4x4x4 parity algorithms

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For those who are just looking for lists of commonly used speedsolving (speedcubing) 4x4x4 parity algorithms and cannot find what they are looking for just based on looking at the Table of Contents, this section is for you! However, it's important to read the Introduction to hopefully make clear the format of this page. (Note: Some of these algorithm lists are LONG. If you don't find the type of algorithms that you are looking for, keep scrolling down and chances are you will find them!)


Last two dedge pairing algorithms:

• There are none on this page, as those are actually not parity cases! See the explanation here in the Introduction and perhaps BIGcubes.com.


PLL Parity:

• For dedges
• 2 corner swaps


OLL Parity (Speedcubing Form):

• (Single) OLL Parity
• (Double) OLL Parity


Single Parity ("Pure" Form):

• (See this section of the Introduction to see why "pure" is an informal way of describing these type of algorithms.)
• Single Dedge Flip
• 2-Gen
• Supercube safe


Double Parity ("Pure" Form):

• Opposite
• Adjacent


Algorithms for other 3x3x3 methods besides the CFOP method (or any 3x3x3 method which solves the cube layer-by-layer, in general):

• Roux
• Petrus


K4 Method

• 2-cycles
• 4-cycles


Cage Method:

• Algorithms which keep the cage intact
• Algorithms which DO NOT keep the cage intact – variants of the Cage Method.


4x4x4 Blindfolded:

• Switching two corners and two Center pieces
• Switching two superube wing edge pieces - Also look for " //Safe" in the K4 2-cycles section for EVERY case.




Introduction

This page attempts to list move optimal algorithms for every common form of parity encountered in popular 4x4x4 (Rubik's Revenge) solving methods.

Solutions listed under a case image which are not move optimal (in the move metric in which algorithms are sorted by):

• are faster to execute,
• demonstrate a notable alternative way to solve the case (including, but not limited to using different types of moves),
• have a different effect on the 4x4x4 supercube, or
• are shorter than "the move optimal algorithms" in other big cube move metrics.

And if you're wondering why is this page so long! and why are there so many algorithms and parity cases when you only technically need to know a handful of cases and one (best) alg for each,

• Yes, oftentimes all (not just some) algs listed under a case image are impractical to actually use (whether it be they're too long or impossible to speedsolve, etc.), but those algs and cases are still included on this page because someone has (very likely) asked if algs for alternate parity cases (to the conventional) are possibly better cases to consider (due to the possibility of faster/shorter algs). (And people will very likely ask similar questions in the future.)
• Therefore, these uncommon cases (and unconventional algs to both common and uncommon cases) are here to:
  • Answer those "what if" questions.
  • Inspire future exploration.
  • And if nothing else, give everyone a deep appreciation for the "standard" (well-known) parity algorithms (to know of other possibilities (cases) and that what can come out of them (algs) is often not as useful/practical as the cases they see displayed on most websites and videos).
    • And make no mistake: most website and video content creators were (and some still are) unaware of the vast collection of cases (and algs) shown on this page.
    • They have essentially copied/pasted from another website or guide from the past, seeing that the parity algs are sufficient to solve the 4x4x4, and therefore felt no need (or even think) to look/research for "what's out there". (What they posted as a guide is all they know.)
• Last but not least, this page represents a lot of cubing history. There are plenty of websites which only list one algorithm for OLL parity and one for PLL parity—This is meant to be the "unabridged authority" on this topic. Links to posts which actually explain the algorithms are plentiful, and original founders are acknowledged.

Lastly, if you carefully look at the table of contents, you may be able to see that (after the PLL parity section) sections are ordered in such a way that cases/algs preserve the most, while each subsequent section's cases preserve less and less (but also, the shortest algs under each case also generally decrease in length) as you go down the page. This comment (and the one that follows) is a perfect demonstration of the goal of this page's section order.




Reduction parity

The term "parity" can be used to describe a number of situations that occur during a 4x4x4 solve which cannot manifest during a 3x3x3 (standard size Rubik's cube) solve. In fact, there has been debate about what situations are considered to be a parity case, but there is one situation of which any cuber who uses the term "parity" for the 4x4x4 identifies as parity: the single dedge flip.




The most popular 2-cycle (a swap of two pieces) besides the single dedge flip case is the following. This 2-cycle of wings is as common during a K4 Method solve as the single dedge flip is, but it should never arise during a solve using the Reduction Method because two dedges are not paired up. However, many who solve the 5x5x5 Rubik's cube using some variant of the Reduction Method will come across this case, and thus several (but not all) of the algorithms listed on this page which solve this case directly can be used for completing the tredge-tripling stage of a 5x5x5 Reduction solve.



An equally well-known form of reduction parity (this term will be defined formally soon) besides the single dedge flip is switching two opposite dedges in the same face.



This parity situation can be transformed into 21 other last layer forms of what is commonly called PLL parity by performing a 3x3x3 PLL and adjusting the upper face (AUF) as needed. That is, there is a total of 22 PLL parity cases. (See the PLL Parity section for details.) The remaining PLL parity cases which involve the fewest number of pieces (besides the most popular case above) are the following.


Despite that one can technically solve all 22 PLL parity cases by executing an algorithm meant to solve any one of them (to any face) and then finish solving the 4x4x4 as if it was a 3x3x3, special algorithms have been developed for every case. This allows one to use fewer moves to solve any given case and gives one more options.


Combining some form of PLL parity and a single dedge flip creates one of the many cases of what's commonly called double parity. For example, performing a swap of dedges to a fully solved 4x4x4 and then flipping the front dedge resulting from that swap gives us the following.




Since the double parity case above and the single dedge flip case both have a single dedge flipped, and since OLL algorithms do not necessarily aim to permute (move) the pieces that they correctly orient in any particular fashion, any 4x4x4 algorithm which solves:

• A case containing an odd number of flipped dedges (which will be called "single parity" on this page)
OR
• A case which additionally has an odd permutation of dedges and an even permutation of corners (or vice versa) (which will be called "double parity" on this page)

is called an OLL parity algorithm.


It is common convention among the speedcubing community to use algorithms which contain wide (double layer) turns to solve OLL parity instead of single inner layer slices. The notation used in Chris Hardwick's Rubik's Revenge Solution signifies a wide right turn as (Rr), for example. In old WCA notation (the notation used on this page), (Rr) is expressed as Rw. (The "w" is short for "wide".) In fact, the most popular speedcubing single parity algorithms perform additional swaps besides flipping a single dedge due to the use of wide turns.




Such an algorithm is called a non-pure algorithm when compared to algorithms which just flip a single dedge, which are often called pure flips. However, the term pure is more formally associated with an algorithm being supercube safe – algorithms which do not permute (move) any centers in the supercube version of a given order.

Most of the algorithms on this page affect some centers of the 4x4x4 supercube: not all algorithms affect the supercube centers in the same manner. Algorithms followed by a " //Safe" are supercube safe.


There are many types of parity cases which can occur during a 4x4x4 solve (when considering the parity cases which can arise in every 4x4x4 method), but the most widely known cases are the cases which can manifest after reducing a fully scrambled 4x4x4 into a pseudo 3x3x3 – an nxnxn cube in which all of its composite edges are complete and all of its centers are complete and are in the correct locations with respect to each other with regards to the cube's Color Scheme – are called reduction parity cases. (Because this is the objective of the Reduction Method and its variants.)

In May 2013, Michael Gottlieb defined reduction parity in detail.

"Reduction parity occurs when you try to reduce the puzzle so it can be solved by a constrained set of moves, putting it into some subset of the positions. However, you can often reach a position which seems like it is in your subset, but which is actually not, and to solve the puzzle you have to briefly go outside your constrained set of moves to bring the puzzle back into the subset you want. Typically the number of positions you can encounter is some small multiple of the number of positions you expect. The obvious example is PLL parity in 4x4x4: all the centers and edges are properly paired, so you expect to be able to finish the puzzle with only outer layer turns, but this isn't quite possible. OLL parity falls under this definition too (so the reduced 4x4x4 has four times as many positions as you would expect)."(Post)
There is a 50/50 chance that PLL parity will be present (assuming that one is doing a solve using reduction). Independently, there is a 50/50 chance to get a single dedge flip. Therefore there is a (1/2)×(1/2) = (chance of PLL parity)×(chance of single parity) = a 1/4 chance of getting all possibilities, separately. That is, for the reduction method (and its variants),
• PLL parity occurs by itself 25% of the time,
• Single parity occurs 25% of the time,
• Double parity occurs 25% of the time, and
• No reduction parity 25% of the time.


This page will keep strong focus on reduction parity (OLL parity and PLL parity) cases, but it will also include a limited number of other parity situations which are also common in other solving methods, as well as cases which share some characteristics with reduction parity algorithms.

The key characteristics of 4x4x4 reduction parity algorithms are:

• They preserve the colors of the centers. (Note that most reduction parity algorithms do not technically preserve the centers themselves, because if they are applied to a fully solved 4x4x4 supercube, one can see that same color centers are swapped with each other.)
• They do not break up (and therefore do not pair up) any dedges. (They preserve the coupling of the dedges, but they may move entire dedges.)

Additionally (with no exceptions),

• All OLL parity algorithms contain an odd number of inner slice quarter turns. (They are called odd parity algorithms.)
• All PLL parity algorithms contain an even number of inner slice quarter turns. (They are called even parity algorithms.)

Other parity cases

It turns out that we are not limited to using well-known dedge-pairing (even parity) algorithms to pair dedges.

Websites such as BIGcubes.com present algorithms with an even number of inner slice quarter turns for the dedge-pairing stage of the Reduction Method.


Algorithms which solve/generate any case in which dedges are broken up (including odd parity algorithms) can be used. For example, algorithms for this parity case (mentioned previously) can be used. This page contains quite a few algorithms which solve that case and other 2-cycle cases like



as well as 4-cycle cases which are contained within two dedges (which also can be used to pair the last two dedges).


There is actually a total of 110 last layer 4-cycles, but since 4-cycles in two dedges are the only ones encountered using the most popular 4x4x4 solving methods, they are the only ones shown on this page. However, this PDF includes all 110 cases and relatively short algorithms to solve each one directly. (All of these cases often occur in the ELL stage of the K4 Method. Algorithms for the Cage Method, as well as algorithms for theoretical purposes and general 4x4x4 exploration are present as well.)




Some short/easy parity fixes

Since all OLL parity algorithms contain an odd number of inner slice quarter turns, one can technically fix any 4x4x4 wing edge odd parity case by executing a single slice quarter turn and then resolve the cube using an even number of inner slice quarter turns. Here's one video tutorial that illustrates the typical process, but this is another (more recent) explanation which also just happens to also preserve the pairing of the dedges.

Similar to doing an inner slice quarter turn like r to technically fix the single dedge flip parity, an inner slice half turn such as r2 is technically all that is needed to fix PLL parity. In fact, only two inner slice quarter turns + 3x3x3 turns is all that is needed to create/solve PLL parity on the 4x4x4 One can split up r2 as r r or as r' r' and insert 3x3x3 moves to obtain the pure form of PLL parity.

This is an example algorithm found by Christopher Mowla in December of 2013.
(r' U' D' R2 U' D' S2 r')(F' U' D' F2 R F2 U2 R E' R F2 L' U2 R D2 R B2 R' x2 y')

However, we can also just use the inner slice turns r and r' as well.

