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Odd parity Algorithms (specifically, single edge "flip")

qqwref

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Dan Cohen said he used ACube to search for all algs in <r,l,U2,B2,D2,F2> and that there were none in fewer than 25 quarter turns. Theoretically there might be a parity alg shorter than that, but I don't think we'll find one with the conventional approach.

EDIT: Lucas's file seems to include "nonpure" algorithms that adjust the last layer by U2, such as the following:
r2 F2 r U2 r U2 x U2 r U2 r' U2 r U2 r2 U2

EDIT: I'm dumb, that was a 25qtm one, but here's a 23:
r' D2 l B2 r' U2 r U2 l' B2 U2 D2 r U2 r'
 
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Stefan

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Nice alg, but that's not what the original post was asking for. ;)
Maybe it's not what he wanted, but I think it still is a valid answer. At least it definitely is an "odd parity algorithm". The OP was a bit ambiguous, making us guess what exactly he means and even what puzzle he's talking about. Also... do you think he counts R2 as two turns but an inner slice quarter turn as one turn? Seems counter-intuitive to me, so I think with his lower case letters he meant double layer turns rather than inner slice turns, and thus the algs he presented do affect other pieces as well, just like mine. Plus he explicitly mentioned "without messing up the centers" but said nothing about the other edges/corners (except "single edge flip" implies the other edges shouldn't be flipped).
 
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Christopher Mowla

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He's correct

Nice alg, but that's not what the original post was asking for. ;)
(Adding R2 U2 R' U2 could qualify, though.)

Yes Lucas, you are correct. That is not what I was looking for. As you probably have realized, I meant a pure edge "flip" which does not affect the cube at all (besides distorting-not discoloring-the centers and "flipping the one edge").
 

Christopher Mowla

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Nice alg, but that's not what the original post was asking for. ;)
Maybe it's not what he wanted, but I think it still is a valid answer. At least it definitely is an "odd parity algorithm". The OP was a bit ambiguous, making us guess what exactly he means and even what puzzle he's talking about. Also... do you think he counts R2 as two turns but an inner slice quarter turn as one turn? Seems counter-intuitive to me, so I think with his lower case letters he meant double layer turns rather than inner slice turns, and thus the algs he presented do affect other pieces as well, just like mine. Plus he explicitly mentioned "without messing up the centers" but said nothing about the other edges/corners (except "single edge flip" implies the other edges shouldn't be flipped).

I am sorry for the confusion. I meant a pure edge flip, much like the ones which you have on your website:

http://www.stefan-pochmann.de/spocc/other_stuff/4x4_5x5_algs/?section=FixOrientationParity

In fact, I am glad that you answered this post. I wanted to ask you about what you say to be Chris Hardwick's pure edge flip algorithm. Did he really come up with it. I noticed on your site that you ask "how was this found". Has he ever told you? I have always been curious about how, that algorithm was found. I have private messaged Chris and emailed him, but he didn't respond.

Lastly, do you think that there is a way to logically come up with a pure edge flip alg, which is around 25q?
 
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Christopher Mowla

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Dan Cohen said he used ACube to search for all algs in <r,l,U2,B2,D2,F2> and that there were none in fewer than 25 quarter turns. Theoretically there might be a parity alg shorter than that, but I don't think we'll find one with the conventional approach.

EDIT: Lucas's file seems to include "nonpure" algorithms that adjust the last layer by U2, such as the following:
r2 F2 r U2 r U2 x U2 r U2 r' U2 r U2 r2 U2

EDIT: I'm dumb, that was a 25qtm one, but here's a 23:
r' D2 l B2 r' U2 r U2 l' B2 U2 D2 r U2 r'

Thanks for the 23q alg. for double parity (even though I was seeking the pure one-edge flip). Please tell me, who found this algorithm and how? Did he/she use a computer, trial and error, or logical reasoning?
 
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cmhardw

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In fact, I am glad that you answered this post. I wanted to ask you about what you say to be Chris Hardwick's pure edge flip algorithm. Did he really come up with it. I noticed on your site that you ask "how was this found". Has he ever told you? I have always been curious about how, that algorithm was found. I have private messaged Chris and emailed him, but he didn't respond.

I did not come up with the alg. In 1998 I learned how to solve the 5x5x5 completely by intuition, except for the "parity" case. I went to www.excite.com to look up algorithms for solving this.

Doing so linked me to a simple text only HTML page that had several algs listed for the parity case. I don't remember how many were listed but it was less than 10. I chose the one that had the fewest number of turns (fewest in written notation). I have not been able to find the same page since.

--edit--
The only parity alg I did come up with on my own is the r2 U2 r2 U2 u2 r2 u2 for 4x4x4 PLL parity - just to be clear. I was trying to adapt the same concept as (R2 U2)*3 on 3x3x3.

Lastly, do you think that there is a way to logically come up with a pure edge flip alg, which is around 25q?

I see two ways to do this. Either we can invent algorithms that are short and are intuitive, or we can try to understand the algorithms already given.

Also, are you certain that every single short algorithm known is not understood by everyone? That's a pretty big assumption.

