Message #4097

From: Marc Ringuette <>
Subject: Re: [MC4D] 2x2x2x2: List of useful algorithms (please add yours)
Date: Wed, 01 Aug 2018 18:48:45 -0700

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.


On 7/31/2018 9:59 PM, Andy F [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 <>.

On 7/28/2018 2:46 PM, Marc Ringuette
[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)