Thread: "Introduction to the 4D_Cubing Group"

Hello, everyone! My name is David Smith, and I hope to
be a contributive member to this unique and highly
interesting group. My interests besides the cube are
mathematics, physics, chess, computer programming,
and retrograde analysis (a very remarkable type of
chess problem).

Reading the posts made by various members has gotten
me very interested in the solving of higher-dimensional
cubes. It seems like a very challenging task, and
congratulations to everyone who has solved a four or
five-dimensional Rubik's Cube! I will definitely try it
soon. What I am interested in now, however, is the
mathematics of the Rubik's Cube. I wonder if any of
you also have an interest in this area?

I am currently working on an interesting problem -
finding a formula for the number of reachable
configurations of the NxNxNxN Rubik's Cube. I
believe I will have an answer to this question
soon, so I would be very glad to share it with
the group, or perhaps only the members who are
interested in mathematics, if any. The paper
written by Eric Balandraud on the MagicCube4D
website has been very helpful, but I am
currently stuck on a minor detail with his
calculation of the number of permutations of
the 5x5x5x5 cube, but I believe I will
discover my error soon.

After this, I want to find the same formula
for 4-dimensional supercubes and super-supercubes.
(see http://www.speedcubing.com/chris/cubecombos.html
for a definition of these terms) After that, I
will (perhaps foolishly!) attempt to find formulae
for cubes, supercubes, and super-supercubes of
any size and any dimension.

I apologize if this post has been too long, but
I wanted to give a detailed introduction of myself
and my current tasks, and I hope that at least some
of you would be interested in discussing these
problems and their solution. I am trying to do
this without any formal mathematics training, so
my solution, when I find it, may be long but
relatively simple to understand.

I wish to thank everyone who has contributed the
the theory of Rubik's Cube knowledge, helped
in the creation of Rubik's Cube software, or
otherwise done amazing things with Rubik's Cubes.

Happy Hypercubing!


Best Regards,

David

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Hi David,

Very nice to make your acquaintance, and welcome! I'm sure others will have
more to say, but I thought I'd point you to two resources on the mathematics
I've seen in the mean time. The first (because it is shorter) is a paper
Melinda pointed me to because it had the calculation for the number of
permutations of the 3^5. It is titled "The Rubik Tesseract" and you can
find it here:

http://udel.edu/~tomkeane/RubikTesseract.pdf

Secondly, there is a book by David Joyner called "Adventures in Group
Theory". It is a course on group theory focused around the 3D Rubik's
cube. Though it doesn't discuss the higher-d puzzles, it could be a useful
source for extending the relevant mathematical skills.

I'll look forward to hearing about your investigations. As someone else who
finds this very interesting but lacks related formal mathematical training,
I've always felt lacking in the group theory aspects of these puzzles...

All the best,

Roice


On Mon, Apr 28, 2008 at 9:29 PM, David Smith wrote:

> Hello, everyone! My name is David Smith, and I hope to
> be a contributive member to this unique and highly
> interesting group. My interests besides the cube are
> mathematics, physics, chess, computer programming,
> and retrograde analysis (a very remarkable type of
> chess problem).
>
> Reading the posts made by various members has gotten
> me very interested in the solving of higher-dimensional
> cubes. It seems like a very challenging task, and
> congratulations to everyone who has solved a four or
> five-dimensional Rubik's Cube! I will definitely try it
> soon. What I am interested in now, however, is the
> mathematics of the Rubik's Cube. I wonder if any of
> you also have an interest in this area?
>
> I am currently working on an interesting problem -
> finding a formula for the number of reachable
> configurations of the NxNxNxN Rubik's Cube. I
> believe I will have an answer to this question
> soon, so I would be very glad to share it with
> the group, or perhaps only the members who are
> interested in mathematics, if any. The paper
> written by Eric Balandraud on the MagicCube4D
> website has been very helpful, but I am
> currently stuck on a minor detail with his
> calculation of the number of permutations of
> the 5x5x5x5 cube, but I believe I will
> discover my error soon.
>
> After this, I want to find the same formula
> for 4-dimensional supercubes and super-supercubes.
> (see http://www.speedcubing.com/chris/cubecombos.html
> for a definition of these terms) After that, I
> will (perhaps foolishly!) attempt to find formulae
> for cubes, supercubes, and super-supercubes of
> any size and any dimension.
>
> I apologize if this post has been too long, but
> I wanted to give a detailed introduction of myself
> and my current tasks, and I hope that at least some
> of you would be interested in discussing these
> problems and their solution. I am trying to do
> this without any formal mathematics training, so
> my solution, when I find it, may be long but
> relatively simple to understand.
>
> I wish to thank everyone who has contributed the
> the theory of Rubik's Cube knowledge, helped
> in the creation of Rubik's Cube software, or
> otherwise done amazing things with Rubik's Cubes.
>
> Happy Hypercubing!
>
> Best Regards,
>
> David
>
>
>

