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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
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" 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
> >> 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
>
>
>

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

px 0px 0.8ex; BORDER-LEFT: #ccc 1px solid">

BOTTOM: 0px; MARGIN: 0px; WIDTH: 470px; PADDING-TOP: 0px">

Congratulations indeed!!
This is definitely a tour de force, Noel. Co=
ngratulations on setting a
record that can never be taken from you. The=
re is no question that you
are the first human being to perform this fe=
at and for all we know this

may be the first time in the universe!

One thing that I'm confus=
ed 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 sai=
d 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 oth=
er 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
fi=
rst 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 r>the shortest 5^5 may simply not be attractive enough for anyone to
at=
tempt it.

So please tell us, Noel. What was it like to battle this b=
east and did

it really only take you a week? Any advice for others thinking about
at=
tempting 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=20

jwgibson3 wrote:
> CONGRATS!!!! Fantast=
ic! 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&#=
39;s shorter than the median solution for the 3^5.

> I'm still pairing up edges and have reached 4000 moves. Roice - th=
e
> finder will be fantastic. It's pretty mind-rotting looking th=
rough
> 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%40yahoogroups.com" target=3D"_blank">4D_Cubing@yahoogroups.coma>, "Roice Nelson" <roice@...> wrote:

>
>> Hey guys,
>>
>> I wanted to let you all=
know the Revenge version of the 5D cube has been
>> solved for th=
e first time! I just uploaded Noel Chalmer's solution to the Hall of In=
sanity if you'd like to take a look.

>>
>> *allofinsanity.html*" target=3D"_blank">www.gravitation3d.com/magiccube5d/ha=
llofinsanity.html*
>> **
>> This puzzle has 2560 stic=
kers and 1024 cubies and as best I can tell from our emails, I think it onl=
y took him about a week! But I'll let him expound if he's intereste=
d.

>>
>> Since they have so many pieces, Noel requested a new f=
eature (a cubie
>> finder) to help him out with the Revenge and Pr=
ofessor 5D puzzles, and this is part of the install now. It is on the optio=
ns menu or you can CTRL+F. Also, thought I'd mention redo was added las=
t year as well (that had been requested a lot so I finally did it, but I ne=
ver mailed out about it).

>>
>> I hope this finds everybody well,
>>
>&=
gt; Roice
>>
>>
>>
>> ---------- Forwar=
ded message ----------
>> From: Noel Chalmers <ltd.dv8r@...>=

>> Date: Wed, Mar 19, 2008 at 1:43 AM
>> Subject: Re: small =
thing
>> To: roice3@...
>>
>>
>> Hey Ro=
ice,
>>
>> 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 abo=
ut 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

lockquote>

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From: "Roice Nelson" <roice@gravitation3d.com>
Date: Tue, 1 Apr 2008 23:40:42 -0500
Subject: Re: [MC4D] Noel conquers the 4^5!

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Yeah, interesting reading for me too. I was swayed by some of the ideas,
but I think I'm sticking with my first thought when I heard the question
posed, which is that 4D beings would have an easier time, at least
conceptually (if the difficulty metric were tedium instead of understanding,
the piece counts could likely reverse my stance).
If MC6D was limited to rotations on coordinate planes like MC5D is for
instance, all the puzzle states would still be available without any real
extra twisting complexity. Even taking into consideration more general
rotations, I think that although the number of them grows combinatorially,
orthogonal components of the more complex rotations are still only limited
to a plane and that doesn't change - in a way understanding the 3D rotations
is kind of all there is to it for this reason.

But my main rationale for leaning towards 4D aliens having the advantage is
that they would have more concept of dimensional analogy just built into
their way of thinking. The extra level does a lot to make dimensional
patterns clear whereas being exposed to only a line, a square, and a cube in
our physical world isn't quite enough to make it easy to extrapolate to the
next level (really I'm just restating Mark's thought that their brains would
be more geared for it).

Having said all that, I still agree with Jenelle too. MC4D would still be
hard, and I bet solving it in that universe would still be a great party
trick that would impress all your friends and be fun to teach :)
Permutation puzzles are just naturally difficult, 3D or not, which is a also
big part of why Rubik's cubes are inexplicably impossible when we are first
exposed to them, and I don't think a 4D brain would have any advantage
there. That is, unless 4D does lead to fundamentally more powerful
computing ;)

Finally, on the viability of such universes, I have no idea but it seems
that the laws of physics would certainly have to be different! Melinda has
made the interesting point to me in the past that when it comes to physical
dimensions, there seems to be nothing precluding multiple time dimensions
entering the picture as well...

On Tue, Apr 1, 2008 at 9:47 AM, Mark Oram wrote:

Thanks guys for the comments/feedback: I found them all very interesting.

