I can readily believe that possible interfaces for 6
(and upward) dimensional simulations would be
workable, based on the 4-D paradigm along with the
ability to selectively hide different layers/faces
etc. Also, I think there is plenty of fun to be had
with a step by step algorithm; with (in the case of
the cubes at least) enough intermediate milestones
such as one face complete, two faces complete etc etc,
to provide sufficient motivation (and satisfaction!)
to keep moving forward.
My main fear would be that this step by step
'orthogonal' approach (I don't know how else to
describe it) misses a lot of the subtleties and
'richness' that the extra dimensions provide. Nor does
it, I firmly believe, give any real insight into how
God's algorithm for each cube might look. Perhaps this
is where software designed to solve the higher
dimensional cubes will have a clear advantage over
human visualisation and imagination, in that it is not
constrained by the 3 dimensions we are familiar with.
--- Melinda Green
> I think the practical skills required to solve these
> puzzles without
> help are the same needed to solve any complex
> problem. That is the
> ability to systematically break a large problem down
> into a series of
> smaller ones.
>
> Maybe someone will solve a 6D cube before too long.
> I suspect that is
> probably inevitable. My real question is whether
> that will be very fun.
> Maybe what we really need is a breakthrough in UI
> design that will allow
> sufficiently patient humans to solve cubes of any
> number of dimensions.
> I have difficulty imagining what might look like but
> I would not be
> surprised if the general design of such an interface
> might also be
> usefully applied to other very practical searching
> and optimizing
> problems. I would doubt that it would look like the
> MC4D interface but
> then I would not have guessed that a workable 5D
> version could be based
> on our design but clearly I was wrong. Perhaps with
> enough controls to
> show and hide carefully selected parts of the
> puzzles, a true
> n-dimensional UI really could be based on our
> design. I just don't know.
>
> -Melinda
>
> Mark Oram wrote:
> > Melinda,
> >
> > I too had the peaks in mind metaphorically. After
> all,
> > if one likens the 3^5 to Everest, one needs other
> > names to invoke for the 4^5 and 5^5, or n^6 etc.
> > Perhaps humans will walk on the REAL Olympus Mons
> > before a 3^6 solution exists??
> >
> > I have no doubt that you, or anyone else reading
> this,
> > could solve the 3^5, or other variants, if you
> wanted
> > to. Maybe the question then becomes (in the
> interests
> > of starting a possible discussion) to what
> practical
> > use if any could these accomplishments be put?
> >
> >
> >
> >
> > --- Melinda Green
> wrote:
> >
> >
> >> Mark,
> >>
> >> When I spoke about still-higher peaks I was still
> >> talking about the
> >> metaphoric types. Most specifically, there are
> still
> >> the 5^4 and the
> >> seductively symmetric 5^5 still waiting to be
> >> climbed. Judging from your
> >> description of the difficulty of solving the 3^5
> I
> >> still stand by my
> >> prediction that we're likely to see exactly one
> >> solution to the 5^5. It
> >> sounds like it could take most of a year to
> >> accomplish that and it's
> >> hard to imagine more than one person finishing it
> >> unless perhaps we end
> >> up with a race. Either way it sounds awful but
> >> remember, any first
> >> solution will only happen once! ;-)
> >>
> >> Thank you for your description of the process. It
> >> made it possible for
> >> me to get an idea of how one might actually solve
> a
> >> 5D cube which until
> >> now just seemed like a miracle.
> >>
> >> -Melinda
> >>
> >> markoram109 wrote:
> >>
> >>> Melinda,
> >>>
> >>> Thank-you very much for your kind words of
> >>>
> >> support: these really
> >>
> >>> make all the difference for me in these crazy
> >>>
> >> undertakings :)
> >>
> >>> Certainly there are many higher peaks out there:
> >>>
> >> Olympus Mons, on
> >>
> >>> the planet Mars, is 3x higher than Everest for a
> >>>
> >> start, and I think
> >>
> >>> there are cliffs on Miranda (a moon of Uranus)
> >>>
> >> even higher. So as
> >>
> >>> you say there are always new peaks to aim for.
> >>>
> >> Still, I'm not sure
> >>
> >>> I'll be emabrking on any of these new challenges
> >>>
> >> just yet....
> >>
> >>> What I will be doing soon is expanding just a
> >>>
> >> little on how this
> >>
> >>> solution worked out for me, with the hope that
> it
> >>>
> >> will be useful
> >>
> >>> (and inspiring?!) for anyone else attempting to
> >>>
> >> solve any of the 5-D
> >>
> >>> versions.