(r' U D S2 U' D' S2 r)(U2 B D2 U2 F' U2 F2 L' R D' B2 L' R D2 F')


Should one wish to induce an odd permutation in the wing edges of the 4x4x4 with a short algorithm without having to restore the cube as much as applying an inner slice quarter turn requires, below are fairly short (and simple) algorithms one can use.

• The shortest half turn 4x4x4 cube odd parity fix which preserves the colors of the centers (essentially found in 2014 by Tom Rokicki and Ed Trice) is f2 r E2 r E2 r f2 (11,7).
• The shortest quarter turn 4x4x4 cube odd parity fix which preserves the colors of the centers was (found by Tom Rokicki on April 27, 2021) is u2 r u M u M' u r' (9,8).
Although these algorithms are not listed under a case image on this page, they would appear in the following format (in an "algorithm bar") if they were.

f2 r E2 r E2 r f2

(11,7)

N

Tom Rokicki and Ed Trice

u2 r u M u M' u r'

(9,8)

N

Tom Rokicki

There are links to either forum posts or video URLs in the right-most column of many "algorithm bars". These links are intended to either show one of the (if not the) first place the algorithm (or a simple conjugation, transformation, and/or a directly related version of it) was first published, and/or show one of the (if not the) earliest date of publish.

Besides just showing parity cases and algorithms for those cases, this page attempts to attribute credit to the original founder of an algorithm as well.

(More will be explained about what other pieces of information in the algorithm bars above mean later in this introduction.)


Since the shortest 4x4x4 (half turn) odd parity fix contains move repetition, it can be written more compactly as f2 (r E2)2 r f2. Clearly this algorithm does not preserve the pairing of dedges, but it does preserve the colors of the centers; and it contains 7 inner slice quarter turns, an odd number.

We can break up this algorithm as (f f r E E r E E r f f) to count 4 f's and 3 r's. At the same time, we can count a total of 11 block quarter turn moves (BQTM). We can count that this algorithm has 7 block half turn moves (BHTM) without breaking it up. In addition to the fact that all parity cases on this page are each represented by a case image,
• The number of moves an algorithm contains in these two big cube move metrics is written next to them in the form of the ordered pair, (BQTM, BHTM). (We can clearly see this in the above algorithm bar.)
• Algorithms with fewer BHTM are listed first in each category.
• Algorithms with fewer BQTM are listed before other algorithms which have the same number of BHTM as they.


The shortest (and well-known) nxnxn cube odd parity fix which preserves the colors of the centers is (r U2)4 r (13,9).

• Chris Hardwick presented an intuitive explanation of how it works (for the centers) here.
• And Christopher Mowla presented a technical explanation of how it works (for both the centers and wing edges) here.
Although this algorithm affects the same number of pieces as the (11,7) move algorithm above on the 4x4x4, it also "works"/does the same for the nxnxn cube. The (11,7) above discolors centers on, say, the 5x5x5 cube.
• Many of the algorithms on this page need to be "adjusted" to work for the nxnxn Rubik's cube.
• Some algorithms may only be translatable to higher order even cubes (6x6x6, and larger).

All algorithms can be applied to the 6x6x6 if instead of turning the outer 2 layers, turn the outer 3 layers; instead of turning 1 inner layer slice, turn 2 inner layer slices.


Finally, one of the simplest OLL parity (more specifically, a double parity) algorithms (found in December of 2017 by Christopher Mowla) to remember also consists of a short repeated sequence:
Rw' (F2 U' Lw' U)5 Rw (27,22)

This was deduced from the same idea that Floyd Newberry came up with for using a short repeated sequence to directly solve a 2-cycle. Below are two single dedge flip (2-cycle) algorithms illustrating the idea.

(Rw B' z')(r' F U2 F')4 r' (z B Rw')

(25,21)

N

Floyd Newberry

(Rw B' z')(r F U2 F')4 r (z B Rw')

(25,21)

N

Floyd Newberry




Additional notes about the algorithms

Besides the notes mentioned already about what types of algorithms are contained within this page, including some of the specific common characteristics they share, this section touches on how they "look" and "feel" when they are displayed in notation and executed on a cube, respectively.

Notation and animations

• All algorithms on this page are written in old WCA notation, where lowercase letters represent inner slice turns, uppercase letters and xyz have the same meaning as standard 3x3x3 notation, and a move like Rw, for example, means to turn both the face and inner slice parallel and adjacent to it simultaneously in the same direction. (Using the same example, Rw = (Rr).)
• Click the cube logo (the cube image to the far left of the algorithm bars), to see the algorithm animated in Old WCA notation at cubedb.net.




Move sets

Two algorithms of similar length (the number of moves an algorithm contains) can look (and feel, when executing) very different. This is especially common if two algorithms are in a different move set (consist only of certain types of turns).

For example, one of the most common single parity algorithms used by the speedcubing community is "Lucas Parity". Notice that it only contains the (and inverses of the) moves Rw, U2, x and Lw.

Rw U2 x Rw U2 Rw U2 Rw' U2 Lw U2 Rw' U2 Rw U2 Rw' U2 Rw' (25,17)

Interesting Fact:
On January 24, 2010, speedsolving.com forum member reThinking the Cube showed that Lucas Garron's "Lucas Parity" (published in mid 2008) is equivalent to Stefan Pochmann's "New Dedge Flip" from his webpage (published in or before early 2007).


One alternate algorithm of the same move length in BHTM (but contains 5 fewer BQTM) is the following. (Clearly this algorithm has much more of a variety of moves than "Lucas Parity". It is also clearly not a speedsolving algorithm as "Lucas Parity" is.)
Lw E Rw2 Uw r' Uw' Rw2 Dw R2 Uw r' Uw' Rw R Dw' E' Lw' (20,17)

This page not only contains commonly practiced speedsolving algorithms: it also contains algorithms which illustrate the veracity of the 4x4x4 cube parity algorithm domain.

Single slices vs. wide turns

Two of the most popular 15 BHTM move algorithms which flip a single dedge on the 4x4x4 are the following.


r' U2 l F2 l' F2 r2 U2 r U2 r' U2 F2 r2 F2 (25,15)
r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 r2 (25,15)

Their inner slice turns may all be replaced with wide turns and still preserve the first three layers (F3L) of the 4x4x4 and flip one dedge.

Rw' U2 Lw F2 Lw' F2 Rw2 U2 Rw U2 Rw' U2 F2 Rw2 F2 (25,15)
Rw2 B2 U2 Lw U2 Rw' U2 Rw U2 F2 Rw F2 Lw' B2 Rw2 (25,15)


The fourth column of algorithm bars (as can be seen in the bar below) is either labelled:

• Y (short for "Yes"), if all of an algorithm's inner slice turns can be substituted with their corresponding wide turns to still preserve F3L (or if an algorithm does not contain any inner layer slices)
OR
• N (short for "No"), if their inner slices cannot be substituted with wide turns and still preserve F3L.
Note that with many algorithms, it's not "all or nothing". A few of the slice turns can be wide to still just flip a single dedge, for example.
For convenience, an algorithm is written with the maximum number of wide turns, should that version of it still preserve as much as the version of it without any wide turns.


The second 15 BHTM algorithm mentioned above could be expressed later on this page with the following algorithm bar, since all of its inner slice turns can be made wide (hence the "Y" instead of an "N") and its first and last moves can be wide and still solve the pure dedge flip case (hence why the algorithm begins and ends with Rw2 instead of r2).

As an example of how algorithm bars are going to be labelled, let us temporarily name it "Old Standard Alg" and called the author "anonymous". (Algorithm names will be explained next.)

Rw2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 Rw2

(25,15)

Old Standard Alg

Y

anonymous

Despite that all (25,15) single dedge flip solutions which begin and end with an l2 or r2 move can instead be Lw2 and Rw2, respectively, all slices will be expressed as single slice (lowercase) turns for simplicity for all (25,15) solutions.

Names/labels

Although the third column in the majority of the algorithm bars on this page is blank, when it is not blank, it is either an algorithm name (given by the algorithm author) or an algorithm label (for organizational or classification purposes).

• Some algorithms like "Lucas Parity" have been named.
• Some algorithms are labelled (named) "Alg1(v1)", "v3", etc., (short for "algorithm 1, version 1" and "version 3", respectively) to:

• represent consecutive ordered version algorithms which are different versions (transformations) of each other

OR

• prove things like "there are actually fewer distinct paths than it appears for an algorithm in this move set and of this length" and also to show that "one algorithm is just another in disguise".

• Algorithms that have been named "5x5x5 Alg" are significantly longer algorithms than their related 4x4x4 algorithm counterparts, but they are algorithms which follow a lot of constraints and manage to work on the 4x4x4 and the 5x5x5.




PLL Parity

• This section shows all 22 possible PLL parity cases. In addition, algorithms for the two cases when two dedges are unoriented (and corners are solved) and the three cases when all four last layer dedges are unoriented (and the corners are solved) are also included.
  • Michael Fung's old page.
  • Frédérick Badie's formula sheet for PLL parity cases
• Of course, all PLL parity cases can be handled by using the first case shown, commonly called "opposite parity" (OP).
• This PDF shows how to handle all 22 cases using a combination of OP and 3x3x3 PLL algorithms.

Dedges

Two dedges (oriented)

I permutation



r2 U2 r2 Uw2 r2 u2

(12,6)

N

Chris Hardwick

r2 U2 r2 U2 d2 r2 d2

(14,7)

N

U2 r2 U2 r2 Uw2 r2 Uw2

(14,7)

N

Chris Hardwick

(Uw2 Rw2 U2) r2 (U2 Rw2 Uw2)

(14,7)

SP01

N

Stefan Pochmann

(Dw2 Rw2 U2) r2 (U2 Rw2 Dw2)

(14,7)

N

(Rw2 F2 U2) r2 (U2 F2 Rw2)

(14,7)

N

Stefan Pochmann

(Rw2 B2 U2) r2 (U2 B2 Rw2)

(14,7)

N

Stefan Pochmann

Rw2 (U2 r U2 S2)2 Rw2 //Safe

(18,10)

Y

Walter Randelshofer

r2 U2 r E2 r2 E2 r' D2 l2 E2 y2 //Safe

(18,10)

Y

r2 D2 l E2 l2 E2 l' U2 l2 E2 y2 //Safe

(18,10)

Y

r2 U2 r' E2 r2 E2 r D2 l2 E2 y2 //Safe

(18,10)

Y

r2 D2 l' E2 l2 E2 l U2 l2 E2 y2 //Safe

(18,10)

Y

r2 U2 l S2 r2 S2 l' D2 l2 E2 y2 //Safe

(18,10)

Y

r2 D2 r S2 l2 S2 r' U2 l2 E2 y2 //Safe

(18,10)

Y

r2 U2 l' S2 r2 S2 l D2 l2 E2 y2 //Safe

(18,10)

Y

r2 D2 r' S2 l2 S2 r U2 l2 E2 y2 //Safe

(18,10)

Y

r2 U2 B2 l r U2 M' U2 r2 B2 U2 //Safe

(19,11)

N

Stefan Pochmann

M U2 l2 B2 D2 r2 D2 B2 r l U2 //Safe

(19,11)

Y

M' U2 r2 B2 U2 r2 U2 B2 r l U2 //Safe

(19,11)

Y

M' U2 l' r' D2 B2 r2 B2 D2 l2 U2 //Safe

(19,11)

Y

M U2 l' r' U2 B2 r2 B2 U2 r2 U2 //Safe

(19,11)

