Message #4102

From: Jay Berkenbilt <ejb@ql.org>
Subject: Re: [MC4D] 2x2x2x2: List of useful algorithms (please add yours)
Date: Thu, 02 Aug 2018 10:05:46 -0400

I’m enjoying all this discussion, though I don’t have the mindspace for all
of the detail at this point. In another life, maybe I’d be diving in with
the proofs and stuff. While I appreciate the friendly spirit with which the
comment about my hacky scramble was made (and seriously, I take it as a
friendly comment), I thought it was pointing out what I was trying to
achieve.

Though I have a (somewhat rusty) math degree, my goal is not mathematical
rigor. My goal is intuitive understanding. I definitely appreciate the
concept of a full scramble and the fact that it takes a certain number of
moves to ensure that the probability of the minimum path to solution is
sufficiently long and that the twists are sufficiently random, but when it
comes to presenting an approach for how to solve the puzzle and how to
understand why that solution works from an intuitive standpoint, a short,
simple, ad hoc scramble totally does the job, and just having the puzzle
"look" scrambled is sufficient to demonstrate the approach. I like how your
work can probably demonstrate mathematically how certain steps of a
solution are sufficient to cover all possible scrambles. My approach is
more a demonstration of how a solution can be reduced to a very small
number of intuitively clear steps. While I am a strong believer in
mathematical rigor, ultimately I feel that the best test of true
understanding is to be able to explain things in simple terms that a
non-mathematition can understand. I’m not defending myself…just
explaining what my goals are and where I’m coming from. :-) I’m quite
certain that if I were to put this puzzle down for years and pick it up
again, I would still be able to solve it. All I need is the general
strategy of using gyro moves to get certain stickers on the outer corners,
the mod 4 = 0 technique, and the super-general isolate/manipulate/reverse
approach. The rest of the algorithms can be generated from those basic
principles. The combination of this intuitive approach and the more
rigorous approach along with optimizing algorithms, etc., I think
contributes to the broadest understanding of this amazingly elegant puzzle.

–Jay

On Thu, Aug 2, 2018 at 9:26 AM Andrew Farkas ajfarkas12@gmail.com
[4D_Cubing] <4D_Cubing@yahoogroups.com> wrote:

>
>
> P.P.S.
>
>
> (And the unfolded view is equivalent to removing the top half from our
>> standard horizontal orientation and placing it to the left of the bottom
>> half with a *z2*.)
>
>
> This is incorrect; an *Rz Lz’* is required before the *z2*.
>
>
> On Thu, Aug 2, 2018 at 1:17 AM Andrew Farkas <ajfarkas12@gmail.com> wrote:
>
>> Gah, I mistyped
>>
>> Thus by the time I’m encountering the apparent corner twist parity I
>>> don’t need to worry about the *z2* rotation since the *I*/*O* sticker
>>> shouldn’t be on the frame anyway.
>>
>>
>> That should read *x2*, not *z2*.
>>
>> On Thu, Aug 2, 2018 at 1:14 AM Andrew Farkas <ajfarkas12@gmail.com>
>> wrote:
>>
>>> Oh goodness.
>>>
>>> You’ve brought up a lot of things to unravel here! I’ll go in order.
>>>
>>> But referring to them as "clockwise" and "counterclockwise" relative to
>>>> I/O didn’t help me. Aren’t we going to need to be able to recognize 3
>>>> distinct cases? I know I needed 3 cases for my sequences for "RUFI by
>>>> x2/y2/z2 while keeping the rest of In/Out fully solved".
>>>
>>>
>>> I think this is a result of a difference in our solving approaches. I
>>> find it easier (and faster) to first consolidate all stickers of the first
>>> color pair in any position except the frame, and only then form faces from
>>> them. Thus by the time I’m encountering the apparent corner twist parity I
>>> don’t need to worry about the *z2* rotation since the *I*/*O* sticker
>>> shouldn’t be on the frame anyway.
>>>
>>> Instead of performing a net 180 degree flip on the piece, you give it a
>>>> net 120 degree twist on a different axis, while exchanging some twists with
>>>> other pieces. So, for instance, applying either sequence 2 times to the
>>>> solved state does not lead to an aligned state like mine does. This
>>>> baffled me for a few minutes there. It takes applying it 3 times..
>>>
>>>
>>> Again taking a speedsolver’s approach here, I focused solely on
>>> achieving the desired effect at the given stage, without regard for the
>>> true nature of the algorithm. I lazily called them "double twist parity
>>> algorithms" simply because they solved an issue which others have called
>>> "double twist parity." I appreciate your analysis of what these algorithms
>>> actually accomplish, though I think I need some more hands-on testing of my
>>> own to fully grasp what you’re saying.