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Roice,

Thank you very much for your reply! I really appreciate the two
resources you kindly pointed out to me. I have actually already read
the paper "The Rubik Tesseract", which is what got me very interested
in the generalized problems of n^4 and n^k Rubik's Cubes. I got
the idea to also try supercubes and super-supercubes from the
n^3 formulas page I referred to in my previous post. That page
inspired me to rediscover the same formulas that Chris Hardwick did.
An interesting thing about 4D super and super-supercubes I
realized is that any hypercubie with more than 1 hyperfacelet can be=20
twisted in the ways Keane and Kamack showed in their paper. However,=20
center hypercubies, with only one hyperfacelet, can be oriented in=20
24 different positions, and they undoubtedly have certain=20
restrictions related to the other hypercubies. I want to figure out
the regular cube first though!

As for the other resource you mentioned, the author of that book has=20
a preprint version available for download (at=20
http://web.usna.navy.mil/~wdj/books.html) which appears to be
very complete, despite the fact that it is not the actual book.
I have already studied it, and it was a good introduction to group=20
theory and how it relates to the Rubik's Cube.

The difficult part of this task is not discovering the formula,
but proving it is correct and not just an upper bound. That is,
once I have ruled out the impossible permutations, I must show
that all of the remaining permutations are actually possible.
(Keane and Kamack actually admit they did not do this for the
3^5 calculation, but state that they are very confident it is
correct.) I suppose I could use the results from that paper, the
ones where they show using a computer program, that all of
the remaining permutations are possible. However, this would
be difficult to expand to higher dimensions. Therefore, I am
writing my own computer program, whose sole purpose is to convert
4-dimensional Rubik's cubes into cycle notation, that is, labeling
every hyperfacelet with a unique number and listing, in cycles,
where each hyperfacelet goes when each hyperface is rotated in each
necessary direction. (I tried doing it by hand at first - not
recommended!) I am currently sorting out bugs in the program.
When it is working, I pan to take the output it provides and
put that into the Computer Algebra System GAP. I would then be
able to directly calculate the number of permutations of any
specific 4D cube, but more importantly, I will be able to show
how different types of hypercubies can interplay with the rest
of the cube. (example: On the 3x3x3x3, can I actually swap
two hyperfacelets of a 3-colored hypercubie without affecting
the rest of the cube?) I then plan to generalize those results
to any sized cube, perhaps by induction.

Once again, thank you Roice, for your quick and detailed reply!
I am glad I have someone else to discuss these things with.


