun, May 6, 2012 at 9:44 PM, David Vanderschel <DvdS@austin.rr.com>wrote:

> Still out of phase. Significant redundancy between my last

> post and Roice’s. :-(

>

> It dawns on me that the problem the cube20.org folks were

> addressing is deeper than the one David Smith was addressing

> (and which I was defending) and more like what Melinda was

> probably driving at. I.e., given two arrangements that are

> visually different when the pile is viewed in standard

> orientation and based on the sticker IDs inferred from

> initial state as in Smith’s calculations, when can those two

> still be regarded as the ‘same’ based on a symmetry of the

> pile? This is where conjugation by a symmetry enters the

> picture, and that can be viewed in a sense which remaps axis

> IDs (along with the corresponding sticker ‘color’ pairs).

> As may be seen from the ‘94 Cube Lovers article, the precise

> issue gets very messy and David Smith was not trying to

> address this more complex version. What Smith was

> calculating is much more straightforward, is still

> meaningful, and is more easily understood. The deeper view

> reduces the "real size" by a factor less than n!*2^n (but

> not quite n!*2^n), which is actually rather minor compared

> to the sizes of the numbers Smith was computing. In my

> view, the main point of his efforts is not the precise

> values (which are relatively useless) but to get some feel

> for just how immense these numbers are. They are

> incomprehensibly large with or without the factor of n!*2^n.

> The little ol’ order 3 3D problem was close enough to being

> tractible that a factor of nearly 48 actually did make a big

> difference in the ultimate analysis there.

>

> One thing that should be pointed out is that, if you admit

> mirroring transformations, then it does create problems for

> the purpose of counting distinct arrangements. The

> problem is that there exist (a small minority of)

> arrangements which possess mirror symmetry, so that the very

> same arrangement would occur for two different symmetry

> transformations of the pile. Accounting for the number of

> distinct positions possessing such symmetry is not at all

> easy; so the basic counting problem is much easier if you do

> not admit reflecting transformations for achieving standard

> orientation. But the cube20.org folks were not interested

> so much in _counting_ arrangements as in analyzing relations

> between explicit instances of state, so the extra complexity

> that gained another factor of 2 was well worth it for them.

>

> Regards,

> David V.

>