>>>
>>> I’ll call your two algs TTA and TTB (for Triple Twist A and B).
>>>
>>>
>>> Sounds great, if not a bit arbitrary. Generally when mirror cases of an
>>> algorithm are followed by "a" or "b," no one remembers which is which (e.g.
>>> A, G, J, N, R, and U PLL algorithms
>>> <https://www.speedsolving.com/wiki/index.php/PLL>), so if there’s a
>>> better way to distinguish the two, I’d be more satisfied. That said, the
>>> only solution I have (CW/CCW) is based on a very limited view of their
>>> effect.
>>>
>>> …
>>>> and note the colors of the piece that sits at RUFO (Red R, White U,
>>>> Green F, Pink O).
>>>
>>>
>>> Interesting that we both chose this as the "default" orientation. I
>>> suppose the red/white/green stems from the standard WCA scramble
>>> orientation, and pink just falls into place (unless one of our puzzles were
>>> rotated through a real fourth dimension!). Even pentaquark394’s scrambler
>>> (and thus my own) use red/orange on the frame with white on the "top"
>>> (really *O*) and green in front, making it easy to reach from the
>>> WCA-inspired horizontal orientation. (And the unfolded view is equivalent
>>> to removing the top half from our standard horizontal orientation and
>>> placing it to the left of the bottom half with a *z2*.) Anyway, back to
>>> the cube theory!
>>>
>>> There are two choices of CW and CCW in this alg, in step 1 and step 2,
>>>> and I think we’ll find that we need to use 3 of those 4 combinations in our
>>>> 3 cases. At least, that’s what ended up happening when I created my
>>>> similar sequences. It looks like the 3rd case can be handled by applying
>>>> the inverse of the 1st alg.
>>>
>>>
>>> You know, as I was developing these I noticed that the *I*/*O* sticker
>>> landed on the frame (*R*/*L* face) if I rotated *I* in the opposite
>>> direction or applied the wrong algorithm for either case; I dismissed this
>>> at the time as an undesired result, but in retrospect it could certainly be
>>> useful. In my solution video, I accidentally left an *I*/*O* sticker on
>>> the frame, and had to spend quite some time resolving it when clearly a
>>> single short algorithm would have sufficed.
>>>
>>> The next step is to see how these conjugates with *I[…]* can be used
>>> to efficiently orient several pieces at once! I’m not sure whether this
>>> would be any faster than simple 3D moves, but perhaps even some existing 3D
>>> algorithms could take advantage of the extra dimension that the 2^4 has to
>>> offer.
>>>
>>> *TTA*: Twist *OFRU* counterclockwise (relative to its Front
>>>> tetrahedron):
>>>> *[ R[ U’ R’ U2 ]: Iy Ix ]**TTB*: Twist *OBRU* clockwise (relative to
>>>> its Back tetrahedron): *[ R[ U R U2 ]: Iy’ Ix’ ]*
>>>
>>>
>>> In my 3D mindset at this stage, I prefer to think of these as clockwise
>>> and counterclockwise respectively around the *R* hypersticker, hence my
>>> original naming.
>>>
>>> Recognition: put misaligned piece on *OFRU*. If the I/O color is on
>>>> the Front face, perform *Rx* and then *TTB*. Otherwise, the piece can
>>>> be aligned via a twist of the Front tetrahedron. Apply *TTA* (if a
>>>> counterclockwise twist is needed, i.e. the I/O color is on U) or *TTA’* (if
>>>> a clockwise twist is needed, i.e. the I/O color is on the R corner).
>>>
>>>
>>> I prefer to combine this into one, slightly more complicated step: Hold
>>> left and right subcubes such that all oriented pieces are on the *I*
>>> and *O* faces, and that the misaligned piece is in *ROFU* or *ROBU*
>>> with the *I*/*O* sticker facing *U*. If the piece is in *ROFU*, apply
>>> TTA; if it is in *ROBU*, apply TTB.
>>>
>>> The same result is achieved either way.
>>>
>>> Thanks for being such a fun co-conspirator.
>>>
>>>
>>> Right back at ya. 🙃
>>>
>>> Theorem: Every combination of three corner twists is equal to one of
>>>> the eight possible single corner twists (clockwise and anticlockwise around
>>>> any of the 4 colors) or the identity. Every combination of two corner
>>>> twists is equal to one of the three monoflips or the identity. (OK, OK,
>>>> this is still just a hypothesis until I enumerate the damn things or
>>>> otherwise prove it more thoroughly than I have done in my head so far.)