Y

M U2 l r D2 F2 r2 F2 D2 l2 U2 //Safe

(19,11)

Y

M' U2 l r U2 F2 r2 F2 U2 r2 U2 //Safe

(19,11)

Y

(y Rw2 U2) Rw U2 (Rw2 U2)2 Rw (U2 Rw2 y')

(20,11)

v1

Y

(y Rw2 U2) Rw' U2 (Rw2 U2)2 Rw' (U2 Rw2 y')

(20,11)

v2

Y

(Rw2 U2) (E r2 E' r)2 (U2 Rw2) //Safe

(15,12)

Y

Werner Randelshofer, Walter Randelshofer, and André Boulouard

(Rw2 U2)(E' r2 E r)2 (U2 Rw2) //Safe

(15,12)

Y

Werner Randelshofer, Walter Randelshofer, and André Boulouard

(Rw2 U2)(E r2 E' r')2 (U2 Rw2) //Safe

(15,12)

Y

Werner Randelshofer, Walter Randelshofer, and André Boulouard

(Rw2 U2)(E' r2 E r')2 (U2 Rw2) //Safe

(15,12)

Y

Werner Randelshofer, Walter Randelshofer, and André Boulouard

r2 U2 r U2 r2 U2 r2 U2 r U2 r2 U2 //Safe

(22,12)

Y

r' F U' R F' U l r U' F R' U F' l' //Safe

(14,14)

N

(Rw2 U2)(r E2)6 (U2 Rw2) //Safe

(24,15)

N

Christopher Mowla

(Rw2 U2)(r' E2)6 (U2 Rw2) //Safe

(24,15)

N

Christopher Mowla

y' Rw2 3Uw2 Rw 3Uw Rw2 3Uw2 Rw2 3Uw' Rw2 3Uw Rw2 3Uw' Rw2 3Uw' Rw2 3Uw Rw2 3Uw Rw2 3Uw' Rw 3Uw2 Rw2 y

(36,23)

Y

Ben Whitmore

M2 Uw M Uw M' Uw' M' Uw M Uw M Uw2 M' Uw2 M' Uw' M' Uw M Uw2 M Uw' M' Uw2 M' Uw' M Uw' M' Uw' M Uw M2

(38,33)

N

Ben Whitmore

Double slash (//) permutation



(R2 D' x) r2 U2 r2 Uw2 r2 u2 (x' D R2)

(18,10)

N

Chris Hardwick

(R2 D' x Uw2 Rw2 U2) r2 (U2 Rw2 Uw2 x' D R2)

(20,11)

SP02

N

Stefan Pochmann

(R2 D' Rw2 U2 F2) r2 (F2 U2 Rw2 D R2)

(20,11)

FB02

N

Frédérick Badie

(F2 U Rw2 U2 F2) r2 (F2 U2 Rw2 U' F2)

(20,11)

N

Stefan Pochmann

(R2 D' x Rw2 F2 U2) r2 (U2 F2 Rw2 x' D R2)

(20,11)

N

Stefan Pochmann

(R2 D' x Rw2 B2 U2) r2 (U2 B2 Rw2 x' D R2)

(20,11)

N

Stefan Pochmann

(F U' F') r2 U2 r2 Uw2 r2 u2 (F U F')

(18,12)

N

Chris Hardwick

(F U' F' Rw2 F2 U2) r2 (U2 F2 Rw2 F U F')

(20,13)

N

Stefan Pochmann

(y' R' F) l E F2 E' l' r' E F2 E' r (F' R y) //Safe

(16,14)

Alg.1

N

Christopher Mowla

(y' R' F) r' E F2 E' l r E F2 E' l' (F' R y) //Safe

(16,14)

Inverse[Alg.1]

N

Christopher Mowla

(y' R' F) l E' F2 E l' r' E' F2 E r (F' R y) //Safe

(16,14)

Alg.2

N

Christopher Mowla

(y' R' F) r' E' F2 E l r E' F2 E l' (F' R y) //Safe

(16,14)

Inverse[Alg.2]

N

Christopher Mowla

(R B') l S U2 S' l' r' S U2 S' r (B R') //Safe

(16,14)

Alg.3

N

Walter Randelshofer

(R B') r' S U2 S' l r S U2 S' l' (B R') //Safe

(16,14)

Inverse[Alg.3]

N

Walter Randelshofer

(R B') l S' U2 S l' r' S' U2 S r (B R') //Safe

(16,14)

Alg.4

N

Walter Randelshofer

(R B') r' S' U2 S l r S' U2 S l' (B R') //Safe

(16,14)

Inverse[Alg.4]

N

Walter Randelshofer

(R U R' U') r2 U2 r2 Uw2 r2 Uw2 (U' R U' R')

(19,14)

N

Chris Hardwick

(R U R') U r2 U2 r2 Uw2 r2 Uw2 U (R U' R')

(19,14)

N

Chris Hardwick

(Uw' R U' R' Uw' Rw2 U2) r2 (U2 Rw2 Uw R U R' Uw)

(20,15)

N

xyzzy

(y Uw' R U R' Uw' Rw2 U2) r2 (U2 Rw2 Uw R U' R' Uw y')

(20,15)

N

xyzzy

(y2 Uw R U' R' Uw Rw2 U2) r2 (U2 Rw2 Uw' R U R' Uw' y2)

(20,15)

N

xyzzy

(y' Uw R U R' Uw Rw2 U2) r2 (U2 Rw2 Uw' R U' R' Uw' y)

(20,15)

N

xyzzy

(y2 Uw2 R U R' U' Rw2 U2) r2 (U2 Rw2 U R U' R' Uw2 y2)

(22,15)

N

xyzzy

(R2 D' x) r2 U2 B2 l r U2 M' U2 r2 B2 U2 (x' D R2) //Safe

(24,15)

N

(Rw' U R U Lw' U2 Rw' U2) r2 (U2 Rw U2 Lw U' R' U' Rw)

(22,17)

v1

N

Christopher Mowla

y' (Rw U' R' U' Rw B2) (Rw B2 r2 B2 Rw') (B2 Rw' U R U Rw') y

(22,17)

v2

N

Christopher Mowla &Robert Yau

y2 Rw U Rw' R U' Rw' U' Rw U Rw U' Rw' U' Rw' R U Rw U R' U' R' U y2

(20,20)

Y

António Gomes

y2 R' U' R' U Rw U R Rw' U' Rw' U' Rw U Rw U' Rw' U' R Rw' U Rw U y2

(20,20)

Y

António Gomes

y R' U Rw U R Rw' U' Rw' U' Rw U Rw U' Rw' U' R Rw' U Rw U R' U' y'

(20,20)

Y

António Gomes

(y2 Bw2 R2 Bw2) Rw U Rw' U' Lw' U' Lw U2 Rw U' Rw' U' Lw' U Lw (Bw2 R2 Bw2 y2)

(28,21)

Y

Moritz Karl

y Rw' U2 Rw U2 Rw' U2 Rw' U' Rw U' Rw U Rw' U' Rw U Rw2 U Rw U' Rw U Rw' U Rw U y'

(30,26)

Y

Ben Whitmore

Two dedges (unoriented)

I permutation



F2 l E F2 E' l' r' E F2 E' r F2 //Safe

(16,12)

Alg.1

N

Christopher Mowla

F2 r' E' F2 E l r E' F2 E l' F2 //Safe

(16,12)

Mirror[Alg.1]

N

Christopher Mowla

F2 r' E F2 E' l r E F2 E' l' F2 //Safe

(16,12)

Inverse[Alg.1]

N

Christopher Mowla

F2 l E' F2 E l' r' E' F2 E r F2 //Safe

(16,12)

Mirror[Inverse[Alg.1]

N

Christopher Mowla

F2 l' S U2 S' l r S U2 S' r' F2 //Safe

(16,12)

Alg.2

N

Walter Randelshofer

F2 r S' U2 S l' r' S' U2 S l F2 //Safe

(16,12)

Mirror[Alg.2]

N

Walter Randelshofer

F2 r S U2 S' l' r' S U2 S' l F2 //Safe

(16,12)

Inverse[Alg.2]

N

Walter Randelshofer

F2 l' S' U2 S l r S' U2 S r' F2 //Safe

(16,12)

Mirror[Inverse[Alg.2]

N

Walter Randelshofer

(y R' U F') r2 U2 r2 Uw2 r2 u2 (F U' R y')

(18,12)

N

Chris Hardwick

r U2 l D2 l' U2 M U2 r D2 r' U2 l' //Safe

(19,13)

N

Kenneth Gustavsson

(y' R' F U' Rw2 U2 F2) r2 (F2 U2 Rw2 U F' R y)

(20,13)

N

Stefan Pochmann

(y R' U F' Rw2 F2 U2) r2 (U2 F2 Rw2 F U' R y')

(20,13)

N

(F E' F Rw2 F2 U2) r2 (U2 F2 Rw2 F' E F')

(20,13)

N

Per Kristen Fredlund

F' U R d2 R' U' F U2 F' U R d2 R' U' F U2 //Safe

(20,16)

N

Per Kristen Fredlund

l U2 M l U2 M' l' U2 M' U2 Rw M' U2 M r' U2 Rw' //Safe

(23,17)

N

Kenneth Gustavsson

(F r U' R U' Lw U2 Rw U2) r2 (U2 Rw' U2 Lw' U R' U r' F')

(24,19)

N

Christopher Mowla

(Rw U2 Rw' U l' U' Lw' U2 Rw' U' l2 U') r2 (U l2 U Rw U2 Lw U l U' Rw U2 Rw')

(32,25)

N

Christopher Mowla

Double slash (//) permutation



(R B) r2 U2 r2 Uw2 r2 u2 (B' R')

(16,10)

N

Chris Hardwick

(R B Rw2 F2 U2) r2 (U2 F2 Rw2 B' R')

(18,11)

N

Stefan Pochmann

(3Lw U Rw2 U2 F2) r2 (F2 U2 Rw2 U' 3Lw')

(18,11)

N

Stefan Pochmann

(R2 D) l' E F2 E' l r E F2 E' r' (D' R2) //Safe

(16,14)

N

Per Kristen Fredlund

F' R d2 R' F U F' U2 R d2 R' U2 F U' //Safe

(16,14)

N

Per Kristen Fredlund

R' U2 B u2 B' U2 R U' R' B u2 B' R U //Safe

(16,14)

N

Walter Randelshofer

R B U2 r2 U2 B2 l r U2 M' U2 r2 B R' //Safe

(21,14)

N

Stefan Pochmann

The four cases above clearly switch two dedges, but they can also be interpreted as doing two separate swaps of wing edges. Recalling that the term "2-cycle" is interchangeable with the common term "swap", these cases perform 2 2-cycles of wing edges. (They are called "2 2-cycles" for short.) There are actually 58 of these cases in the last layer, in general. However, the other 54 will only be encountered during a K4 Method solve. This PDF includes all 58 cases and short algorithms to solve each one directly.

