# Message #4099

From: Andrew Farkas <ajfarkas12@gmail.com>

Subject: Re: [MC4D] 2x2x2x2: List of useful algorithms (please add yours)

Date: Thu, 02 Aug 2018 01:31:44 -0400

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