> >>>
> >>>
> >>> Mark.
> >>>
> >>>
> >>>
> >>> --- In 4D_Cubing@yahoogroups.com, Melinda Green
> >>>
> >>
> >>
> >>>
> >>>
> >>>> Hey, congratulations! That's beyond amazing to
> >>>>
> >> have solved a 5D cube. I haven't even solved the
> 4D
> >> version!
> >>
> >>>> You did right by sending your log file to Roice
> >>>>
> >> and I see that he's added you to the
> >> hall-of-insanity though I don't see your log file
>
> >>
> >>>> listed there. BTW, even though you have clearly
> >>>>
> >> conquered Everest, there are still higher peaks
> >> waiting for the first person to conquer them too!
>
> >> ;-)
> >>
> >>>> -Melinda
> >>>>
> >>>
> >>>
> >
> >
> >
> >
>
___________________________________________________________
> > Yahoo! Answers - Got a question? Someone out there
> knows the answer. Try it
> > now.
> > http://uk.answers.yahoo.com/
> >
> >
> >
> > Yahoo! Groups Links
> >
> >
> >
> >
> >
>
___________________________________________________________
Yahoo! Answers - Got a question? Someone out there knows the answer. Try it
now.
http://uk.answers.yahoo.com/
Mark,
Do you have a feeling for how god's algorithm works for even the 3D
cube? I'm not sure that is possible even for the 2^3. That's because I
view god's algorithm as a high dimensional problem where each cubie
represents a single dimension that is at some distance from where it
needs to be. I can visualize the shortest path between two points in 3
dimensions but that's my limit. I don't think that I can even "feel" my
way to a 3D cube solution. Maybe some of the best speed solvers can do
that. To me that would not mean that they approach god's algorithm but
that they would abandon any step by step approach and place one or two
cubies at a time based on how quickly each one can be moved into place,
somewhat similar to how people solve jigsaw puzzles.
Computer solutions really are mostly just human solutions and are
usually much simpler than the sorts of things people do. Computers just
do them quickly and flawlessly. They are definitely not constrained by
our 3-dimensional visualization limitations however. Computers are only
limited by algorithmic complexity. Creating the algorithms is the real
creative part regardless of whether they're performed by humans or
machines. In my mind Don's N-dimensional computer solution proves that
he's solved the N-dimensional cube even if he never solves a puzzle by
hand. An existence proof is still a proof. In other words, with enough
patience, he could follow his own instructions without a computer. In
principal you could do the same thing by creating a sufficiently large
pyramid of macros on top of macros until you could take a fully
scrambled cube, find and click on each cubie in order, and sit back and
let the computer do all the work. I would consider your master macro to
be a solution even though I probably wouldn't hold a speed-solving
contest that includes macro and non-macro solvers.
-Melinda
Mark Oram wrote:
> I can readily believe that possible interfaces for 6
> (and upward) dimensional simulations would be
> workable, based on the 4-D paradigm along with the
> ability to selectively hide different layers/faces
> etc. Also, I think there is plenty of fun to be had
> with a step by step algorithm; with (in the case of
> the cubes at least) enough intermediate milestones
> such as one face complete, two faces complete etc etc,
> to provide sufficient motivation (and satisfaction!)
> to keep moving forward.
>
> My main fear would be that this step by step
> 'orthogonal' approach (I don't know how else to
> describe it) misses a lot of the subtleties and
> 'richness' that the extra dimensions provide. Nor does
> it, I firmly believe, give any real insight into how
> God's algorithm for each cube might look. Perhaps this
> is where software designed to solve the higher
> dimensional cubes will have a clear advantage over
> human visualisation and imagination, in that it is not
> constrained by the 3 dimensions we are familiar with.
>
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Wow, that's pretty interesting. Having never solved a higher-than-3D cube,
and being a programmer, myself, it makes me want to try to write my own
program. However, it seems like the program would use "dumb" methods that
would make the solution quite long... but perhaps not. If it did, then a
human probably wouldn't have the patience to do it.
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Wow, that's pretty interesting. Having never solved a higher-than-3D cube, and being a programmer, myself, it makes me want to try to write my own program. However, it seems like the program would use "dumb" methods that would make the solution quite long... but perhaps not. If it did, then a human probably wouldn't have the patience to do it.