Four dedges (oriented)

O permutation

Oa



r2 Uw2 b2 U' b2 r2 U b2 r2 Uw2 r2

(20,11)

CG03

N

Clément Gallet

f2 Uw2 r2 U' r2 f2 U r2 f2 Uw2 f2

(20,11)

N

Clément Gallet

y2 M2 U' M2 u Uw f2 M2 Dw2 b2 U2 b2

(20,11)

N

Clément Gallet

M' U2 Uw2 r2 Uw2 r2 U2 l2 U2 M' U M2 U M2

(22,13)

N

M U2 Uw2 l2 Uw2 l2 U2 r2 U2 M U M2 U M2

(22,13)

N

y2 D2 M E2 M U M2 U' l2 U2 r2 Uw2 r2 Uw2

(22,13)

PKF03

N

Per Kristen Fredlund

(E2 M)2 U M2 U' l2 U2 r2 Uw2 r2 u2

(22,13)

PKF03

N

Per Kristen Fredlund

l2 U2 f2 U' l2 f2 U u2 f2 u2 l2 U2 l2

(23,13)

N

Clément Gallet

l2 U2 f2 U' l2 f2 U u2 f2 l2 U2 l2 u2

(23,13)

N

Clément Gallet

r2 Uw2 b2 U' b2 r2 U b2 R2 Uw2 Rw2 Uw2 R2

(24,13)

N

Clément Gallet

M2 U M U2 r' l' U2 F2 r2 F2 U2 r2 U M2

(23,14)

N

M2 U S2 l2 S2 U2 r S2 l2 S2 r' U2 r2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 l2 S2 U2 r' S2 l2 S2 r U2 r2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 l2 S2 U2 l E2 l2 E2 l' U2 r2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 l2 S2 U2 l' E2 l2 E2 l U2 r2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 r2 S2 D2 r S2 l2 S2 r' D2 l2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 r2 S2 D2 r' S2 l2 S2 r D2 l2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 r2 S2 D2 l E2 l2 E2 l' D2 l2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 r2 S2 D2 l' E2 l2 E2 l D2 l2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 l2 D2 l S2 r2 S2 l' D2 S2 r2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 l2 D2 l' S2 r2 S2 l D2 S2 r2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 l2 D2 r E2 r2 E2 r' D2 S2 r2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 l2 D2 r' E2 r2 E2 r D2 S2 r2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 r2 U2 l S2 r2 S2 l' U2 S2 l2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 r2 U2 l' S2 r2 S2 l U2 S2 l2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 r2 U2 r E2 r2 E2 r' U2 S2 l2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U S2 r2 U2 r' E2 r2 E2 r U2 S2 l2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U' F2 M' F2 M U2 r2 D2 F2 r2 F2 D2 r2 U' M2 //Safe

(28,16)

N

Christopher Mowla

F' L B D2 (R d L R' E2 M' f' M E2 L') D2 B' L' F //Safe

(21,17)

N

Walter Randelshofer

l' r' 3Dw L' R' u L R 3Dw' M2 3Dw' L R u L' R' 3Dw l' r' //Safe

(20,19)

N

Christopher Mowla

l r 3Dw L R u' L' R' 3Dw' M2 3Dw' L' R' u' L R 3Dw l r //Safe

(20,19)

N

Christopher Mowla

l' r' 3Dw' L' R' u' L R 3Dw M2 3Dw L R u' L' R' 3Dw' l' r' //Safe

(20,19)

N

Christopher Mowla

l r y' U' L R u L' R' 3Dw M2 3Dw L' R' u L R y' U' l r //Safe

(20,19)

N

Christopher Mowla

U2 (R' U' R' F R F' U R) (Uw2 r2 Uw2 r2 U2 r2) (F U R U' R' F')

(28,21)

N

Antoine Cantin

(F2 U2 M U Fw2 r2 u S' Rw2 u2) M (u2 Rw2 S u' r2 Fw2 U' M' U2 F2)

(33,21)

N

Christopher Mowla

Rw2 U R2 U R U' R' U R Rw' U2 Rw2 U2 Rw2 U2 Rw' U' R U R' U2 Rw2 U'

(31,22)

Y

António Gomes

Rw2 U2 R' U R U' Rw' U2 Rw2 U2 Rw2 U2 R Rw' U R' U' R U R2 U Rw2 U'

(31,22)

Y

António Gomes

Rw R U' Rw' R' U' Rw' R U Rw U' R2 U R U R U' Rw U Rw' R U' Rw' U' Rw

(24,23)

Y

António Gomes

Rw2 U R2 U R U' R U Rw U R' Rw2 U2 R Rw2 U2 R' Rw2 U Rw' U2 Rw2 U'

(29,23)

Y

António Gomes

Rw2 U2 Rw' U R' Rw2 U2 R Rw2 U2 R' Rw2 U Rw U R U' R U R2 U Rw2 U'

(29,23)

Y

António Gomes

Rw2 U' R' U' R U R U' Rw U2 Rw2 U2 Rw2 U2 Rw U2 R U' R U R2 Rw2 U'

(30,23)

Y

António Gomes

Rw2 U' R' U' R U R U' Rw' U2 Rw2 U2 Rw2 U2 Rw' U2 R U' R U R2 Rw2 U'

(30,23)

Y

António Gomes

Rw2 R2 U R U' R U2 Rw U2 Rw2 U2 Rw2 U2 Rw U' R U R U' R' U' Rw2 U'

(30,23)

Y

António Gomes

Rw2 R2 U R U' R U2 Rw' U2 Rw2 U2 Rw2 U2 Rw' U' R U R U' R' U' Rw2 U'

(30,23)

Y

António Gomes

Rw2 R2 U R2 Rw2 U R2 U' Rw2 U' Rw2 U' Rw U' Rw2 U' R2 U Rw2 U Rw' U' R2

(30,23)

Y

António Gomes

Rw2 R2 U R2 Rw2 U R2 U' Rw2 U' Rw2 U' Rw' U' Rw2 U' R2 U Rw2 U Rw U' R2

(30,23)

Y

António Gomes

R U' R' Rw2 U2 Rw U2 Rw2 U2 Rw2 U2 Rw U2 R2 Rw2 U R U R U' R' U' R2

(31,23)

Y

António Gomes

R U' R' Rw2 U2 Rw' U2 Rw2 U2 Rw2 U2 Rw' U2 R2 Rw2 U R U R U' R' U' R2

(31,23)

Y

António Gomes

Rw2 U R2 U R U' R' U R Rw U2 Rw2 U2 Rw2 U2 Rw U' R U R' U2 Rw2 U'

(32,33)

Y

António Gomes

Rw2 U2 R' U R U' Rw U2 Rw2 U2 Rw2 U2 R Rw U R' U' R U R2 U Rw2 U'

(32,33)

Y

António Gomes

R U2 R U R' Rw2 U2 Rw U2 Rw2 U2 Rw2 U2 Rw U2 R2 Rw2 U R2 U' R' U' R2 //Safe

(33,23)

Y

António Gomes

R U2 R U R' Rw2 U2 Rw' U2 Rw2 U2 Rw2 U2 Rw' U2 R2 Rw2 U R2 U' R' U' R2 //Safe

(33,23)

Y

António Gomes

Ob



r2 Uw2 r2 b2 U' r2 b2 U b2 Uw2 r2

(20,11)

CG03

N

Clément Gallet

f2 Uw2 f2 r2 U' f2 r2 U r2 Uw2 f2

(20,11)

N

Clément Gallet

b2 U2 b2 Dw2 M2 f2 Uw' u' M2 U M2 y2

(20,11)

N

Clément Gallet

M2 U' M2 U' M U2 l2 U2 r2 Uw2 r2 Uw2 U2 M

(22,13)

N

M2 U' M2 U' M' U2 r2 U2 l2 Uw2 l2 Uw2 U2 M'

(22,13)

N

Uw2 r2 Uw2 r2 U2 l2 U M2 U' M' E2 M' D2 y2

(22,13)

PKF03

N

Per Kristen Fredlund

u2 r2 Uw2 r2 U2 l2 U M2 U' (M' E2)2

(22,13)

PKF03

N

Per Kristen Fredlund

l2 U2 l2 u2 f2 u2 U' f2 l2 U f2 U2 l2

(23,13)

N

Clément Gallet

u2 l2 U2 l2 f2 u2 U' f2 l2 U f2 U2 l2

(23,13)

N

Clément Gallet

R2 Uw2 Rw2 Uw2 R2 b2 U' r2 b2 U b2 Uw2 r2

(24,13)

N

Clément Gallet

M2 U' r2 U2 F2 r2 F2 U2 l r U2 M' U' M2

(23,14)

N

M2 U r2 U2 r S2 l2 S2 r' U2 S2 l2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U r2 U2 r' S2 l2 S2 r U2 S2 l2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U r2 U2 l E2 l2 E2 l' U2 S2 l2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U r2 U2 l' E2 l2 E2 l U2 S2 l2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U l2 D2 r S2 l2 S2 r' D2 S2 r2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U l2 D2 r' S2 l2 S2 r D2 S2 r2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U l2 D2 l E2 l2 E2 l' D2 S2 r2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U l2 D2 l' E2 l2 E2 l D2 S2 r2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U r2 S2 D2 l S2 r2 S2 l' D2 l2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U r2 S2 D2 l' S2 r2 S2 l D2 l2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U r2 S2 D2 r E2 r2 E2 r' D2 l2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U r2 S2 D2 r' E2 r2 E2 r D2 l2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U l2 S2 U2 l S2 r2 S2 l' U2 r2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U l2 S2 U2 l' S2 r2 S2 l U2 r2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U l2 S2 U2 r E2 r2 E2 r' U2 r2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U l2 S2 U2 r' E2 r2 E2 r U2 r2 S2 U' M2 //Safe

(26,15)

N

Christopher Mowla

M2 U r2 D2 F2 r2 F2 D2 r2 U2 M' F2 M F2 U M2 //Safe

(28,16)

N

Christopher Mowla

F' L B D2 (L E2 M' f M E2 L' R d' R') D2 B' L' F //Safe

(21,17)

N

Walter Randelshofer

l r 3Dw' L R u' L' R' 3Dw M2 3Dw L' R' u' L R 3Dw' l r //Safe

(20,19)

N

Christopher Mowla

l' r' 3Dw' L' R' u L R 3Dw M2 3Dw L R u L' R' 3Dw' l' r' //Safe

(20,19)

N

Christopher Mowla

l r 3Dw L R u L' R' 3Dw' M2 3Dw' L' R' u L R 3Dw l r //Safe

(20,19)

N

Christopher Mowla

l' r' U y L' R' u' L R 3Dw' M2 3Dw' L R u' L' R' U y l' r' //Safe

(20,19)

N

Christopher Mowla

(F R U R' U' F') (r2 U2 r2 Uw2 r2 Uw2) (R' U' F R' F' R U R) U2

(28,21)

N

Antoine Cantin

(F2 U2 M U Fw2 r2 u S' Rw2 u2) M' (u2 Rw2 S u' r2 Fw2 U' M' U2 F2)

(33,21)

N

Christopher Mowla

U Rw2 U2 R U' R' U Rw U2 Rw2 U2 Rw2 U2 Rw R' U' R U R' U' R2 U' Rw2

(31,22)

Y

António Gomes

U Rw2 U' R2 U' R' U R U' Rw R' U2 Rw2 U2 Rw2 U2 Rw U R' U' R U2 Rw2

(31,22)

Y

António Gomes

Rw' U Rw U R' Rw U' Rw' U R' U' R' U' R2 U Rw' U' R' Rw U R Rw U R' Rw'

(24,23)

Y

António Gomes

U Rw2 U2 Rw U' Rw2 R U2 Rw2 R' U2 Rw2 R U' Rw' U' R' U R' U' R2 U' Rw2

(29,23)