Best Regards,

David


--- In 4D_Cubing@yahoogroups.com, "Roice Nelson" wrote:
>
> Hi David,
>=20
> Very nice to make your acquaintance, and welcome! I'm sure others=20
will have
> more to say, but I thought I'd point you to two resources on the=20
mathematics
> I've seen in the mean time. The first (because it is shorter) is=20
a paper
> Melinda pointed me to because it had the calculation for the=20
number of
> permutations of the 3^5. It is titled "The Rubik Tesseract" and=20
you can
> find it here:
>=20
> http://udel.edu/~tomkeane/RubikTesseract.pdf
>=20
> Secondly, there is a book by David Joyner called "Adventures in=20
Group
> Theory". It is a course on group theory focused around the 3D=20
Rubik's
> cube. Though it doesn't discuss the higher-d puzzles, it could be=20
a useful
> source for extending the relevant mathematical skills.
>=20
> I'll look forward to hearing about your investigations. As=20
someone else who
> finds this very interesting but lacks related formal mathematical=20
training,
> I've always felt lacking in the group theory aspects of these=20
puzzles...
>=20
> All the best,
>=20
> Roice
>=20
>=20
> On Mon, Apr 28, 2008 at 9:29 PM, David Smith =20
wrote:
>=20
> > Hello, everyone! My name is David Smith, and I hope to
> > be a contributive member to this unique and highly
> > interesting group. My interests besides the cube are
> > mathematics, physics, chess, computer programming,
> > and retrograde analysis (a very remarkable type of
> > chess problem).
> >
> > Reading the posts made by various members has gotten
> > me very interested in the solving of higher-dimensional
> > cubes. It seems like a very challenging task, and
> > congratulations to everyone who has solved a four or
> > five-dimensional Rubik's Cube! I will definitely try it
> > soon. What I am interested in now, however, is the
> > mathematics of the Rubik's Cube. I wonder if any of
> > you also have an interest in this area?
> >
> > I am currently working on an interesting problem -
> > finding a formula for the number of reachable
> > configurations of the NxNxNxN Rubik's Cube. I
> > believe I will have an answer to this question
> > soon, so I would be very glad to share it with
> > the group, or perhaps only the members who are
> > interested in mathematics, if any. The paper
> > written by Eric Balandraud on the MagicCube4D
> > website has been very helpful, but I am
> > currently stuck on a minor detail with his
> > calculation of the number of permutations of
> > the 5x5x5x5 cube, but I believe I will
> > discover my error soon.
> >
> > After this, I want to find the same formula
> > for 4-dimensional supercubes and super-supercubes.
> > (see http://www.speedcubing.com/chris/cubecombos.html
> > for a definition of these terms) After that, I
> > will (perhaps foolishly!) attempt to find formulae
> > for cubes, supercubes, and super-supercubes of
> > any size and any dimension.
> >
> > I apologize if this post has been too long, but
> > I wanted to give a detailed introduction of myself
> > and my current tasks, and I hope that at least some
> > of you would be interested in discussing these
> > problems and their solution. I am trying to do
> > this without any formal mathematics training, so
> > my solution, when I find it, may be long but
> > relatively simple to understand.
> >
> > I wish to thank everyone who has contributed the
> > the theory of Rubik's Cube knowledge, helped
> > in the creation of Rubik's Cube software, or
> > otherwise done amazing things with Rubik's Cubes.
> >
> > Happy Hypercubing!
> >
> > Best Regards,
> >
> > David
> >
> >=20
> >
>

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Hi David,

The computer program sounds like a cool project :) After reading this last
post, I thought I would also point you to Don's solving code page just in
case you hadn't seen it.

http://www.plunk.org/~hatch/MagicCubeNdSolve/

His solver is actually hooked up to the MC4D UI under "Edit->Solve (For
Real)" (we've wanted to do the same for MC5D, but haven't). The reason I
mention it in the context of your email is that I've used it to
answer similar questions to the example you gave. By manually editing the
log files, then seeing if the resulting position is solvable, one can test
whether certain states are reachable in the 4D cube (I recently looked at
possible types of checkerboards). So again, just mentioning in case it
could help you with your current goals, but it sounds like you are already
well on your way since you are already taking down software bugs.

Btw, I have some 3^3 puzzles with the pictures on them, though I didn't know
they were called supercubes, and I hadn't heard of super-supercubes before.
That's a pretty neat extension!

cya,

Roice

On Tue, Apr 29, 2008 at 9:20 PM, David Smith wrote:

> Roice,
>
> Thank you very much for your reply! I really appreciate the two
> resources you kindly pointed out to me. I have actually already read
> the paper "The Rubik Tesseract", which is what got me very interested
> in the generalized problems of n^4 and n^k Rubik's Cubes. I got
> the idea to also try supercubes and super-supercubes from the
> n^3 formulas page I referred to in my previous post. That page
> inspired me to rediscover the same formulas that Chris Hardwick did.
> An interesting thing about 4D super and super-supercubes I
> realized is that any hypercubie with more than 1 hyperfacelet can be
> twisted in the ways Keane and Kamack showed in their paper. However,
> center hypercubies, with only one hyperfacelet, can be oriented in
> 24 different positions, and they undoubtedly have certain
> restrictions related to the other hypercubies. I want to figure out
> the regular cube first though!
>
> As for the other resource you mentioned, the author of that book has
> a preprint version available for download (at
> http://web.usna.navy.mil/~wdj/books.html) which appears to be
> very complete, despite the fact that it is not the actual book.
> I have already studied it, and it was a good introduction to group
> theory and how it relates to the Rubik's Cube.
>
> The difficult part of this task is not discovering the formula,
> but proving it is correct and not just an upper bound. That is,
> once I have ruled out the impossible permutations, I must show
> that all of the remaining permutations are actually possible.
> (Keane and Kamack actually admit they did not do this for the
> 3^5 calculation, but state that they are very confident it is
> correct.) I suppose I could use the results from that paper, the
> ones where they show using a computer program, that all of
> the remaining permutations are possible. However, this would
> be difficult to expand to higher dimensions. Therefore, I am
> writing my own computer program, whose sole purpose is to convert
> 4-dimensional Rubik's cubes into cycle notation, that is, labeling
> every hyperfacelet with a unique number and listing, in cycles,
> where each hyperfacelet goes when each hyperface is rotated in each
> necessary direction. (I tried doing it by hand at first - not
> recommended!) I am currently sorting out bugs in the program.
> When it is working, I pan to take the output it provides and
> put that into the Computer Algebra System GAP. I would then be
> able to directly calculate the number of permutations of any
> specific 4D cube, but more importantly, I will be able to show
> how different types of hypercubies can interplay with the rest
> of the cube. (example: On the 3x3x3x3, can I actually swap
> two hyperfacelets of a 3-colored hypercubie without affecting
> the rest of the cube?) I then plan to generalize those results
> to any sized cube, perhaps by induction.
>
> Once again, thank you Roice, for your quick and detailed reply!
> I am glad I have someone else to discuss these things with.
>
> Best Regards,
>
> David
>
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Hi Roice,

Thank you very much for pointing me to that website! I had
not seen that website before. As for my program, it is not
going well; I have very limited coding experience. For one
thing, the program works but the data it generates is
incorrect. It is overly complicated. It is currently specific to the=20
4^4 cube. And when I run it and generate the data, it takes GAP over 5
hours to process the data! In short, I need to find a way
to either write a program that tests whether a specific
position is possible, for a 4D (preferably, N-D) cube of any size,
or do the same with an existing program. I have tried to analyze
the code for Don's N-dimensional Rubik's Cube solver, but
it is beyond me.

I would be extremely grateful if you or any others in
this group could help me understand the algorithms involved
when testing whether a position is possible for higher-dimensional
cubes. The reason I can't simply use Don's program is that
I need to study larger cubes of 4 (preferably any) dimensions.
I definitely need to study 4^4, and perhaps 5^4, I am not
sure yet. And eventually I hope to tackle cubes of more
than four dimensions. As I said, I tried to understand Don's
code for 2^k and 3^k cubes, but could not understand it.

The only functionality of my program that I need is
the ability to test whether a position is solvable. Any
help you provide, no matter how little, would be greatly
appreciated. I hope I'm not asking for too much. If
there is anything I could do to reciprocate your kindness,
please let me know. Of course if I find the answers I am
looking for, this group would be the first to know!

Once again, thank you for all of your help. I will let
you know if I make any more progress.

Best Regards,

David


PS: I just recently realized that you helped write the code for the
5-dimensional Rubiks Cube program. Amazing!


