Message #633

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Parity on MC m^n
Date: Sun, 01 Feb 2009 17:23:57 -0600

A little more roice spam, this time inline :) (and a little of it from the
earlier post).

> I’m still retaining my nontraditional definition of parity errors
>
essentially as odd parity (and in my n^d solution double odd as
>
well). Ignoring this definition, check out the "single flipped"
>
parity error on a 4^3. It will take an odd number of inner slice
>
quarter twists to solve this.
>

> On a 3^3, a single swapped pair has odd parity. This cannot happen
>
due to the even (aka double odd) parity of a single quarter twist on
>
a 3^3. However, using reduction, the "single swapped edge pair" (with
>
correct orientation) parity err on a 4^3 can seem to occur. This will
>
alway take an even number of quarter twists to solve.
>

Furthermore, we can say that some of those twists will have to be of the
inner slice. In fact, both inner and outer twists will be required, an even
number of both. I can provide my reasoning for this conclusion if desired.

You can see a similar phenomenon with a 3^3. If you have an even
>
number of corner pair-swaps to perform, you will have an even number
>
of edge pair-swaps also. It took an even number of quarter twists to
>
generate this position, and it will take an even number of quarter
>
twists to solve. The same goes for odd. An odd # of Corner pair-swaps
>
will always be accompanied by an odd # of Edge pair-swaps and an odd
>

of twists. This is my double odd parity.

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.

I’ve also uploaded four log files to the same folder. Two of these
>
are cases that I’d consider parity errors. I cannot see how these
>
could be generated on a 3^4 or a 4^4. One involves a pair of 4C
>
pieces swapped. The other, a pair of 3C swapped so they appear to
>
have 2 stickers flipped. I’d be shocked to see a solution for either
>
of these.

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.

The final log (Faked2CHalf) I believe is the other case you refered
>
to in 4_4Roice1.log. I would agree/argue that this isn’t a parity
>
error because you hadn’t reached a 3^4 state yet. If this is a parity
>
error, then any position before you reach a 3^4 is because they’re
>
all impossible on a 3^4. The spirit of parity errors are situations
>
you wouldn’t know were a problem until you got to the end, and "Wait
>
a minute, I can’t solve this position!" (like Roice2)
>

That is exactly what happened to me though ;) But using a mental parity
model of a 4^3, not a 3^4. Considering two adjacent, identically colored
and oriented pieces as behaving like a single edge piece on a 4^3, I reached
this state and did not know how to solve it (because the solution
necessitated breaking these already solved, artificially joined pieces back
in two). 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.

Overall, I’d say the different usages of the word parity is still clouding
the discussion here. I am conceding that my use of the phrase "parity
problem" is perhaps too general. After all, a problem is simply a
configuration you don’t know how to solve using a tool set of sequences.
And parity just means even or odd, which can be applied in a number of
ways. I promise it’s clear in my mind though :) I like the reductionist
approach of analyzing parities and deriving what that says about the
solution. I like the concept of "parity problem" defined in the context of
mismatches between the different parity characteristics of various puzzles,
that they are mental surprises (perceived impossibilities) when one tries to
use solution methods across multiple puzzles.

All the best,
Roice