Y

António Gomes

U Rw2 U' R2 U' R' U R' U' Rw' U' Rw2 R U2 Rw2 R' U2 Rw2 R U' Rw U2 Rw2

(29,23)

Y

António Gomes

U Rw2 R2 U' R' U R' U2 Rw' U2 Rw2 U2 Rw2 U2 Rw' U R' U' R' U R U Rw2

(30,23)

Y

António Gomes

U Rw2 R2 U' R' U R' U2 Rw U2 Rw2 U2 Rw2 U2 Rw U R' U' R' U R U Rw2

(30,23)

Y

António Gomes

U Rw2 U R U R' U' R' U Rw' U2 Rw2 U2 Rw2 U2 Rw' U2 R' U R' U' Rw2 R2

(30,23)

Y

António Gomes

U Rw2 U R U R' U' R' U Rw U2 Rw2 U2 Rw2 U2 Rw U2 R' U R' U' Rw2 R2

(30,23)

Y

António Gomes

R2 U Rw U' Rw2 U' R2 U Rw2 U Rw' U Rw2 U Rw2 U R2 U' Rw2 R2 U' Rw2 R2

(30,23)

Y

António Gomes

R2 U Rw' U' Rw2 U' R2 U Rw2 U Rw U Rw2 U Rw2 U R2 U' Rw2 R2 U' Rw2 R2

(30,23)

Y

António Gomes

R2 U R U R' U' R' U' Rw2 R2 U2 Rw' U2 Rw2 U2 Rw2 U2 Rw' U2 Rw2 R U R'

(31,23)

Y

António Gomes

R2 U R U R' U' R' U' Rw2 R2 U2 Rw U2 Rw2 U2 Rw2 U2 Rw U2 Rw2 R U R'

(31,23)

Y

António Gomes

U Rw2 U2 R U' R' U Rw' U2 Rw2 U2 Rw2 U2 Rw' R' U' R U R' U' R2 U' Rw2

(32,33)

Y

António Gomes

U Rw2 U' R2 U' R' U R U' Rw' R' U2 Rw2 U2 Rw2 U2 Rw' U R' U' R U2 Rw2

(32,33)

Y

António Gomes

R2 U R U R2 U' Rw2 R2 U2 Rw' U2 Rw2 U2 Rw2 U2 Rw' U2 Rw2 R U' R' U2 R' //Safe

(33,23)

Y

António Gomes

R2 U R U R2 U' Rw2 R2 U2 Rw U2 Rw2 U2 Rw2 U2 Rw U2 Rw2 R U' R' U2 R' //Safe

(33,23)

Y

António Gomes

B permutation



• Note: This is commonly called the "W" permutation.
• A list of algorithms for the inverse case are unnecessary. Just do U2 (alg) U2 or y2 (alg) y2.

M2 U M' U2 M U l2 U2 r2 Uw2 r2 Uw2 U2

(20,12)

N

u2 S2 Uw' u' b2 R2 b2 Rw2 f2 r2 U S2

(21,12)

N

Clément Gallet

F2 D' f2 d2 b2 D' B2 Dw2 3Uw' M2 3Uw F2 Uw2

(21,13)

N

Clément Gallet

(r2 U2 r2 Uw2 r2 u2) (F2 U M' U2 M U F2)

(22,13)

PKF04

N

Per Kristen Fredlund

R2 Uw2 B2 R2 Uw2 B2 R2 U R2 B2 R2 U B2 Uw2

(26,14)

CG04

Y

Clément Gallet

F2 Uw' u' M2 U' F2 D S2 Dw d Bw2 u2 b2 U2 y2

(22,14)

N

Clément Gallet

(Rw2 F2 U2 r2 U2 F2 Rw2) (F2 U M' U2 M U F2) //Safe

(24,14)

N

Christopher Mowla

l2 D2 B2 Rw2 B2 D2 l2 F2 U' F2 L2 F2 U' L2

(26,14)

N

Clément Gallet

y' M2 U' r2 b2 Rw2 Bw2 R2 Bw2 L2 U2 M2 U2 F2 D M2 x2 y

(28,15)

N

Doug

R' U R' U' R' U' R' U R U' Uw2 r2 Uw2 r2 U2 Rw2

(21,16)

SP04

N

Stefan Pochmann

R' U R' U' R' U' R' U R U' Uw2 r2 Uw2 r2 U2 Rw2

(21,16)

SP04

N

Stefan Pochmann

r2 (U2 r U2 S2)2 Rw2 S2 D' F2 D S2 D' F2 D R2 //Safe

(32,19)

N

Walter Randelshofer

R2 U R' U' R2 U R U Rw2 U2 R' Rw U2 R' Rw2 U2 Rw2 U2 Rw U2 Rw2 //Safe

(30,20)

Y

António Gomes

R2 U R' U' R2 U R U R' Rw2 U2 R' Rw U2 Rw2 U2 Rw2 U2 Rw U2 Rw2 //Safe

(30,20)

Y

António Gomes

R2 U R' U' R2 U R U R' Rw2 U2 R' Rw' U2 Rw2 U2 Rw2 U2 Rw' U2 Rw2 //Safe

(31,21)

Y

António Gomes

y R2 U' R' U' R U2 Rw U R Rw' U' Rw' U' Rw U Rw U' Rw' U' R Rw' U R Rw y'

(24,22)

Y

António Gomes

y' U' Rw2 U2 Rw' U Rw2 U' Rw2 U Rw' U Rw U Rw' U' Rw' U Rw U Rw' U Rw2 U' Rw2 U Rw U2 Rw2 y

(36,28)

Y

Ben Whitmore

(M2 U Fw2 r2 u r2 Uw2 S' U' B R B' R2 U) M' (U' R2 B R' B' U S Uw2 r2 u' r2 Fw2 U' M2)

(41,29)

N

Christopher Mowla

Four Dedges (unoriented)

O permutation

Oa



F2 M F2 U' l2 U2 r2 Uw2 r2 U' Uw2 B2 M B2

(23,14)

N

Christopher Mowla

y' U2 Rw2 U2 r2 Uw2 r2 y Uw2 M U' M2 U' F2 U2 M

(24,14)

N

Christopher Mowla

R2 S' (u2 r2 Uw2 r2 U2) Rw2 U' S2 U L2 S' L2

(24,14)

N

Per Kristen Fredlund

R2 S (u2 r2 Uw2 r2 U2) Rw2 U S2 U' L2 S L2

(24,14)

N

Per Kristen Fredlund

L2 S L2 U S2 U' Rw2 (U2 r2 Uw2 r2 u2) S R2

(24,14)

N

Per Kristen Fredlund

L2 S' L2 U' S2 U Rw2 (U2 r2 Uw2 r2 u2) S' R2

(24,14)

N

Per Kristen Fredlund

M2 D R' L' u' R L D2 R L y M2 u' M2 y' R' L' D M2 //Safe

(22,17)

N

Christopher Mowla

M2 D R L u R' L' D2 R' L' y M2 u M2 y' R L D M2 //Safe

(22,17)

N

Christopher Mowla

M2 D' R' L' y M2 d' M2 y' R L D2 R L d' R' L' D' M2 //Safe

(22,17)

N

Christopher Mowla

M2 D' R L y M2 d M2 y' R' L' D2 R' L' d R L D' M2 //Safe

(22,17)

N

Christopher Mowla

M2 U F2 D2 r F2 r2 F2 D2 l2 F2 D2 r2 D2 l U' M2

(30,17)

N

Christopher Mowla

(M2 U Fw2 r2 u S' Rw2 u2) M' (u2 Rw2 S u' r2 Fw2 U' M2)

(27,17)

N

Christopher Mowla

(M2 U Fw2 r2 u' S' Rw2 u2) M' (u2 Rw2 S u r2 Fw2 U' M2)

(27,17)

N

Christopher Mowla

Rw U' Rw2 U' r2 U' Rw' U R' U2 R' U Rw' U' r2 U' Rw2 U' Rw U2

(26,20)

N

xyzzy

Ob



B2 M' B2 Uw2 U r2 Uw2 r2 U2 l2 U F2 M' F2

(23,14)

N

Christopher Mowla

M' U2 F2 U M2 U M' Uw2 y' r2 Uw2 r2 U2 Rw2 U2 y

(24,14)

N

Christopher Mowla

L2 S L2 U' S2 U Rw2 (U2 r2 Uw2 r2 u2) S R2

(24,14)

N

Per Kristen Fredlund

L2 S' L2 U S2 U' Rw2 (U2 r2 Uw2 r2 u2) S' R2

(24,14)

N

Per Kristen Fredlund

R2 S' (u2 r2 Uw2 r2 U2) Rw2 U S2 U' L2 S' L2

(24,14)

N

Per Kristen Fredlund

R2 S (u2 r2 Uw2 r2 U2) Rw2 U' S2 U L2 S L2

(24,14)

N

Per Kristen Fredlund

M2 D' L R y M2 u M2 y' L' R' D2 L' R' u L R D' M2 //Safe

(22,17)

N

Christopher Mowla

M2 D' L' R' y M2 u' M2 y' L R D2 L R u' L' R' D' M2 //Safe

(22,17)

N

Christopher Mowla

M2 D L R d L' R' D2 L' R' y M2 d M2 y' L R D M2 //Safe

(22,17)

N

Christopher Mowla

M2 D L' R' d' L R D2 L R y M2 d' M2 y' L' R' D M2 //Safe

(22,17)

N

Christopher Mowla

M2 U l' D2 r2 D2 F2 l2 D2 F2 r2 F2 r' D2 F2 U' M2

(30,17)

N

Christopher Mowla

(M2 U Fw2 r2 u S' Rw2 u2) M (u2 Rw2 S u' r2 Fw2 U' M2)

(27,17)

N

Christopher Mowla

(M2 U Fw2 r2 u' S' Rw2 u2) M (u2 Rw2 S u r2 Fw2 U' M2)

(27,17)

N

Christopher Mowla

Rw' U Rw2 U r2 U Rw U' R U2 R U' Rw U r2 U Rw2 U Rw' U2

(26,20)

N

xyzzy

B permutation



• Note: This is commonly called the "W" permutation.
• A list of algorithms for the inverse case are unnecessary. Just do U2 (alg) U2 or y2 (alg) y2.

y' M2 U' D2 r2 S2 l S2 r2 E2 r' U' M2 y' //Safe

(19,12)

N

Christopher Mowla

M2 U' r U D L2 U D S2 r S2 D' U' L2 D' M2 //Safe

(22,16)

N

Christopher Mowla

y M2 U r' U' D' L2 U' D' S2 r' S2 D U L2 D M2 y' //Safe

(22,16)

N

Christopher Mowla

M2 D R2 D U r U' D' R2 U' D' S2 r S2 U M2 //Safe

(22,16)

N

Christopher Mowla

y M2 D' R2 D' U' r' U D R2 U D S2 r' S2 U' M2 y' //Safe

(22,16)

N

Christopher Mowla

(M2 U')(r' E2 l E2)3 (U M2) //Safe

(24,16)

N

Christopher Mowla

(M2 U')(E2 r' E2 l)3 (U M2) //Safe

(24,16)

N

Christopher Mowla

(y x' M2 Fw f r2 u r2 S' u2) M (u2 S r2 u' r2 f' Fw' M2 x y')

(25,17)

N

Christopher Mowla

(y x' M2 Fw f r2 u' r2 S' u2) M (u2 S r2 u r2 f' Fw' M2 x y')