--- In 4D_Cubing@yahoogroups.com, "Roice Nelson" wrote:
>
> Hi David,
>=20
> The computer program sounds like a cool project :) After reading=20
this last
> post, I thought I would also point you to Don's solving code page=20
just in
> case you hadn't seen it.
>=20
> http://www.plunk.org/~hatch/MagicCubeNdSolve/
>=20
> His solver is actually hooked up to the MC4D UI under "Edit->Solve=20
(For
> Real)" (we've wanted to do the same for MC5D, but haven't). The=20
reason I
> mention it in the context of your email is that I've used it to
> answer similar questions to the example you gave. By manually=20
editing the
> log files, then seeing if the resulting position is solvable, one=20
can test
> whether certain states are reachable in the 4D cube (I recently=20
looked at
> possible types of checkerboards). So again, just mentioning in=20
case it
> could help you with your current goals, but it sounds like you are=20
already
> well on your way since you are already taking down software bugs.
>=20
> Btw, I have some 3^3 puzzles with the pictures on them, though I=20
didn't know
> they were called supercubes, and I hadn't heard of super-supercubes=20
before.
> That's a pretty neat extension!
>=20
> cya,
>=20
> Roice

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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


On Wed, Apr 30, 2008 at 6:11 PM, David Smith wrote:

> Hi Roice,
>
> Thank you very much for pointing me to that website! I had
> not seen that website before. As for my program, it is not
> going well; I have very limited coding experience. For one
> thing, the program works but the data it generates is
> incorrect. It is overly complicated. It is currently specific to the
> 4^4 cube. And when I run it and generate the data, it takes GAP over 5
> hours to process the data! In short, I need to find a way
> to either write a program that tests whether a specific
> position is possible, for a 4D (preferably, N-D) cube of any size,
> or do the same with an existing program. I have tried to analyze
> the code for Don's N-dimensional Rubik's Cube solver, but
> it is beyond me.
>
> I would be extremely grateful if you or any others in
> this group could help me understand the algorithms involved
> when testing whether a position is possible for higher-dimensional
> cubes. The reason I can't simply use Don's program is that
> I need to study larger cubes of 4 (preferably any) dimensions.
> I definitely need to study 4^4, and perhaps 5^4, I am not
> sure yet. And eventually I hope to tackle cubes of more
> than four dimensions. As I said, I tried to understand Don's
> code for 2^k and 3^k cubes, but could not understand it.
>
> The only functionality of my program that I need is
> the ability to test whether a position is solvable. Any
> help you provide, no matter how little, would be greatly
> appreciated. I hope I'm not asking for too much. If
> there is anything I could do to reciprocate your kindness,
> please let me know. Of course if I find the answers I am
> looking for, this group would be the first to know!
>
> Once again, thank you for all of your help. I will let
> you know if I make any more progress.
>
> Best Regards,
>
> David
>
> PS: I just recently realized that you helped write the code for the
> 5-dimensional Rubiks Cube program. Amazing!
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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=20
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=20
discovering the required algorithms, so I do not think I will
discover a general formula for any-sized any-dimensional cubes=20
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=20
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" wrote:
>
> Hi David,
>=20
> I'm afraid I'm not going to be as much help as I would like since=20
I haven't
> been through the process of trying to write a solver yet. But I=20
had a few
> short thoughts on how one would do it on the way home from work=20
today.
>=20
> A dumb brute force solver could theoretically verify any given=20
state as
> valid or not, but that is intractable because the state spaces are=20
so
> unbelievable huge.
>=20
> That means the solver must be smart, and to write such a program=20
one would
> have to code a toolkit of sequences to place pieces and the=20
knowledge of how
> to apply the sequences in various situations. If I were attacking=20
this
> then, I would literally try to code in the sequences I use to=20
solve MC4D.
> Until the toolkit is verified to be complete, the solver will not=20
be good at
> being sure if a puzzle state is unsolvable (maybe it was, but=20
maybe the
> toolkit was just incomplete or maybe the code wasn't smart enough=20
to handle
> troublesome situation like parities in the 4^4). But it still=20
could be
> useful to verify solvable puzzle states, and if you had an=20
enumeration of
> all the sets of groups that needed to be checked and it could=20
solve all of
> them, you would know it was a complete solver (this must be what=20
Keane and
> Kamack did). Only at that point then could the program=20
confidently be used
> to verify unsolvable states.
>=20
> In fact, even though I have solved MC4D a number of times now,=20
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=20
is highly
> effective since I have never rigorously verified my sequences can=20
solve all
> the subgroups.
>=20
> The enumeration proof could be done without a computer too I bet,=20
and I
> figure someone who has become intimate enough with the mathematics=20
to prove
> the number of permutation states by coming up with a provably=20
complete set
> of sequences may not need a computer solver to investigate certain=20
puzzle
> states (I'm sure this person could have reached my checkerboard=20
conclusions
> in this way, and would have been more sure of the answers!).=20=20
Anyway, hope
> this was helpful, even if just a little...
>=20
> Take Care,
>=20
> Roice

