Message #2120

From: Melinda Green <>
Subject: Re: [MC4D] Calculating the number of permutation of 2by2by2by2by2 (2^5)
Date: Mon, 07 May 2012 16:30:44 -0700

It sounds like we are getting closer to closure. We certainly need to
distinguish between exact counts and "not quite" exact. Here is my
understanding from the discussion: For the purpose of getting a sense of
the scale of these big numbers, it is certainly not important to be
exact, though for the cube20 folks it was important for practical
reasons. Still, there is value in knowing the exact, precise count for
particular puzzles and that is simply the pleasure of knowing exactly
what it is. Lots of things in math appear to be "useless" as you David
says, but like ornamental diamonds, they can still be very desirable.
Practicality has nothing to do with this. The main MC4D page claims to
give the exact counts for the 3D and 4D puzzles. Is that not true? If
not, then I need to change it.

Please help me to understand the part that is "not quite". Here is what
I am hearing: The main thing that makes it tricky to be exact seems to
involve mirror symmetry. In particular, we would need to account for
some relatively rare cases of states that happen to have mirror
symmetry, and that counting those special cases appears to be very

If it can’t be done then it can’t be done. If that is the case, then
what is our best estimate of the truly unique positions when accounting
for color and mirror symmetries, and in particular, what is the term
that we need to divide by and how is it derived?

Thanks all,

On 5/7/2012 11:19 AM, Roice Nelson wrote:
> 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 <
> <>> wrote:
> Still out of phase. Significant redundancy between my last
> post and Roice’s. :-(
> It dawns on me that the problem the <>
> 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 <> 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.