Message #488

From: David Smith <djs314djs314@yahoo.com>
Subject: Re: [MC4D] Introduction to the 4D_Cubing Group
Date: Sun, 04 May 2008 03:02:36 -0000

Hi Roice,

Once again, thank you for all of your help! I really appreciate
the time you take to reply with your excellent advice.

Right after I read your post, I had an idea for achieving what
I want to do without writing a program at all! My idea basically
consists of discovering general algorithms (using MagicCube4D) that
can show that any possible permutation within the constraints I
will discover is possible, for any sized cube. I have taken some
algorithms from Keane and Kamack’s paper as given, which will help
me. If I decide to do 5-dimensional cubes after this, I will not
have this luxury! The MagicCube4D program is essential for
discovering the required algorithms, so I do not think I will
discover a general formula for any-sized any-dimensional cubes
without an advanced group theory approach (although I may discover
the upper bound without proving equality).

Right now, I am working out the final details of a general algorithm
that can perform a 3-cycle of any three hypercubies in the same
family on any sized cube. This only produces any even permutation,
but I will also show that for an arbitrarily-sized cube, certain
permutation parity restrictions exist, and will also show that all
of the other parities can be generated. Then, my 3-cycle algorithm
will show that for each possible parity condition, I can generate
any possible permutations for that parity, and this means that all
possible permutations can be reached. If you want the details of
this algorithm, I can email them to you (or post it on this group,
whichever you feel is most appropriate) and send you macro files
showing some specific examples of the general algorithm. I still
have to do something similar for orientations, although Keane
and Kamack’s paper helps me out with the corner and central edge
algorithms they discovered.

I have also discovered what I believe to be two mistakes in the
calculation of the 5x5x5x5 cube’s permutations on the MagicCube4D
website written by Eric Balandraud. They appear to be fairly
obvious mistakes (once you understand the logic of the paper), and
I would not say this if I were not at least 95% certain of it, but
anyone may feel free to correct me if I am wrong. I think that
the term ((3!)^31) should be (((3!)^31)*3) and that the term
(16!) should be ((16!)/2), making the answer given correct if we
multiply it by (3/2). The author of the paper has clearly shown
himself to be very proficent in this area, so I believe these
errors are typos or an oversight, but once again, anyone please
let me know if I am wrong.

Once again, Roice, thank you for your advice and support. I look
forward to hearing from you!

Best Regards,

David

— In 4D_Cubing@yahoogroups.com, "Roice Nelson" <roice3@…> wrote:
>
> Hi David,
>
> I’m afraid I’m not going to be as much help as I would like since
I haven’t
> been through the process of trying to write a solver yet. But I
had a few
> short thoughts on how one would do it on the way home from work
today.
>
> A dumb brute force solver could theoretically verify any given
state as
> valid or not, but that is intractable because the state spaces are
so
> unbelievable huge.
>
> That means the solver must be smart, and to write such a program
one would
> have to code a toolkit of sequences to place pieces and the
knowledge of how
> to apply the sequences in various situations. If I were attacking
this
> then, I would literally try to code in the sequences I use to
solve MC4D.
> Until the toolkit is verified to be complete, the solver will not
be good at
> being sure if a puzzle state is unsolvable (maybe it was, but
maybe the
> toolkit was just incomplete or maybe the code wasn’t smart enough
to handle
> troublesome situation like parities in the 4^4). But it still
could be
> useful to verify solvable puzzle states, and if you had an
enumeration of
> all the sets of groups that needed to be checked and it could
solve all of
> them, you would know it was a complete solver (this must be what
Keane and
> Kamack did). Only at that point then could the program
confidently be used
> to verify unsolvable states.
>
> In fact, even though I have solved MC4D a number of times now,
this forces
> me to admit that my personal toolkit is not proven complete in the
> mathematical sense. All I can say for sure right now is that it
is highly
> effective since I have never rigorously verified my sequences can
solve all
> the subgroups.
>
> The enumeration proof could be done without a computer too I bet,
and I
> figure someone who has become intimate enough with the mathematics
to prove
> the number of permutation states by coming up with a provably
complete set
> of sequences may not need a computer solver to investigate certain
puzzle
> states (I’m sure this person could have reached my checkerboard
conclusions
> in this way, and would have been more sure of the answers!).
Anyway, hope
> this was helpful, even if just a little…
>
> Take Care,
>
> Roice