Message #4066

From: Andy F <>
Subject: Re: [MC4D] 2x2x2x2: a few algs and an example solve
Date: Sat, 14 Jul 2018 15:48:08 -0400

Well that was fun!

NCM = Net (modulus) Clockwise Movement* before* executing the algorithm.
After executing the correct algorithm, NCM will be zero.

While HTM <> is the standard
for (l>3)^n, I think ATM <> is
more appropriate for 2^n, so that’s what I’ve used here.

Hopefully the formatting holds up.

PBL Case NCM Algorithm Move count Explanation

0 solved 0
2 (R2 F2 R2 U’)2 8

ADJ swap
1 (U) (R’ L’) U2 R U R’ U2 L U’ R 10 inverse Ja permutation
3 R’ U L’ U2 R U’ R’ U2 (R L) (U’) 10 Ja permutation

OPP swap
1 y [alg below without AUF] (U’) 13
3 R’ U L’ U2 R U’ x’ U L’ U2 R U’ L (U) 13 Nb permutation

double OPP swap
0 (U’) R2 F2 R2 (U’) 5 standard Ortega alg cancels completely with skip-2
2 R2 F2 R2 3 standard Ortega alg

double ADJ swap
(BL ↔ BR)
0 R2 U’ F2 U2 R2 D’ R2 7 standard Ortega alg
2 y R2 F2 U’ F2 U2 R2 D’ R2 8 R2 F2 R2 rotates case while solving parity;
cancels nicely with alg above

hybrid swap
0 (D) [alg below] (D) 9
2 L’ U R’ D2 R U’ L 7 standard Ortega alg (half Aa permutation)

This algset has an average movecount of 7.75 including AUF; pure Ortega,
taking the shortest algorithm from each pair above plus a 50% chance of
U2P, yields 6.83+(8/2)=10.83. PBL-P saves an average of ~3 moves per solve.

So that covers U2P, but Marc also mentioned corner twists between faces
that manifest during Ortega. I would think this is a significant issue,
happening 2/3 of the time, so here’s a crazy idea to remedy that:

Use PBL-P only for solving the second half. If there’s a +2 NCM during PBL
on the first half, carry it over when solving the second half. For the
first half, we’d need variants of OCLL and PBL that do the following:
OCLL-[something] - Orient corners of last layer, but "solve" (into some
consistent position) a badly oriented corner in the process. These
algorithms would be strictly 3D, and would have to be executed on both
halves. They might require some lookahead into PBL. I don’t know how large
they would have to be, but ignoring permutation leaves 10 additional cases
over OCLL by my count (only 5 unique cases due to symmetry).
PBL-[something] - Permute both layers of one half while exchanging a corner
twist with the opposite half. These algorithms would be 4D, and I have no
idea how many there would be or how difficult they would be to execute.

A simpler, less glamorous solution is to simply solving the corner twist
before doing either Ortega half. Recognizing a twisted corner isn’t too
difficult on an n^3, even when scrambled, especially if one is already
familiar with the seven OCLLs. This is the 4D equivalent of step 4 in
the Guimond
Beginner Method
Here’s a short alg for a solving a clockwise twist on R: I[y2] R[U’ R’ U’ R
U2] I[y2] (Brackets indicate ROIL-style 3D moves – let me know if my
notation is difficult to understand.)

But what if we could solve the twisted corner *and* OLL of both halves all
at the same time? There are only a few possibilities there:
No corner parity:
Mirror OLLs - 7 cases; execute 3D algorithm using 4D twists
Non-mirror OLLs - 21 unique cases (26 including mirrors; like
sune/antisune); can use specialized 4D algorithms or two separate OCLLs
Corner parity:
6 cases for clockwise half * 6 cases for counterclockwise half = 36 cases;
must use specialized 4D algorithms

Most, if not all, of these specialized 4D algorithms would probably consist
of the "Orient All" stage of the 2^3 SOAP method
<> (algorithms here
<>) sandwiched by

Using I[y2] like this gave me an idea for a 3-stage method, after 4D

  1. Solve one face of each half (strictly 3D; just like Ortega)
    2.1. Orient the L and R halves so that their unsolved layers are in LI and
    RO respectively
    2.2. Perform I[y2]
  2. 3D-orient all pieces on the R half (strictly 3D; perhaps using something
    akin to Guimond <>
  3. Solve PBL on both faces

Here’s another crazy idea to go along with the above proposition: PBBL
(permute both "both layers"s). These would be 4D algorithms that solve PBL
on both halves of the puzzle at once. I think there would only be 10 unique
cases, not counting those with one half solved (i.e. normal PBL) or where
both halves have the same case (for these, execute the normal PBL algorithm
using the 4D subset <RUFLBD>). This set might not be too useful in physical
solves using Marc’s moveset, where 3D algs are easy, but in a virtual solve
it may be worth considering. Those 10 unique cases are:

I hope this gave you all something to think about, and I might start to
look for PBBL algorithms (especially some that would be easy to perform on
the physical puzzle).