>>>
>>>
>>> Well, so much for sleeping tonight. 😛 I have a strong feeling that both
>>> of those are true, but of course a proof is necessary. Enumeration is
>>> pretty trivial at this scale – there’s probably only a dozen or so cases
>>> after removing mirrors and the like – but of course a rational argument is
>>> much more appealing. I’ll give it a shot.
>>>
>>> Random idea: at the beginning of a solve, if we notice that there’s a
>>>> color pair with exactly 1 piece on the corners, we should just probably
>>>> just go ahead and align the other 15 pieces of that color pair, then apply
>>>> one of these algs. Now that it’s so easy to fix this kind of misalignment,
>>>> futzing around with additional gyros doesn’t seem worth it if we’re only 1
>>>> piece off from having a color pair off of the corners.
>>>
>>>
>>> Certainly! It might even be worth it for two, if we can account for
>>> double tw– er, corner twist (?) parity along the way. I would still like
>>> to develop a general intuitive strategy and/or algorithm set for this
>>> stage; I think it’s the least consistent part of Fourtega and thus the one
>>> that could use the most improvement.
>>>
>>> Thank you very much for continued analysis and discussion! It’s fun to
>>> be exploring new territory.
>>>
>>> - Andy
>>>
>>> P.S. Counterexample to the first hypothesis: (execute on *ROFU*) CW
>>> around *R *+ CW around *U* + CW around *R* results in *x2*. The second
>>> hypothesis contradicts corner twist parity: two corner twists in the same
>>> direction violates corner twist parity, while monoflips and the identity do
>>> not.
>>>
>>> On Wed, Aug 1, 2018 at 9:57 PM Marc Ringuette ringuette@solarmirror.com
>>> [4D_Cubing] <4D_Cubing@yahoogroups.com> wrote:
>>>
>>>>
>>>>
>>>> Hey, Andy,
>>>>
>>>> I love your maybe-the-shortest-possible monoflip aligners. But
>>>> referring to them as "clockwise" and "counterclockwise" relative to I/O
>>>> didn’t help me. Aren’t we going to need to be able to recognize 3 distinct
>>>> cases? I know I needed 3 cases for my sequences for "RUFI by x2/y2/z2
>>>> while keeping the rest of In/Out fully solved".
>>>>
>>>> Your algs are a bit more confusing for me to think about than mine
>>>> were, because they do three distinct corner twists on the misaligned piece,
>>>> whereas mine do two. Instead of performing a net 180 degree flip on the
>>>> piece, you give it a net 120 degree twist on a different axis, while
>>>> exchanging some twists with other pieces. So, for instance, applying
>>>> either sequence 2 times to the solved state does not lead to an aligned
>>>> state like mine does. This baffled me for a few minutes there. It takes
>>>> applying it 3 times.
>>>>
>>>> I’ll call your two algs TTA and TTB (for Triple Twist A and B).
>>>>
>>>> In tracing through your first alg, TTA, I found it useful to start from
>>>> my standard solved state and note the colors of the piece that sits at RUFO
>>>> (Red R, White U, Green F, Pink O).
>>>>
>>>> Step 1. R [ U’ R’ U2 ] – twists RUFO CCW around the Right
>>>> center (the red corner of the piece) and then places the piece on RUBI with
>>>> R[ U2 ]
>>>> Step 2. Iy Ix – twists RUBI CW around the In
>>>> center (the anti-green corner of the piece) and does not permute it
>>>> Step 3. R [ U2 R U ] – places the RUBI piece on RUFO with R [
>>>> U2 ] and then twists it CW around the Right center (the pink corner of the
>>>> piece)
>>>>
>>>> There are two choices of CW and CCW in this alg, in step 1 and step 2,
>>>> and I think we’ll find that we need to use 3 of those 4 combinations in our
>>>> 3 cases. At least, that’s what ended up happening when I created my
>>>> similar sequences. It looks like the 3rd case can be handled by applying
>>>> the inverse of the 1st alg.
>>>>
>>>> Note that in TTA three different colors on the piece get twists applied
>>>> (Red CCW, Green CCW, Pink CW). The net result is Green CCW (!), the color
>>>> that was originally Front and still remains Front.
>>>>
>>>> Tracing similarly, TTB twists the Back tetrahedron of OBRU (Blue in
>>>> this case) CW.