(25,17)

N

Christopher Mowla

(M2 U Fw2 r2 d r2 Uw2 S') M (S Uw2 r2 d' r2 Fw2 U' M2)

(27,17)

N

Christopher Mowla

(M2 U Fw2 r2 d' r2 Uw2 S') M (S Uw2 r2 d r2 Fw2 U' M2)

(27,17)

N

Christopher Mowla

Two corner swaps

Underscore ( _ ) permutation



F2 R2 B' D' B R2 F' U Fw2 F L2 f2 Lw2 f2 l2 U'

(24,16)

N

Clément Gallet

L' U Lw2 L B2 l2 Bw2 l2 b2 U' L2 F2 R' D' R F2

(24,16)

N

Clément Gallet

y' B2 L U L' B2 R D' r2 F2 r2 Fw2 r2 f2 R D R2 y

(25,16)

PKF10

N

Per Kristen Fredlund

R U' R B2 L' D L B2 R2 U r2 F2 r2 Fw2 r2 f2

(25,16)

N

F2 L2 B D B' L2 F U' F f2 U2 L2 b2 L2 U2 f2 U

(26,17)

N

Clément Gallet

R U' 3Lw U2' L' B L U2' 3Lw2' B Lw2 F2 U2' l2' U2' F2 Lw2' x'

(27,17)

N

Clément Gallet &Christopher Mowla

(F2 D R2 U' R2 F2 U' F2 U F2 D')(r2 U2 r2 Uw2 r2 u2)

(29,17)

N

Per Kristen Fredlund

z r2 U2 R' U2 R' U2 R x U2 Rw2 U2 B2' L U2 L' U2 Rw2 U2' z' y'

(29,17)

N

Christopher Mowla

y' R Uw Rw' U' Rw' F r R' Fw' U f r U Rw' Uw' R Uw r Uw' y

(19,19)

N

Ben Whitmore

z' Rw2 x U2 R' U2 x' U2 R' U2 R U2 L' x U2 Rw2 U2 Rw2 U2 Rw2 x' U2 Rw2 z U

(33,19)

Y

Christopher Mowla

r2 U2 r2 Uw2 r2 u2 y' R U R' U' R' F R2 U' R' U' R U R' F' y

(27,20)

N

y' (R U R' F') U' r2 U2 r2 Uw2 r2 Uw2 U' (R U R' U') R' F R2 U' R' U' y

(28,22)

N

Stefan Lidström

z' x Rw2 R' U2 Rw' U2 Rw2 U2 Rw2 U2 L Rw' U2 R' U2 R U2 L' R U2 R U2 Rw2 x' z

(35,22)

Y

Christopher Mowla

z' x Rw2 R' U2 Rw U2 Rw2 U2 Rw2 U2 L Rw U2 R' U2 R U2 L' R U2 R U2 Rw2 x' z

(35,22)

Y

Christopher Mowla

z' (Rw U L U' Rw' U L' U' F)9 z

(81,81)

Y

Christopher Mowla

Slash (\) permutation



Rw2 f2 U2 Fw2 D Rw2 U2 Fw2 U' Fw2 L2 U2 B2 Lw2 U

(27,15)

N

Clément Gallet

Rw2 f2 U2 Bw2 D' Rw2 U2 Bw2 U Fw2 R2 D2 B2 Lw2 U

(27,15)

v1

N

Clément Gallet

Rw2 f2 U2 Fw2 U' Rw2 U2 Fw2 U Fw2 R2 U2 F2 Rw2 U

(27,15)

v2

N

Michael Fung

Rw2 f2 D2 Fw2 U' Rw2 U2 Bw2 U Bw2 R2 U2 F2 Rw2 U'

(27,15)

N

Clément Gallet

Rw2 f2 D2 Fw2 D' Rw2 U2 Bw2 U Bw2 L2 U2 B2 Lw2 U

(27,15)

N

Clément Gallet

Rw2 f2 D2 Bw2 U Rw2 U2 Fw2 U' Bw2 L2 D2 F2 Rw2 U'

(27,15)

N

Clément Gallet

Fw2 r2 D2 Lw2 U Bw2 D2 Lw2 D' Rw2 B2 U2 L2 Bw2 U'

(27,15)

N

Clément Gallet

Fw2 r2 D2 Rw2 U Bw2 D2 Rw2 D' Rw2 F2 D2 L2 Bw2 U

(27,15)

N

Clément Gallet

Rw2 F2 U2 y Rw2 U' Rw2 U D Lw2' U' Lw2' y' r2 U2 F2 Rw2 U

(27,16)

N

Christopher Mowla

Rw2 F2 U2 y Lw2' U Lw2' U' D' Rw2 U Rw2 y' r2 U2 F2 Rw2 U

(27,16)

N

Christopher Mowla

r2 F2 L' U2 Lw2 F2 R' D2 R2 U2 B2 R D2 B2 Rw2 U2

(29,16)

Y

Clément Gallet

y' z (R2' Uw2' R2 U R2') y (R2 U2 R2' U' R2) y' (R2' Uw2' R2 3Dw' R2) Uw2 U2 x

(29,16)

reCornerX(BD) v1

N

reThinking the Cube

y' U2 Rw2 U2 R U2 F2 R2 F2 R' F2 U2 Rw2 U2 R' F2 r2 y

(29,16)

reCornerX(BD) v2

N

reThinking the Cube

y' Rw' r' S2 U R U' B2 D L' D Lw l D2 F2 Lw l D2 z2 y

(22,17)

N

Frédérick Badie &Clément Gallet

y' R2 D Rw2 U2' B2 r2 B2 U2' Lw2 R U' 3Lw U2 3Rw' U 3Rw U2 x' y

(26,17)

N

Mario Laurent

F2 L' U2 L R U2 R' F2 Lw2 F2 U2 Lw2' L' U2 F2 Lw2 U2

(28,17)

Y

Clément Gallet

y' r2 F2 R' F2 U2 R2 U2 R' F2 R2 U2 Rw' r' U2 F2 Rw2 U2 y

(30,17)

N

Christopher Mowla

y' r2 B2 R' U2 Rw2 U2 B2 R' B2 r2 U2 Rw2 B2 U2 R' U2 Rw2 y

(31,17)

N

Christopher Mowla

L2 u2 F2 U2 L2 u2 F2 U' F2 L2 U2 L2 U L2 U2 F2 u2 U

(32,18)

N

Clément Gallet

y' R Rw2 F2 L U L' B D L D' B2 U B r2 U2 F2 Rw2 R' U2 y

(25,19)

FB14

N

Frédérick Badie

y' r2 U2 r2 Uw2 r2 Uw2 L' U R' U2 L U' L' R U R' U2 L U' R U y

(29,21)

N

Rw2 R U2 R U2 R' U2 Rw U2 Rw2 U2 Rw2 U2 L Rw U2 R' U2 R U2 L' Rw2

(35,22)

Y

Christopher Mowla

Rw2 R U2 R U2 R' U2 Rw' U2 Rw2 U2 Rw2 U2 L Rw' U2 R' U2 R U2 L' Rw2

(35,22)

Y

Christopher Mowla

l2 U2 F2 l R U2 R' U2 L2 F2 L F2 Lw F2 U2 Lw2 U2 F2 U2 F2 L' U2

(38,22)

N

Christopher Mowla

(F R U') (R' U' R U) (R' F') U r2 U2 r2 Uw2 r2 Uw2 U (R U R' U') (R' F R F')

(30,25)

N

Stefan Lidström

F' (Rw U' L' U Rw' U' L U2)9 F

(83,74)

N

Christopher Mowla




Two corner swaps (only 2 supercube center piece exchange)

Underscore ( _ ) permutation



x' Uw l S' D2 B d2 B' D2 S L F l2 F' D' Fw' D' Fw D2 Lw' Uw' Rw F' Lw'

(28,23)

N

Tom Rokicki

U Rw2 Bw2 Rw F Rw' Bw2 Rw F' L2 r U' b2 U L' F' L U' b2 U L' F M x

(29,23)

N

Christopher Mowla

z' Fw' L2 Uw F' U R U' F r2 F' U R' U' F r2 Fw Uw' L Uw Fw' Uw' L Fw L z

(27,24)

N

Christopher Mowla

z' Fw' L2 Uw r2 F' U R U' F r2 F' U R' U' F Fw Uw' L Uw Fw' Uw' L Fw L z

(27,24)

N

Christopher Mowla

y' (Rw' R2 U' R' U R' Rw U' R U2' R' U' R' Rw U R Rw' U' Rw' F Rw2 U' Rw' U') (Rw U Rw' F') y

(27,25)

Y

Robert Yau

y' Rw' U' R U r U' R' U R2 U R' U' r U r' F' Rw U Rw' U' Rw' F Rw2 U' Rw' U' y

(28,26)

N

Robert Yau

y' Rw' U' R U r U' R' U R2 U R' U' r U r' U' Rw' F Rw2 U' Rw' U' Rw U Rw' F' y

(28,26)

N

Robert Yau

y' Rw' U' R U r U' R' U R2 U R' U' r U R U' Lw Uw2 Rw' U' Rw Dw2 Rw' U Rw' F' x y'

(29,26)

N

Robert Yau

U Rw2 Bw2 Rw F Rw' Bw2 Rw F' D' f' D F2 D' f D F2 Rw L F' L' f' L F L' f

(31,26)

N

Chris Hardwick

y' Rw U Rw' U' Rw' F Rw2 U' Rw' U' Rw U Rw' R' F' r' F R F' r2 B' R B r' B' R' B y

(29,27)

N

y' Rw U R' U' r' U R U' R2 U' R U r' U' R' Lw U Lw' U' Rw U Lw U' Lw' Rw' U Rw U y

(29,28)

N

Christopher Mowla

Slash (\) permutation



F r2 F' U R' U' F r2 Fw Uw' L Uw Fw' Uw' L Fw L Fw' L2 Uw F' U R U'

(27,24)

N

Christopher Mowla

Fw R Fw' U' Fw2 Rw2 Fw' R' Fw Rw2 Fw' U2 R2 U Fw L2 Fw' U' R2 U' R' Fw L2 Fw2

(33,24)

N

Tom Rokicki

F Rw' F Rw2 U' Rw' U' Rw' U' R U r U' R' U R2 U R' U' r U r' F' Rw U Rw' U' F'

(30,28)

N

Robert Yau

Two X-center piece swap algorithms



• These algorithms, and other algorithms like them, are used to make two corner swap algorithms which only swap two X-center pieces on the 4x4x4 supercube.

y Rw U' Lw Uw2 Rw' U Rw Uw2 x Rw2 U y'

(13,10)

Y

Fw' U' Rw Fw Rw' U' Rw Fw' Rw' U2 Fw U'

(13,12)

Y

Christopher Mowla

Rw U Rw' U' Rw' F Rw2 U' Rw' U' Rw U Rw' F'

(15,14)

Y

Rw' U' Lw U Lw' U' Rw U Lw U' Lw' Rw' U Rw U

(15,15)

Y

Christopher Mowla




Two corner swap and a dedge 3-cycle

D permutation

Da



z2 F2 R2 Dw2 F2 U' R2 Uw2 U R2 U F2 Uw2 U M2 F2 U'

(25,16)

Y

Clément Gallet

F2 U Rw2 U' B2 U' F2 U B2 U' r2 F2 U2 Rw2 U' F2 U'

(27,17)