------=_Part_43955_33000316.1186920316660--
Melinda,
Thank-you for your (as always) thought-provoking
comments. To be honest don't feel I have any handle on
God's algorithm in 3 (or more) dimensions (I actually
do have for the 2-D 'cube', but is this too trivial to
be useful or meaningful?) to answer your first
question.
It is a good way I think to visualise each cubie as
being a certain distance from Start, in a space with
as many dimensions as needed, and then conceive of
moving each piece home by the shortest and most
efficient route. Your analogy with a jig-saw puzzle
makes sense in this light. The key difference seems to
be that moving one jig-saw puzzle piece home along the
shortest route would not in general affect the routes
of any other piece: the intriguing (and infernally
frustrating!) difference with Rubik's paradigm is that
moving one piece NECESSARILY drags other cubies along
with it - perhaps moving some even further from Start.
So every move of a cubie in a cube (regardless of
which dimensional version) alters the landscape for at
least some of the other cubies, in a (is it fair to
say?) non-deterministic way. If so, it is not possible
then to devise an algorithm to find a given set of
moves that is the most efficient route home for all
the pieces affected. I had fallen into a probably
classic trap by imagining a computer program having
the ability to do just that.
As you pointed out, however, we humans still need to
write the algorithm in the first place. Writing one
with the ability to find, up front, the shortest route
for any given position seems tantamount to finding the
route oneself - so why write the progam at all?
(except, as you allude to, for practical reasons of
speed, infallibility, time saved etc etc) I also
wonder if parrallel computing approaches such as
quantum computing, DNA computing or others might have
some milage here? Any thoughts I have on such matters
are still very nebulous, but I'll happily discuss them
further.
--- Melinda Green
> Mark,
>
> Do you have a feeling for how god's algorithm works
> for even the 3D
> cube? I'm not sure that is possible even for the
> 2^3. That's because I
> view god's algorithm as a high dimensional problem
> where each cubie
> represents a single dimension that is at some
> distance from where it
> needs to be. I can visualize the shortest path
> between two points in 3
> dimensions but that's my limit. I don't think that I
> can even "feel" my
> way to a 3D cube solution. Maybe some of the best
> speed solvers can do
> that. To me that would not mean that they approach
> god's algorithm but
> that they would abandon any step by step approach
> and place one or two
> cubies at a time based on how quickly each one can
> be moved into place,
> somewhat similar to how people solve jigsaw puzzles.
>
> Computer solutions really are mostly just human
> solutions and are
> usually much simpler than the sorts of things people
> do. Computers just
> do them quickly and flawlessly. They are definitely
> not constrained by
> our 3-dimensional visualization limitations however.
> Computers are only
> limited by algorithmic complexity. Creating the
> algorithms is the real
> creative part regardless of whether they're
> performed by humans or
> machines. In my mind Don's N-dimensional computer
> solution proves that
> he's solved the N-dimensional cube even if he never
> solves a puzzle by
> hand. An existence proof is still a proof. In other
> words, with enough
> patience, he could follow his own instructions
> without a computer. In
> principal you could do the same thing by creating a
> sufficiently large
> pyramid of macros on top of macros until you could
> take a fully
> scrambled cube, find and click on each cubie in
> order, and sit back and
> let the computer do all the work. I would consider
> your master macro to
> be a solution even though I probably wouldn't hold a
> speed-solving
> contest that includes macro and non-macro solvers.
>
> -Melinda
>
> Mark Oram wrote:
> > I can readily believe that possible interfaces for
> 6
> > (and upward) dimensional simulations would be
> > workable, based on the 4-D paradigm along with the
> > ability to selectively hide different layers/faces
> > etc. Also, I think there is plenty of fun to be
> had
> > with a step by step algorithm; with (in the case
> of
> > the cubes at least) enough intermediate milestones
> > such as one face complete, two faces complete etc
> etc,
> > to provide sufficient motivation (and
> satisfaction!)
> > to keep moving forward.
> >
> > My main fear would be that this step by step
> > 'orthogonal' approach (I don't know how else to
> > describe it) misses a lot of the subtleties and
> > 'richness' that the extra dimensions provide. Nor
> does
> > it, I firmly believe, give any real insight into
> how
> > God's algorithm for each cube might look. Perhaps
> this
> > is where software designed to solve the higher
> > dimensional cubes will have a clear advantage over
> > human visualisation and imagination, in that it is
> not
> > constrained by the 3 dimensions we are familiar
> with.
> >
>
___________________________________________________________
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