Chris
 
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Christopher Mowla

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Also, are you certain that every single short algorithm known is not understood by everyone? That's a pretty big assumption.
Chris

To be more specific, I was not so much referring to all algorithms, only odd parity (especially the one edge "flip").

I know that it is possible for someone to study an initially unobvious algorithm long enough until he/she believes that he/she has a good understanding of the algorithm; however, this involves a timely study, and, is that understanding a real understanding?

I might be missing it, but think of math. Now, I have not taken advanced calculus yet, but tell me if I am wrong (and I am serious, because I know you majored in math). Take for example the formal definition of the limit. I honestly do not know any students who fully understand what a limit is. They usually just think of "plug in the number which x approaches" into f(x): if none of the indeterminate forms is not obtained, then the limit is found...otherwise do L'hospital's rule, algebra, trig identities, etc..to evaluate it...if it exists at all.
From an advanced calculus student's perspective, one who has understood all of the proofs and logic behind the limit, would define a limit much differently than a freshman year calculus student.

Similarly, studying a brief "one edge correction" algorithm long and hard might help develop a light undertanding of what is happening to a cube, but not an understanding that is good enough to be confident in.

If that confidence was great, then that person should be able to write 100s of algorithms for the single "edge flip". Obviously all could not be the same length in moves, but, many algorithms should intuitively come out of that person's head (many of them very brief). Furthermore, that person should be able to prove what is the briefest possible algorithm for the one edge "flip" (without tools such as a computer).

If I understand one integration problem, does that make me an expert at integration?
 
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qqwref

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Thanks for the 23q alg. for double parity (even though I was seeking the pure one-edge flip). Please tell me, who found this algorithm and how? Did he/she use a computer, trial and error, or logical reasoning?

A computer of course. Did you look at his file? There are hundreds of algorithms there... nobody would waste their time looking for algs like this by hand when computers can do it much faster and guarantee optimality (assuming you entered in the right assumptions).

If that confidence was great, then that person should be able to write 100s of algorithms for the single "edge flip". Obviously all could not be the same length in moves, but, many algorithms should intuitively come out of that person's head (many of them very brief). Furthermore, that person should be able to prove what is the briefest possible algorithm for the one edge "flip" (without tools such as a computer).

I don't think the things you say are possible except for certain easy cases, unless you are willing to do a non-humanly-possible amount of work. Cubing is not as easy as mathematics in a way - there are no simple theorems that guarantee a solution of a given length (or any algorithms that always generate a reasonably short solution without brute force trial and error). A few things such as parity considerations and certain subgroups are guaranteed but beyond that you are really on your own. No matter how much you know of cube theory, I don't think there is any known way at all to prove optimality without a certain measure of brute force (and the less brute force you want, the more algorithms you need to store in the form of pruning tables).

Here is one intuitive way to deal with parity, though: Do a z rotation and use only R2, u slice, and d slice moves. Then the middle two layers can be thought of as two layers which each have four corners and four "edges". Flipping a 4x4 edge in this layer is a matter of swapping the two corners lying directly above one another. (However, since an even permutation of the middle layers will cause the rest of the cube to be an R2 off, in practice you also need to swap one or three pairs of identical 'edges'.) So algs can be generated in this way. I could personally generate a lot of different move sequences for this but there would be no point since a computer could generate more and more efficient sequences (and I would, again, have no way to prove optimality without an impossible amount of work).
 

Stefan

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From an advanced calculus student's perspective, one who has understood all of the proofs and logic behind the limit, would define a limit much differently than a freshman year calculus student.
What?! Why?

If that confidence was great, then that person should be able to write 100s of algorithms for the single "edge flip".
Well, many of us easily could.

Furthermore, that person should be able to prove what is the briefest possible algorithm for the one edge "flip" (without tools such as a computer).
No. I'm afraid some things in life simply require brute force, and some too much to do by a human.
 

Christopher Mowla

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A computer of course. Did you look at his file? There are hundreds of algorithms there... nobody would waste their time looking for algs like this by hand when computers can do it much faster and guarantee optimality (assuming you entered in the right assumptions).

Which site? Can you show me. I would like to add it to my bookmarks!:eek:
 

qqwref

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It isn't a site, it's a program, called ACube (the version I use is JACube, same thing though). This program only does 3x3 but you can add any move constraints you want, so in this case you can simulate the 4x4 middle slices by only allowing R, L, U2, D2, F2, B2 and ignoring the middle slice of the 3x3.

There is a more general program out there called ksolve which can be used to solve pretty much anything (you could even optimally solve 4x4 positions if you had enough computing power/time) but it takes more work to set up and I don't think you would find any good solutions that are not in <r,l,U2,F2,B2,D2>.
 

Christopher Mowla

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I don't think you would find any good solutions that are not in <r,l,U2,F2,B2,D2>.
Well, here's one of length 28 that I just made up:
Dw' L' U F (r U2 r U2 r U2 r U2 r) F' U' L d L' d' L D L' d L
Ok, so it's a bit longer. But I'm only human.

I think that's the record without the use of a computer. I am impressed. ;).

Most other algorithms one-edge flip algorithms (not computer generated) that I have seen are not in the sub 30 range. But, as always, I am suprised of how little I know (there could be more out there besides Stefan's).:fp
 
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