>>>>
>>>> So I guess here’s how I’d have described your algs and the recognition:
>>>>
>>>> * TTA*: Twist *OFRU* counterclockwise (relative to its Front
>>>> tetrahedron): *[ R[ U’ R’ U2 ]: Iy Ix ]*
>>>> *TTB*: Twist *OBRU* clockwise (relative to its Back tetrahedron): *[
>>>> R[ U R U2 ]: Iy’ Ix’ ]*
>>>>
>>>> Recognition: put misaligned piece on *OFRU*. If the I/O color is on
>>>> the Front face, perform *Rx* and then *TTB*. Otherwise, the piece can
>>>> be aligned via a twist of the Front tetrahedron. Apply *TTA* (if a
>>>> counterclockwise twist is needed, i.e. the I/O color is on U) or *TTA’*
>>>> (if a clockwise twist is needed, i.e. the I/O color is on the R
>>>> corner).
>>>>
>>>> What do you think?
>>>>
>>>> (The three cases above could also be recognized as the ones where a y2
>>>> flip, z2 flip, and x2 flip are needed, respectively; although we do not
>>>> actually perform that flip, so it would seem a bit odd to do recognition by
>>>> figuring out what 180 degree flip we "could" use, and then not using it.
>>>> I might do it that way anyway.)
>>>>
>>>>
>>>>
>>>> I absolutely love this part of the puzzle-figuring-out process, because
>>>> I’m starting to get the hang of the 12 orientations, and how they divide up
>>>> into 4’s and 3’s, and how corner twists can combine into monoflips, etc.
>>>> Your triple twister algorithms are reminding me that I don’t fully grok it
>>>> yet, but I feel like I’m making good progress. Thanks for being such a fun
>>>> co-conspirator.
>>>>
>>>>
>>>> Theorem: Every combination of three corner twists is equal to one of
>>>> the eight possible single corner twists (clockwise and anticlockwise around
>>>> any of the 4 colors) or the identity. Every combination of two corner
>>>> twists is equal to one of the three monoflips or the identity. (OK, OK,
>>>> this is still just a hypothesis until I enumerate the damn things or
>>>> otherwise prove it more thoroughly than I have done in my head so far.)
>>>>
>>>>
>>>> Random idea: at the beginning of a solve, if we notice that there’s a
>>>> color pair with exactly 1 piece on the corners, we should just probably
>>>> just go ahead and align the other 15 pieces of that color pair, then apply
>>>> one of these algs. Now that it’s so easy to fix this kind of misalignment,
>>>> futzing around with additional gyros doesn’t seem worth it if we’re only 1
>>>> piece off from having a color pair off of the corners.
>>>>
>>>>
>>>> Cheers
>>>> Marc
>>>>
>>>>
>>>> On 7/31/2018 9:59 PM, Andy F legomany3448@gmail.com [4D_Cubing] wrote:
>>>>
>>>> I’ll include my "double twist" algorithms here. The rest are trivial or
>>>> simply 4D use of 3D methods. These algorithms preserve I/O orientation for
>>>> the other seven pieces, but do not preserve orientation on other axes or
>>>> permutation at all.
>>>>
>>>> Twist *OFRU* clockwise (relative to I/O): *[ R[ U’ R’ U2 ]: Iy Ix ]*
>>>> Twist *OBRU* counterclockwise (relative to I/O): *[ R[ U R U2 ]: Iy’
>>>> Ix’ ]*
>>>>
>>>> The *Iy Ix* and *Iy’ Ix’* moves can be executed by moving the right
>>>> and left endcaps around the inner face, as can be seen in my solution
>>>> video <https://youtu.be/Fd9NUaO5AYo?t=5m58s>.
>>>>
>>>>
>>>> On 7/28/2018 2:46 PM, Marc Ringuette ringuette@solarmirror.com
>>>> [4D_Cubing] wrote:
>>>>
>>>> Monoflip, solving In+Out faces only: (12 moves physical using ROIL
>>>> Zero, 3 cases)
>>>> RUFI by x2: Rzy I [ U F U’ F U F2 U’ ] Iy Lx2 Iy’ Rz’
>>>> RUFI by y2: Ry’z’ I [ U F U’ F U F2 U’ ] Iy Lx2 Iy’ Ry
>>>> RUFI by z2: Rzy Iy Lx2 Iy’ I [ U F2 U’ F’ U F’ U’ ] Rz’
>>>> (those are sideways Sune, Sune, and Antisune, inside the brackets)
>>>>
>>>>
>>>
>>> –
>>>
>>> "Machines take me by surprise with great frequency." - Alan Turing
>>>
>>
>>
>> –
>>
>> "Machines take me by surprise with great frequency." - Alan Turing
>>
>
>
> –
>
> "Machines take me by surprise with great frequency." - Alan Turing
>
>