# 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