FB07

N

Frédérick Badie

y2 U' R2 b2 l2 Fw2 l2 F2 r2 L D L' B2 L D' L' F2 R2

(27,17)

N

Clément Gallet

R2 U' f2 D2 L2 b2 D' R2 D L2 D' R2 D' f2 U R2 U'

(27,17)

N

Clément Gallet

y2 U' Uw' R U R' Uw' Rw2 U2' Rw2 R D' R U R' D R U Rw2 Uw R U' R' Uw y2

(27,23)

N

xyzzy

Db



U F2 M2 Uw2 U' F2 U' R2 Uw2 U' R2 U F2 Dw2 R2 F2 z2

(25,16)

Y

Clément Gallet

U F2 U Rw2 U2 F2 r2 U B2 U' F2 U B2 U Rw2 U' F2

(27,17)

FB07

N

Frédérick Badie

R2 F2 L D L' B2 L D' L' r2 F2 l2 Fw2 l2 b2 R2 U y2

(27,17)

N

Clément Gallet

U R2 U' f2 D R2 D L2 D' R2 D b2 L2 D2 f2 U R2

(27,17)

N

Clément Gallet

y2 Uw' R U R' Uw' Rw2 U' R' D' R U' R' D R' Rw2 U2 Rw2 Uw R U' R' Uw U y2

(27,23)

N

xyzzy

K permutation

Ka



U R U' L U2 R' U L l2 F2 Lw2 Fw2 l2 Fw2

(20,14)

N

Clément Gallet

Dw2 R2 D B2 Dw2 D' B2 D' R2 Dw2 D' B2 L2 U' L2

(23,15)

Y

Clément Gallet

u2 r2 Uw2 r2 U2 Rw2 D R D' R F2 L' U L F2

(23,15)

PKF06

N

Per Kristen Fredlund

d2 D' B2 L2 U d2 L2 D' B2 U R2 U R2 Dw2 L2

(24,15)

N

Clément Gallet

y (Rw2' D' Rw U2 Rw' D Rw U2 Rw) U (Rw U2 Rw D Rw' U2 Rw D' Rw2') U2 y'

(27,20)

Y

Daniel Sheppard

R U R' F' R U R' U' R' F R2 U' R' U (r2 U2 r2 Uw2 r2 Uw2)

(27,20)

N

Stefan Lidström




Kb



Fw2 l2 Fw2 Lw2 F2 l2 L' U' R U2 L' U R' U'

(20,14)

N

Clément Gallet

L2 U L2 B2 Dw2 D R2 D B2 Dw2 D B2 D' R2 Dw2

(23,15)

Y

Clément Gallet

F2 L' U' L F2 R' D R' D' Rw2 U2 r2 Uw2 r2 u2

(23,15)

PKF06

N

Per Kristen Fredlund

L2 Dw2 R2 U' R2 U' B2 D L2 d2 U' L2 B2 D d2

(24,15)

N

Clément Gallet

y' (Rw2 D Rw' U2 Rw D' Rw' U2 Rw') U' (Rw' U2 Rw' D' Rw U2 Rw' D Rw2') y U2

(27,20)

Y

Daniel Sheppard

(r2 U2 r2 Uw2 r2 Uw2 U2) R U R' F' R U R' U' R' F R2 U' R' U'

(27,20)

N

Stefan Lidström

P permutation

Pa



U' z' U2 B2 Lw2 U2 L' B2 Lw2' L B2 L U2 Lw2' L U2 z

(23,15)

CG05

Y

Clément Gallet

y U L2 3Uw' L2 U L2 y u2 l2 Uw2 l2 U2 Lw2 3Uw L2 U' L2 y2

(27,16)

N

Stefan Pochmann

R2 U R2 U' R2 F2 U' (Fw2 r2)2 F2 r2 D R2 D'

(27,16)

PKF05

N

Per Kristen Fredlund

y2 L' U' L U L F' L2 U L (r2 U2 r2 Uw2 r2 Uw2) U' L' U' L F y2

(26,20)

N

Stefan Lidström

Pb



z' U2 Lw2 L' U2 L' B2 Lw2' L' B2 L U2 Lw2' B2 U2 z U

(23,15)

CG05

Y

Clément Gallet

y2 L2 U L2 3Uw' Lw2 U2 l2 Uw2 l2 u2 y' L2 U' L2 3Uw L2 U' y'

(27,16)

N

Stefan Pochmann

R2 D' r2 F2 (r2 Fw2)2 U F2 R2 U R2 U' R2 D

(27,16)

PKF05

N

Per Kristen Fredlund

R U R' U (r2 U2 r2 Uw2 r2 Uw2) R' F R2 U' R' U' R U R' F'

(27,20)

N

Stefan Lidström

b permutation

Ba



z' U2 Lw2' L U2 L F2 Lw2' L F2 L' U2 Lw2 F2 U2 z U'

(23,15)

CG05

Y

Clément Gallet

R2' U' R2 3Uw Rw2 U2 r2 Uw2 r2 u2 y R2 U R2' 3Uw' R2 U y'

(27,16)

N

Stefan Pochmann

R2 D r2 B2 (r2 Bw2)2 U' B2 R2 U' R2 U R2 D'

(27,16)

PKF05

N

Per Kristen Fredlund

y2 L' U' L U' (r2 U2 r2 Uw2 r2 Uw2) L F' L2 U L U L' U' L F y2

(27,20)

N

Stefan Lidström

Bb



U z' U2 F2 Lw2' U2 L F2 Lw2 L' F2 L' U2 Lw2 L' U2 z

(23,15)

CG05

Y

Clément Gallet

y U' R2 3Uw R2 U' R2 y' u2 r2 Uw2 r2 U2 Rw2 3Uw' R2 U R2

(27,16)

N

Stefan Pochmann

R2 U' R2 U R2 B2 U (Bw2 r2)2 B2 r2 D' R2 D

(27,16)

PKF05

N

Per Kristen Fredlund

R U R' U' R' F R2 U' R' (r2 U2 r2 Uw2 r2 Uw2) U R U R' F'

(25,20)

N

Stefan Lidström

Q permutation

Qa



y' z U2 R' U2 Rw2 B2 R U2 Rw2 R' U2 R2 B2 Rw2 R U2 z' y U

(25,16)

CG11

Y

Clément Gallet

B2 L2 U2 L2 d2 D' R2 U2 F2 d2 D B2 U2 R2 U L2 R2 u2

(30,17)

N

Clément Gallet

B2 d2 L2 U2 L2 D' R2 U2 F2 d2 D B2 U2 R2 U L2 R2 u2

(30,17)

N

Clément Gallet

R' U' R F2 R' U R U F2 U' F2 U' Fw2 U2 f2 Uw2 f2 u2

(27,18)

PKF11

N

Per Kristen Fredlund

y F' U' F L2 F' U F U L2 U' L2 U' Lw2 U2 l2 Uw2 l2 u2 y'

(27,18)

N

Per Kristen Fredlund

Qb



y' z U2 R U2 Rw2 F2 R' U2 Rw2 R U2 R2 F2 Rw2 R' U2 z' y U

(25,16)

CG11

Y

Clément Gallet

u2 F2 B2 U B2 U2 R2 D d2 L2 U2 B2 D' d2 F2 U2 F2 R2

(30,17)

N

Clément Gallet

u2 F2 B2 U B2 U2 R2 D d2 L2 U2 B2 D' F2 U2 F2 d2 R2

(30,17)

N

Clément Gallet

u2 l2 Uw2 l2 U2 Lw2 U' L2 U' L2 3Dw L U L' F2 L U' L' y

(27,18)

PKF11

N

Per Kristen Fredlund

y2 u2 r2 Uw2 r2 U2 Rw2 U' R2 U' R2 U F U F' R2 F U' F' y2

(27,18)

N

Per Kristen Fredlund




Two corner swap and a dedge 2 2-cycle

C permutation

Ca



y' Rw2 F2 U2 r2 U' B2 U' F2 U B2 U' Rw2 U y

(21,13)

N

Clément Gallet

Lw2 U F2 U' B2 U F2 U l2 U2 B2 Lw2 U'

(21,13)

AO09

N

Alexander Ooms

U Lw2 B2 U2 l2 U' F2 U' B2 U F2 U' Lw2

(21,13)

Inv of AO09

N

Alexander Ooms

y' Rw r B' R F2 R' B R U2 r2 U2 F2 Rw2 U y

(20,14)

N

Frédérick Badie &Clément Gallet

r2 F2 R2 D2 l2 D L2 D R2 D' L2 D F2 r2 U'

(25,15)

N

Clément Gallet

U Dw2 Lw2 B2 U F2 U' B2 U F2 U l2 U2 Lw2 Dw2

(24,15)

N

Frédérick Badie

l r U2 x U' L U' R2 U L' U' Rw2 U2 x' U2 l' r' U'

(21,16)

N

Christopher Mowla

Cb



y Rw2 B2 U2 r2 U F2 U B2 U' F2 U Rw2 U' y'

(21,13)

N

Clément Gallet

Lw2 U' B2 U F2 U' B2 U' l2 U2 F2 Lw2 U

(21,13)

AO09

N

Alexander Ooms

U' Lw2 F2 U2 l2 U B2 U F2 U' B2 U Lw2

(21,13)

Inv of AO09

N

Alexander Ooms

y Rw2 R F R' B2 R F' R' U2 r2 U2 B2 Rw2 U' y'

(20,14)

N

Frédérick Badie &Clément Gallet

r2 B2 R2 D2 l2 D' L2 D' R2 D L2 D' B2 r2 U

(25,15)

N

Clément Gallet

U Dw2 Rw2 B2 U' F2 U B2 U' F2 U' r2 U2 Rw2 u2 y2

(24,15)

N

Frédérick Badie

l' r' U2 x' U L' U R2 U' L U Rw2 U2 x U2 l r U

(21,16)

N

Christopher Mowla

H permutation



L2 Uw2 L2 U' L2 D' B2 R2 d2 U' R2 D B2 u2 U

(24,15)

N

Clément Gallet

L2 Uw2 L2 U L2 3Uw L2 F2 U z l2 F2 L' U2 Lw2 z' U y

(25,15)

CG08

N

Clément Gallet

R2 D' F2 D R2 B2 D L2 D' Bw2 U2 b2 Uw2 b2 Uw2 U'

(26,16)

N

Kenneth Gustavsson

Lw2 U F2 L D' L D L' U L U' L' U l2 U2 F2 Lw2

(23,17)

AO08

N

Alexander Ooms

Rw2 U2 r2 Uw2 r2 u2 D' F2 U F2 R2 U R2 U' R2 D

(27,16)

N

Per Kristen Fredlund

Theta (θ) permutation



R2 U B2 Uw2 R2 U' B2 Uw2 U B2 U2 R2 Uw2 U' R2 U'

(25,16)

Y

Clément Gallet

z F2 Rw2 R' F2 R2 U2 Rw2 R U2 R' F2 Rw2 U2 R F2 z' U

(25,16)

CG12

Y

Clément Gallet

u2 D F2 U R2 U2 u2 F2 D' F2 U L2 D2 B2 U' u2 R2 y2

(29,17)

N

Clément Gallet

R' B' R F D' R' D B R Fw' f' R2 f2 Rw2 f2 Rw2 U R2

(24,18)

N

Per Kristen Fredlund &Clément Gallet

F U F' R2 F U' F' U' R2 U f2 R2 (f2 Rw2)2 U R2

(27,18)

