Message #866

From: David Vanderschel <DvdS@Austin.RR.com>
Subject: Re: [MC4D] Fractal cubes
Date: Thu, 11 Mar 2010 22:03:05 -0600

As far as I can tell, Melinda is only talking about stickers
on cubies that are in external slices. Yes, the previously
hidden stickers on those cubies, though now visible, do
remain ‘inside’ (i.e., never facing outwards); and their
orientation is determined by that of the stickers which do
face outwards.

What is interesting about the order-9-with-Menger-holes is
that you can now see cubies that are not in _any_ external
slice, making it possible to see what is happening to some
permutations and orientations of cubies down in the interior
of the puzzle. The stickers on these interior cubies are
also stickers that I would consider to be "inside stickers".
What I was questioning is whether these deeply interior
stickers (indeed, the interior cubies themselves) will
necessarily be correct when the outer stickers are
correct. That is still not obvious to me, and I am inclined
to doubt that it holds. For me, that makes this variation
interesting in a new sense that we have not previously
concerned ourselves with.

Regards,
David V.

—– Original Message —–
From: "Melinda Green" <melinda@superliminal.com>
To: <4D_Cubing@yahoogroups.com>
Sent: Thursday, March 11, 2010 8:43 PM
Subject: Re: [MC4D] Fractal cubes

David,

Indeed I failed. I do that a lot. ;-)

I didn’t have a proof of my guess, but that’s what my "I think"
equivocation meant: It was just my guess. It seems entirely
clear to me
that inner stickers cannot mix with outer stickers given the
twists
we’ve talked about so far. I’ll go even further and say that I’m
pretty
sure that inner stickers can never be adjacent to any outer
stickers
that are not on its own cubie. As to a proof? Well, let’s see.
Level-1
cubies with inner stickers will always have two inner and two
outer
stickers arranged in a ring: outer, outer, inner, inner. Given a
uniform
pristine coloring like we’ve been considering, every level-1
edge cubie
with outer colors O1,O2 will have inner colors I1,I2 that are
identical
to every other cubie with the same pair of outer colors. Since
all edge
cubies with the same pair of outer stickers are identical and
there is
only one correct orientation for any edge piece, it seems to
follow that
once all of the outer stickers are placed correctly, all inner
stickers
will also have to also be correct.

-Melinda

David Vanderschel wrote:
> Melinda, relative to my previous post in this thread (which was
> primarily a response to your preceding post), you failed to
> answer a direct question for which I was eager to see your
> answer. So I will repeat the question.
>
> You had written:
>
>>> Regarding the coloring of inside stickers, I didn’t
>>> completely follow what you [Chris] proposed, but it
>>> doesn’t seem like an important issue because I don’t think
>>> that the inside stickers can contribute to the puzzle. I
>>> think they should automatically be correct when the rest
>>> of the puzzle is complete.
>>>
>
> and I had asked:
>
>> This is not obvious to me. Do you have proof, reasoning,
>> reference? I thought this was the interesting thing about
>> the order-9 variation.
>>
>
> Regards,
> David V.