Message #456

From: Roice Nelson <roice@gravitation3d.com>
Subject: Re: [MC4D] Re: Noel conquers the 4^5!
Date: Tue, 01 Apr 2008 20:28:24 -0500

I had hoped to hear some discussion about Melinda’s parity question. I have
never really understood the parity thing as much as I would like, but that
won’t stop me from blabbing about it anyway :)

I think Noel was guessing the situation would be similar to the 3D case (and
4D case? - I’m not sure since I haven’t solved the 5^4). But in the 3D case
at least, you see parity problems in the 4^3 but not the 5^3, even though
the latter contains the smaller puzzle within it in some sense.

I think this is because the even puzzles are unique in that their centers
are not fixed by a single central piece that never moves. I have gathered
that this extra freedom allows all the central pieces to be placed with
either an even or odd number of twists (relative to the scramble), which
isn’t possible in the odd puzzles, and that using an odd number of twists
leads to the parity problem.

Actually, the issue is not confined to only the center 1-colored cubies, and
in the 4^3 there are two parity problems that can happen. One is associated
with setting up the centers. The other can be encountered depending on how
one ends up placing the 2-colored pieces, which also have no "unmovable
center" with which to align pieces. In the 3D Revenge puzzle, each of the 2
parity problems happen 50% of the time, and taken together the solver only
gets lucky enough to not have to deal with either a quarter of the time.

In the 4^4, I’d venture to guess there are 3 possible parity problems, one
associated with 1C centers, one with 2C cubies, and one with 3C cubies, and
that the solver is really lucky only an eighth of the time. In the 4^5,
I’ll unjustifiably extrapolate this guess further to say there is yet one
more possible problem and the chance of having no parities at all is halved
once more. I might be wrong on this though (for example, I could see the
individual parity problems interacting in a combinatorial way instead of a
multiplicative one as suggested). I’d love for someone to set the record
straight about how all of it works!

One last comment on these is that parity problems are frustrating in that
they don’t manifest themselves immediately. In the 4^3, you don’t know if
you were lucky placing the centers until you are finishing up setting the
edges. "Doh! The last one is flipped!" And likewise the second problem
only manifests itself as you try to place the last few corners. Since they
can involve a significant backtracking of the work done on the puzzle, I’ve
heard some describe the Rubik’s Revenge as more difficult than the
Professor’s Cube. Only Noel knows if that is true or not in 5D ;)

Roice

On Sat, Mar 22, 2008 at 1:42 AM, Melinda Green <melinda@superliminal.com>
wrote:

> Congratulations indeed!!
> This is definitely a tour de force, Noel. Congratulations on setting a
> record that can never be taken from you. There is no question that you
> are the first human being to perform this feat and for all we know this
> may be the first time in the universe!
>
> One thing that I’m confused about is why you say that you won’t face
> parity problems in the 5^5. Doesn’t the 5^5 contain the 4^5? I know that
> it’s been said that after 5^d there are no more new combinatorial
> elements for all dimensions. Can someone please spell out exactly what
> the issues are and why this is true?
>
> I can certainly understand why Noel says that he’ll never do the 4^5
> again. I can imagine that other people might accomplish that if they
> feel that they can turn in a shorter solution, and I’ll also make a
> guess that this is the year that someone will solve the 5^5 for the
> first time. I’m also on record for predicting that it will be a very
> long time before a second person slays that monster if ever! The lure of
> the shortest 5^5 may simply not be attractive enough for anyone to
> attempt it.
>
> So please tell us, Noel. What was it like to battle this beast and did
> it really only take you a week? Any advice for others thinking about
> attempting to repeat your achievement? And are you *really* going to
> take a break before attempting the 5^5 or are you just trying to make
> other would-be solvers *think* that they don’t need to hurry to be the
> first? No need to answer that last question, BTW. :-)
>
> -Melinda
>
>
> jwgibson3 wrote:
> > CONGRATS!!!! Fantastic! Although, I must admit, I’m a little
> > jealous. I was hoping I could be first ;) And Noel, your solution is
> > a great length; it’s shorter than the median solution for the 3^5.
> > I’m still pairing up edges and have reached 4000 moves. Roice - the
> > finder will be fantastic. It’s pretty mind-rotting looking through
> > 1000 pieces with the shift key held down hoping the next one is it.
> > Congrats Noel! And thanks for the new features, Roice!
> >
> > Best wishes,
> >
> > John
> >
> > — In 4D_Cubing@yahoogroups.com <4D_Cubing%40yahoogroups.com>, "Roice
> Nelson" <roice@…> wrote:
> >
> >> Hey guys,
> >>
> >> I wanted to let you all know the Revenge version of the 5D cube has
> been
> >> solved for the first time! I just uploaded Noel Chalmer’s solution to
> the Hall of Insanity if you’d like to take a look.
> >>
> >> *www.gravitation3d.com/magiccube5d/hallofinsanity.html*
> >> **
> >> This puzzle has 2560 stickers and 1024 cubies and as best I can tell
> from our emails, I think it only took him about a week! But I’ll let him
> expound if he’s interested.
> >>
> >> Since they have so many pieces, Noel requested a new feature (a cubie
> >> finder) to help him out with the Revenge and Professor 5D puzzles, and
> this is part of the install now. It is on the options menu or you can
> CTRL+F. Also, thought I’d mention redo was added last year as well (that had
> been requested a lot so I finally did it, but I never mailed out about it).
> >>
> >> I hope this finds everybody well,
> >>
> >> Roice
> >>
> >>
> >>
> >> ———- Forwarded message ———-
> >> From: Noel Chalmers <ltd.dv8r@…>
> >> Date: Wed, Mar 19, 2008 at 1:43 AM
> >> Subject: Re: small thing
> >> To: roice3@…
> >>
> >>
> >> Hey Roice,
> >>
> >> Well, I said I’d do it and it’s done. This thing was HARD. I had a
> parity error at every step of solving it like a 3x3 and I had to kind of
> make up a sequence on the fly that would correct it. I’m looking forward to
> the 5^5 where I won’t have to worry about parity! But I think I’ll take a
> bit of a break before that.
> >> This is by no means a shortest move solution, I gave up on trying to
> have a low move count when I hit the parities. I suspect that if someone
> else
> >> solves it they’ll have less moves but that’s fine by me, I don’t plan
> on
> >> doing this ever again. Thanks again for the update to the program, I
> >> couldn’t have done it without your help!
> >>
> >> Cheers,
> >> Noel
>
>
>