Message #636

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Parity on MC m^n
Date: Sun, 01 Feb 2009 23:25:18 -0600

One last trimmed down reply for me as well :)

> > It is interesting to note you can have an odd number of corner pair
>
> swaps with an even number of edge pair swaps if the orientations of
>
> the corners also come into play, the most simple example being a
>
> cube that is solved except for two corners. In this case I guess
>
> the double odd parity is shared with the orientations instead of
>
> the edges.
>

> Are you refering to two corners that aren’t oriented correctly? If
>
not I’m not sure what you mean. On a 3^3, you cannot swap a single
>
pair of corners, regardless of orientation, unless an odd # of pairs
>
of edges are swapped as well.

I just learned something! And had to pull out my cube to sway myself :)
After thousands of solves of the 3^3 over the years, I never realized the
situation where there are two unsolved corners meant those corners had to be
in their correct positions, but incorrect orientations. That’s what I love
about the cube though, that it seems there are an endless stream of little
facts like this to learn which I can then integrate into my cube "world
view". And it is amazing what details one can gloss over. Sorry for the
incorrect claim (often I learn by making assertions then seeing the
counterexample, or having it pointed out to me).

> I think we can deduce the 4C case is impossible because corners are
>
> only permuted/oriented by outer twists (that is, by the same twist
>
> set as the 3^4), and this is not a possible configuration on the
>
> 3^4. The 3C case I’m unsure of, but I plan to experiment with it
>
> some.
>

> I’m not so sure. The same thing is possible on the 4^3, yet not on
>
the 3^3….
>

Ah yes, there is a hole in the reasoning! I think it can be filled though
with the extra observation you’ve made about the 4^4, which is that no
twists create a single odd-parity condition. And a single set of swapped
corners is odd, so it still looks impossible. What do you think?

In fact, this convinces me the 3C situation you uploaded is impossible as
well. The manifestation of that would have to be the result of one of two
cases:

(1) The two pieces are mirrored in place (impossible due to enantiomorphic
constraints).
(2) The pieces are exchanged and flipped, but a single swapped pair of edges
is a single odd parity condition, again impossible.

The scenario in 4x4x4x4_roice2.log differs from both of these in that it is
two pair of swaps, an even parity condition. So now it makes sense to me
why that is possible whereas the above two situations are not.

> Since I had
>
> already mentally recategorized those sets of two as single merged
>
> pieces, the puzzle model I was trying to use to my advantage left
>
> me stuck.
>

> Interesting. Essentially you went 4^4, to 4*3^3, to 3^4? This parity
>
case happened between 4*3^3 and 3^4? Is this close to accurate?
>

I’m not following what you mean by 4*3^3, but I was trying to apply the 4^3
pairing of 2C stickers approach/behavior to the 4^4 (the analogue was
pairing up pairs of 2C stickers). And yep, this was an interim step in the
effort to go from 4^4 -> 3^4).

Thanks for the lively discussion.
>

Absolutely. I really feel like I learned a lot the past few days reading
the discussion and struggling over these scenarios. I’m really happy to
have the concepts more crystallized in my mind, so thank you very much as
well! And thanks for your patience with my mistakes.

Good night,
Roice