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Excellent, this sounds like a great approach. If you are interested in
writing it up, I'm sure some would enjoy reading about your general
algorithm to cycle 3 pieces of any type, but not worries too if you don't
feel like doing it (if you'd rather just focus on the problem :))

I'll let Eric and/or others comment on the possible 5^4 permutation
calculation issues for now because I'll still need to hunker down and do
some study myself before I'm in a position to contribute there.

All the best,

Roice

P.S. As a teaser, I'm going to present a new permutation calculation problem
soon (hopefully this evening) that I hope you guys will be able to help me
figure out!

On Sat, May 3, 2008 at 10:02 PM, David Smith wrote:

> 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
> Visit Your Group
>
> Earth Day 2008
>
> Get things and
>
> get things for free.
>
> Find out how.
> 10 Day Club
>
> on Yahoo! Groups
>
> Share the benefits
>
> of a high fiber diet.
> Yahoo! Groups
>
> Join a program
>
> to help you find
>
> balance in your life.
> .
>
>
>

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Hi all,

I got pretty bored one day so I decided to retry solving the 120-cell for
the heck of it. My previous attempts got me to layer 4, but I decided to
start from scratch again because my 4D CFOP, 120-cell edition, improved a
little since then (2011). =P Method's here (for 3^4), though I might edit
it in the next few days or weeks.
http://wiki.superliminal.com/wiki/3%5E4#Sheerin-Zhao_Method_.28Hybrid.29_V1

It was finally solved on Aug 30, 2014. Started Aug 13, 2014. Including the
1k move scramble, it's just over 20k moves.

Of course, the fact that I didn't solve it a long time ago meant that I
couldn't break the age record of 15 since I'm 16 now, but at least I got to
beat the shortest. I also beat myself; by the time I was on layer 4, the
solution seemed to have become almost 40% more efficient than the 2011
attempt though I probably have to check that number. Half of the reason for
the efficiency was the use of RKT, or limiting moves to only two cells,
pretending one of them is a megaminx (a bit hard to explain. the wiki link
mentions it as ). The other half would be because I strived to
learn how the 3c and 4c turns work, and pressed undo many times just to
find the smoothest insert of an f2l pair.

Because CFOP uses pretty intuitive F2L, what's macros? Roice's program
works nicely, and I have to admit it probably has the easiest and most
comfortable controls, but that's probably because I used it for weeks
straight. Its similar colours have got me into trouble multiple times,
though, but how do you pick an arrangement with 120 contrasting colours so
that you don't get a case with a 3c grey-grey-grey piece? (I hated that)
Perhaps even a variation of 4-colour rule would work here. ^_~

As for during the solve, the beginning was pretty frustrating since pieces
would seem to be everywhere; most of the time was spent finding the piece
since I'd rotate the puzzle, turn on the layer, and forget the exact
position that the piece was in.

By the time layer 4 was half-solved, things felt extremely restricting
since some turns would have the same effect as exposing a solved slot on
the 3^3. In fact, I screwed up multiple times and had to refix multiple
pairs because I had taken them out by accident. That by itself probably
added around 2k moves.

The worst part was the very end, when there seemed to be only one flipped
edge; the other one was actually wedged between two greys in layer 5. @_@
At the same time, it was kind of exciting to be able to find a solution to
that case; filled up a little piece of scrap paper with the steps and all.
=P

So that's pretty much all about the solve. .-. Roice encouraged me to post
so yeah. Not sure what puzzles I'll go for next (aka improve on),
considering that Uni registration is this year (w00t) though duoprisms and
penteracts sound nice. :)

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