Certainly there are more things to keep track of as the dimensions go up,
but I wonder if that necessarily implies that the moves HAVE to be more
"complex", or just that there are longer sequences needed? Obviously one can
solve, say, the 5-D cube by small steps; moving each class of cubie into
place one by one with small sequences of moves. It just takes longer and
needs more patience. Also, it may not be the quickest (or most elegant)
means. If nothing else, I assume a 4-D (or 5-D etc) being could always
resort to such an approach. Crucially, however, this does not exclude the
possibility that a 4-D being could have a fundamentally superior insight,
and might be able to see the most efficient sets of moves more intuitively
than us every time.

I hadn't considerd the practicalities of how the 3-D or 4-D cubes would be
assembled physically in 4-D space, and again it may be that the engineering
hurdles are equivalent regardless of the actual dimension of the space in
question. It reminds me of the initial confusion form many people (myself
included) when they had seen the 3-D cube for the first time, wondering how
it could even be made without everything falling apart once it was turned!
Still, clearly the 3-D cube is physically possible,a nd I have no doubt that
equivalent ones are possible - at least in principle.

Of course, it is easy to imagine that 4-D and 5-D hypercomputer screens
could represent any of these cubes with ease, but again the issue of how to
usefully represent extra dimensions on a screen with a fixed maximum number
of dimensions would occur there as well. (I have always felt that a large
part of the genius of Charlie, Melinda, Remigiusz and Roice was finding a
way to do this for the 5-D cube on a 2-D screen, so thanks again guys!)

One final point: are the 4-D/5-D hypercomputers still just Turing machines
at heart, or is it possible that there can be some fundamentally more
powerful means of computing in the higher dimensions? (Or am I now wandering
too far from the main group topic???)

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Yeah, interesting reading for me too. I was swayed by some of the ideas, but I think I'm sticking with my first thought when I heard the question posed, which is that 4D beings would have an easier time, at least conceptually (if the difficulty metric were tedium instead of understanding, the piece counts could likely reverse my stance).

If MC6D was limited to rotations on coordinate planes like MC5D is for instance, all the puzzle states would still be available without any real extra twisting complexity. Even taking into consideration more general rotations, I think that although the number of them grows combinatorially, orthogonal components of the more complex rotations are still only limited to a plane and that doesn't change - in a way understanding the 3D rotations is kind of all there is to it for this reason.

But my main rationale for leaning towards 4D aliens having the advantage is that they would have more concept of dimensional analogy just built into their way of thinking. The extra level does a lot to make dimensional patterns clear whereas being exposed to only a line, a square, and a cube in our physical world isn't quite enough to make it easy to extrapolate to the next level (really I'm just restating Mark's thought that their brains would be more geared for it).

Having said all that, I still agree with Jenelle too. MC4D would still be hard, and I bet solving it in that universe would still be a great party trick that would impress all your friends and be fun to teach :) Permutation puzzles are just naturally difficult, 3D or not, which is a also big part of why Rubik's cubes are inexplicably impossible when we are first exposed to them, and I don't think a 4D brain would have any advantage there. That is, unless 4D does lead to fundamentally more powerful computing ;)

Finally, on the viability of such universes, I have no idea but it seems that the laws of physics would certainly have to be different! Melinda has made the interesting point to me in the past that when it comes to physical dimensions, there seems to be nothing precluding multiple time dimensions entering the picture as well...

On Tue, Apr 1, 2008 at 9:47 AM, Mark Oram <markoram109@yahoo.co.uk> wrote:

Thanks guys for the comments/feedback: I found them all very interesting.

Certainly there are more things to keep track of as the dimensions go up, but I wonder if that necessarily implies that the moves HAVE to be more "complex", or just that there are longer sequences needed? Obviously one can solve, say, the 5-D cube by small steps; moving each class of cubie into place one by one with small sequences of moves. It just takes longer and needs more patience. Also, it may not be the quickest (or most elegant) means. If nothing else, I assume a 4-D (or 5-D etc) being could always resort to such an approach. Crucially, however, this does not exclude the possibility that a 4-D being could have a fundamentally superior insight, and might be able to see the most efficient sets of moves more intuitively than us every time.

I hadn't considerd the practicalities of how the 3-D or 4-D cubes would be assembled physically in 4-D space, and again it may be that the engineering hurdles are equivalent regardless of the actual dimension of the space in question. It reminds me of the initial confusion form many people (myself included) when they had seen the 3-D cube for the first time, wondering how it could even be made without everything falling apart once it was turned! Still, clearly the 3-D cube is physically possible,a nd I have no doubt that equivalent ones are possible - at least in principle.

Of course, it is easy to imagine that 4-D and 5-D hypercomputer screens could represent any of these cubes with ease, but again the issue of how to usefully represent extra dimensions on a screen with a fixed maximum number of dimensions would occur there as well. (I have always felt that a large part of the genius of Charlie, Melinda, Remigiusz and Roice was finding a way to do this for the 5-D cube on a 2-D screen, so thanks again guys!)

One final point: are the 4-D/5-D hypercomputers still just Turing machines at heart, or is it possible that there can be some fundamentally more powerful means of computing in the higher dimensions? (Or am I now wandering too far from the main group topic???)

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