On Fri, Jul 13, 2018 at 11:11 PM, Andy F <> wrote:

> Hey all!
> I haven’t been keeping up with the activity here, but I thought I’d give
> my thoughts on Marc’s solve anyway.
> The double Ortega method makes a lot of sense, so I just tried it on a
> virtual 2^4 since I don’t have a physical puzzle yet. It took 132 moves,
> which was on-par with my 130-move first-layer/OLL/PLL (FLOP?) solve.
> Instead of using the <RI> subset for permuting the first layer (PFL?), I
> used <RUFLBD>, which cut the move count on that step by half at the cost
> since the opposite layer’s permutation didn’t matter.
> As for solving the U2 parity, I was introduced to a much shorter algorithm
> at a recent 3D competition:
> (R’ L’ U2 R L U’)2
> This can be adapted to an even shorter 2^n algorithm, which is quite
> ergonomic on a 2^3:
> (R2 F2 R2 U’)2
> Since PBL is so small, I think it may even be possible to combine PBL with
> U2 parity (U2P? Am I going too crazy with these acronyms?). I’ll start
> looking through the PBL algs
> <> and see if I can put
> together a PBL-P set.
> - Andy
> On Fri, Jul 13, 2018 at 12:42 PM, Ty Jones
> [4D_Cubing] <> wrote:
>> Wow! That’s exciting that someone with so many subscribers got a hold of
>> your puzzle!
>> On Fri, Jul 13, 2018 at 12:32 AM Melinda Green
>> [4D_Cubing] <> wrote:
>>> That’s so cool, Marc! Best of all it puts you among the first 10
>>> official solvers at number 8. And so much goodness within these videos. I
>>> love your single piece flip and the way you so casually perform twists
>>> in the spooky projection
>>> <>
>>> which hurts my brain. I’m also curious about that crazy solve you did where
>>> you placed all the pieces and then oriented them all in place and hope
>>> you’ll make a video walk through of that method. It may not be a useful
>>> solution method but it’s certainly a good exercise of a handy technique.
>>> In unrelated news, a crazy YouTube celebrity Miguel Gimenez in Spain
>>> bought a puzzle and made a wonderful video introduction in Spanish
>>> <>. I had seen his work
>>> before but didn’t realize this was him until he told me about his new
>>> video. He has an astounding 1.2 million subscribers and his video already
>>> has over 100,000 views and 1,000 comments! He appears to have done a very
>>> careful job though I can’t tell how accurate it is. If one of you
>>> understands the puzzle and also speaks Spanish, please let me know what you
>>> think. He told me he plans to allow community subtitle submissions, so if
>>> anyone wants to attempt that, please let him know, and of course let us
>>> know because I’m eager to read it myself..
>>> Now back to Marc with congratulations on securing his prime spot in the
>>> 2x2x2x2 HOF! Very nicely done, Marc!
>>> Happy puzzling!
>>> -Melinda
>>> On 7/12/2018 6:51 PM, Marc Ringuette
>>> [4D_Cubing] wrote:
>>> Hi gang! I’m back to studying the physical 2^4, after a 6 month
>>> break. You know what that means: lots of little YouTube videos with
>>> no production values whatsoever! I’ve uploaded six videos to the new
>>> YouTube channel I’ve created for my puzzle stuff. Five short videos
>>> describe the move set I’m using and a few algorithms, then I give a 10
>>> minute solution video to put me on the HOF list.
>>> Probably the most generally interesting of these videos is the third,
>>> #23 in my numbering sequence, showing how to perform a 180 degree twist
>>> on a single piece (in 30 moves using ROIL Zero). I shorten this to 10
>>> moves in the case of orienting two opposite faces only, in video #24. I
>>> used the shortened version in my example solve at this point 6m45s in:
>>> 21 ROIL Zero moveset 1m11s
>>> 22 Sune+Antisune+twist2 3m17s
>>> 23 Doubletwist algorithm 2m17s
>>> 24 Doubletwist, align two faces only, shorter, 3 cases 2m59s
>>> 25 Alg for final U2 on R cube 1m16s
>>> 26 Example solve 10m42s
>>> Just now, I tried an additional goofy experiment: a wildly different
>>> solution method where I permute all pieces first, then orient them
>>> afterwards! It only took half an hour or so. It was a hoot, try
>>> it! I’m fascinated by the 12 orientations of these 4d puzzle pieces,
>>> and the second half of this method is a way to focus in on orientation
>>> exclusively.
>>> Cheers
>>> Marc
>>> ————————————
>>> Posted by: Marc Ringuette <> <>
>>> ————————————
>>> ————————————
>>> Yahoo Groups Links
> –
> "Engineers like to solve problems. If there are no problems handily
> available, they will create their own problems." - Scott Adams

"Engineers like to solve problems. If there are no problems handily
available, they will create their own problems." - Scott Adams