# Message #631

From: rev_16_4 <rev_16_4@yahoo.com>

Subject: Re: Parity on MC m^n

Date: Sun, 01 Feb 2009 22:02:43 -0000

Thanks, Roice. Impressive you found CD’s from 2000! I’ll take a look

at these, and see what I think. (Actually, I’m loading them right

now…)

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.

The third is the 2C case I believe you’re refering to in

4_4Roice2.log. I’ve included my solution to this case in the Faked2C

file. I solved this using the same technique I’d use to solve a pair

of fliped 2C’s on the 3^4. The only difference is I had a 4th layer

on the U(?) axis to work with instead of 3, which allowed for a quick

conjugate. I use this same sort of technique on the 4^3 for the

single edge pair-swap parity.

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)

However, I consider this one slightly more difficult (at least

initially) than the last. It’s also a key to understanding why any

piece with at least 1 identical piece has no parity to me, so I’ll

explain. This is still solvable with two pair-swaps (before or after

you’ve reduced it to a 3^4). I used an algorithm that swaps a set of

three pieces. This is the same algorith I use for all (d-2)C on all

n>3. I needed a conjugate to set it up (first and last six twists),

but the rest was the algorith (I was in a hurry and used at least 8

more twists than my algorith needed for normal solving). Swap an

identical pair, swap the pair with the problem.

I’ve got a major inspection at work all week, so it might be a few

days before any additional responses. Who am I kidding, I’m gonna

need the down time, so I’m sure I’ll be on here!

-Levi

— In 4D_Cubing@yahoogroups.com, Roice Nelson <roice3@…> wrote:

>

> I dug up old cd backups I had and found my log files from April

2000! I

> save solutions along the way out of paranoia, and luckily I had

files at the

> problem points I saw. I just uploaded 2 log files to the a new

folder in

> the files area of the

> group<http://games.groups.yahoo.com/group/4D_Cubing/files/parity%

20error%20logs/>showing

> the parity error situations I encountered on my 4^4 solution.

> quick aside on a program technical issue: These weren’t loading

with the

> current java version (back then the 4^4 was only available in the

linux

> version). I altered the header line to look more current, and they

seems to

> load fine now. However, since I don’t know the format, I’m unsure

of what

> ramifications the editing might have. Here is an example.

> old: MagicCube4D 1 0 857

> new: MagicCube4D 2 2 844 4

>

> Anyway, I’ll do my best now to reconstruct what looks like was

going on - I

> can’t remember last week, much less the details of a decade ago :)

>

> In the first file, I was trying to solve 2C pieces. When matching

up sets

> of four, I found the very final set (orange/yellow) had two in one

> orientation and two in another, as in the puzzle state of the log

file.

> This is not a parity problem in the context of a reduction to a

3^4 since

> the reduction hasn’t even happened yet. Rather it is a parity

problem in

> the context of a 4^3! Because when pairing up the 2Cs on a 4^3,

you will

> never encounter the situation where all are matched up to form

single

> 3^3-like edges except that the last two are flipped in relation to

each

> other. If placed on the same edge, the final two will always be in

the same

> orientation. I’m glad I pulled this out again, because this feels

more

> subtle that what I had written in my last email. In essence, the

problem is

> the same however. I saw an "impossible" configuration when using a

simpler

> mental model of parities on a more complicated puzzle. I hope this

made

> sense because it is a really interesting effect to me.

>

> In the second file, the situation is a parity problem in the

context of a

> 3^4 reduction, again relative to 2C pieces. The final (pink/red)

2C was

> flipped as a whole, so I believe this is the case you wanted to see

an

> example of. Unfortunately, the log file doesn’t easily show how to

generate

> this position from a pristine state. Indeed, the possibility of it

may rely

> on the permutations/orientations of 3C pieces, so it may not be

possible

> without scrambled 3C pieces?

>

> Let me know what you think!

>

> Roice

>

> On Sun, Feb 1, 2009 at 1:20 AM, rev_16_4 <rev_16_4@…> wrote:

>

> > Roice, you bring up a very good point. I wasn’t sure there were

> > positions on a 4^d that, using a reduction method, would generate

> > impossible positions on a 3^d (I’m going to switch to your

notation,

> > it’s been around longer). I thought it might be possible, seeing

that

> > was the gereral consensus. But I hadn’t experienced one myself.

Can

> > someone email me a 4^4 log file with such a position?

> >

> > 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.

> >

> > 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.

> >

> > Now with the case of n=4, d>3, this rule above is not the case.

you

> > can generate any position with an even number of twists, or an odd

> > number of twist. I’d write out all the actual pair-swaps from a

> > single quarter twist, but I’m too lazy right now. In a nutshell

there

> > are 6 corner pairs, 18 edge pairs, 18 face pairs, and 6 center

pairs

> > swapped during an outer slice rotation. An inner slice rotation

has 6

> > edge pairs, 18 face pairs, and 18 center pairs swapped. As you can

> > see, all are even, hence even parity (the faces and centers are

> > irrelevent). You can never generate an odd parity position, hence

my

> > belief there are no parity errors for this puzzle.

> >

> > With the reduction method, sometimes the pairs are swapped in

such a

> > manner that a simple even parity position for the caging method I

> > use, if attempted to be solved using reduction, would result in an

> > unsolvable 3^4 position. (I’m trying to think of a position where

> > this could occur… SOMEONE PLEASE SEND ME AN EXAMPLE LOG FILE!)

If

> > someone can show me a log with a "single 3C w/ two stickers

flipped"

> > 4^4 parity position, and how to generate it, I’d be grateful (and

> > completely shocked!). Other than that, I can’t think of a 4^4

parity

> > that I think would be unsolvable with (non-caging) techniques

similar

> > to a 3^4.

> >

> > -Levi

> >

> > _._,___

> >

>