PKF12

N

Per Kristen Fredlund

F R U' R' U' R U R' F' (r2 U2 r2 Uw2 r2 u2) R U R' U' R' F R F'

(29,23)

N

Stefan Lidström

Xi (Ξ) permutation



Uw d B2 D M2 U L2 B2 U B2 Uw2 R2 D F2 Uw2 3Lw2 y'

(26,16)

N

Clément Gallet

R2 Uw2 U' M2 U R2 F2 Uw2 U' F2 U2 3Rw2 Uw2 U' F2 U2 R2 U

(28,18)

Y

Clément Gallet

Uw2 Lw2 U' B' U B U2 F U' F' U F' L2 F Lw2 U2 Lw2 Uw2 U'

(26,19)

FB13

Y

Frédérick Badie

U L2 U2 F2 D B2 u2 D' F2 L2 F2 D B2 D u2 R2 D2 F2 d2

(33,19)

N

Clément Gallet

Dw2 l' r' F' L' B' L F L' R' F' R B R' F' l2 F2 Lw2 Bw2 F' x

(24,20)

N

Frédérick Badie &Clément Gallet

U Uw2 Rw2 U2 Rw2 R' D R' U' R (U2 D') R' D' R U2 R' (U' D) Rw2 Uw2

(25,20)

N

xyzzy

z Uw2 r2 Uw2 r2 U2 (Lw' Rw' z') U R' U R U' R B2 R' U R' U' R U B2 U y'

(29,21)

N

Matthew Sheerin

z Uw2 r2 Uw2 r2 U2 (Lw' Rw' z') U R' U R U' R B2 R' U R' U' R (z 3Lw) U2 z' U

(29,21)

N

Matthew Sheerin




The shortest PLL parity fixes in BQTM

• Note that these are the only PLL parity algorithms in this section which only contain two inner layer slice quarter turns.
  • This alg set is for theoretical purposes only, as they scramble the pseudo 3x3x3 state of the 4x4x4.

Rw U D L2 U D S2 Rw

(10,8)

Y

Christopher Mowla

Rw' U D L2 U D S2 Rw'

(10,8)

Y

Christopher Mowla

Rw U' D' L2 U' D' S2 Rw

(10,8)

Y

Christopher Mowla

Rw' U' D' L2 U' D' S2 Rw'

(10,8)

Y

Christopher Mowla

Rw U D R2 U D S2 Rw

(10,8)

Y

Christopher Mowla

Rw' U D R2 U D S2 Rw'

(10,8)

Y

Christopher Mowla

Rw U' D' R2 U' D' S2 Rw

(10,8)

Y

Christopher Mowla

Rw' U' D' R2 U' D' S2 Rw'

(10,8)

Y

Christopher Mowla

Rw U D S2 U' D' S2 Rw'

(10,8)

Y

Christopher Mowla

Rw' U D S2 U' D' S2 Rw

(10,8)

Y

Christopher Mowla

Rw U' D' S2 U D S2 Rw'

(10,8)

Y

Christopher Mowla

Rw' U' D' S2 U D S2 Rw

(10,8)

Y

Christopher Mowla




• Taking the first algorithm, for example, and solving back the outer layers, we see that these short parity fixes were made from a supercube safe unoriented 4-cycle of dedges in M:

r U D L2 U D S2 r S2 D' U' L2 D' U'

(18,14)

N

Christopher Mowla




• However, note that this is not the shortest solution to this 4x4x4 position, as the following is a shorter supercube safe algorithm.

(l E2 r' E2)3 //Safe

(18,12)

N

Christopher Mowla




Pure Flips (Single Parity)

• All algorithms in this "pure flips" section are single parity algorithms.

One dedge flip



Shortest half turn move algorithms

• It turns out that the shortest algorithms in the half turn move metric (in both the single slice turn metric and the block turn metric) have a length of 15.
• There are exactly 288 algorithms (with a length of 15 half turns) which flip the upper front dedge and affect the top or front centers of the 4x4x4 supercube.

They all happen to be 25 BQTM and single slice turn-based algorithms (algorithms which predominately consist of single slice turns).

All 288 (25,15) algorithms can be obtained by running the 3x3x3 Classic Setup through a 3x3x3 solver like Cube Explorer and choosing all outputted 15f algorithms which are confined to the move set <U2,F2,D2,B2,L,R>.


Simply convert the turns of the L and R faces into their corresponding inner slice turns and select the resulting algorithms which flip a single dedge on the 4x4x4.

• These 288 (25,15) algorithms have been divided into 36 clusters. There are exactly 8 algorithms in each cluster.
• Every cluster essentially represents a single unique algorithm path, as move transformations (from cube rotations, mirrors, and inverses) can be done to any algorithm in a given cluster to obtain the 7 other algorithms in its cluster. To avoid human error, a custom program was written to perform these move transformations to the first algorithm listed in each cluster and to label the remaining 7 algorithms in its cluster as the transformation they are of the first algorithm.
• Even further, several of these clusters are categorized as being in the same group number. This is because, although transformations are strictly restricted to being in the 36 clusters, some of these clusters of algorithms are directly related to each other by either conjugation and/or by cyclic shifting (a special type of conjugation).

Take for example the first algorithm in the first cluster under Group 1 of non-symmetrical algorithms, r' U2 l F2 l' F2 r2 U2 r U2 r' U2 F2 r2 F2.

Shifting ("rotating the algorithm's moves") 5 times in a clockwise manner gives: F2 r2 U2 r U2 r' U2 F2 r2 F2 r' U2 l F2 l'.

This is a single dedge flip of the upper back dedge.

Rotating about y2 gives B2 l2 U2 l U2 l' U2 B2 l2 B2 l' U2 r B2 r', which is the third algorithm in the first cluster under Group 3 of non-symmetrical algorithms.

• This is why Group 3 is called "Group 3: Cyclic Shift of Group 1", for example.

• Even further, all unique non-symmetrical algorithms are directly related to each other by transformations (resulting from inserting cube rotations in between moves of the algorithms). The same holds true for all unique symmetrical algorithms.

• This implies that, despite that there are 288 15 half turn move algorithms which solve the upper front dedge pure flip case, there are actually only two fundamentally unique (25,15) algorithm paths: the symmetrical approach and the non-symmetrical approach.

• This was shown by Christopher Mowla in 2012.

Additional Terminology

An algorithm is called symmetrical if its first and last moves are inverses of each other (or if it can be rewritten as such, should its first and/or last move cancel with the second and/or second to last move in the algorithm, respectively). Symmetrical algorithms are conjugates.

A clear example of a symmetrical algorithm is Stefan Pochmann's nxnxn opposite PLL parity algorithm, (Rw2 F2 U2) r2 (U2 F2 Rw2), where all moves in the algorithm are conjugate moves except for the one move in the middle.

Although symmetrical algorithms are technically conjugates of non-symmetrical algorithms, non-symmetrical algorithms are algorithms which are solely the result of a composition of one or more separate algorithm pieces, which all together accomplish the desired task. (No "conjugate assistance" is used.)

In practice, human creation of symmetrical algorithms requires more trial and error of different paths in both creation of the base (the base is defined as the move sequence B in A B A') and final setup moves, whereas the creation of non-symmetrical algorithms requires having knowledge of forming different pieces individually and knowing how to combine them.

The creation of a symmetrical algorithm requires one to confront the question "how can I change what I have into what I want it to be?" (video example), whereas the creation of a non-symmetrical algorithm requires one to confront the question "I have already achieved a portion of the task, but what is the missing piece?" (written example).

(Non-symmetrical algorithms)

Group 1

r' U2 l F2 l' F2 r2 U2 r U2 r' U2 F2 r2 F2

(25,15)

Alg.1(v1)

Y

Frédérick_Badie

l U2 r' F2 r F2 l2 U2 l' U2 l U2 F2 l2 F2

(25,15)

Mirror[Alg.1(v1)]

Y

Frédérick_Badie

F2 r2 F2 U2 r U2 r' U2 r2 F2 l F2 l' U2 r

(25,15)

Inverse[Alg.1(v1)]

Y

Frédérick_Badie

l' F2 r U2 r' U2 l2 F2 l F2 l' F2 U2 l2 U2

(25,15)

Rotation[Alg.1(v1)]

Y

Frédérick_Badie

F2 l2 F2 U2 l' U2 l U2 l2 F2 r' F2 r U2 l'

(25,15)

Mirror[Inverse[Alg.1(v1)]]

Y

Frédérick_Badie

r F2 l' U2 l U2 r2 F2 r' F2 r F2 U2 r2 U2

(25,15)

Mirror[Rotation[Alg.1(v1)]]

Y

Frédérick_Badie

U2 l2 U2 F2 l F2 l' F2 l2 U2 r U2 r' F2 l

(25,15)

Inverse[Rotation[Alg.1(v1)]]

Y

Frédérick_Badie

U2 r2 U2 F2 r' F2 r F2 r2 U2 l' U2 l F2 r'

(25,15)

Mirror[Inverse[Rotation[Alg.1(v1)]]]

Y

Frédérick_Badie

r' U2 r U2 l' U2 l2 F2 r F2 l' U2 F2 r2 F2

(25,15)

Alg.1(v2)

Y

Frédérick_Badie

l U2 l' U2 r U2 r2 F2 l' F2 r U2 F2 l2 F2

(25,15)

Mirror[Alg.1(v2)]

Y

Frédérick_Badie

F2 r2 F2 U2 l F2 r' F2 l2 U2 l U2 r' U2 r

(25,15)

Inverse[Alg.1(v2)]

Y

Frédérick_Badie

l' F2 l F2 r' F2 r2 U2 l U2 r' F2 U2 l2 U2

(25,15)

Rotation[Alg.1(v2)]

Y

Frédérick_Badie

F2 l2 F2 U2 r' F2 l F2 r2 U2 r' U2 l U2 l'

(25,15)

Mirror[Inverse[Alg.1(v2)]]

Y

Frédérick_Badie

r F2 r' F2 l F2 l2 U2 r' U2 l F2 U2 r2 U2

(25,15)

Mirror[Rotation[Alg.1(v2)]]

Y

Frédérick_Badie

U2 l2 U2 F2 r U2 l' U2 r2 F2 r F2 l' F2 l

(25,15)

Inverse[Rotation[Alg.1(v2)]]

Y

Frédérick_Badie

U2 r2 U2 F2 l' U2 r U2 l2 F2 l' F2 r F2 r'

(25,15)

Mirror[Inverse[Rotation[Alg.1(v2)]]]

Y

Frédérick_Badie

r B2 r' U2 r U2 r2 F2 r' D2 l D2 B2 r2 F2

(25,15)

Alg.1(v3)

Y

Frédérick_Badie

l' B2 l U2 l' U2 l2 F2 l D2 r' D2 B2 l2 F2

(25,15)

Mirror[Alg.1(v3)]

Y

Frédérick_Badie

F2 r2 B2 D2 l' D2 r F2 r2 U2 r' U2 r B2 r'

(25,15)

Inverse[Alg.1(v3)]

Y

Frédérick_Badie

l D2 l' F2 l F2 l2 U2 l' B2 r B2 D2 l2 U2

(25,15)

Rotation[Alg.1(v3)]

Y

Frédérick_Badie

F2 l2 B2 D2 r D2 l' F2 l2 U2 l U2 l' B2 l

(25,15)

Mirror[Inverse[Alg.1(v3)]]

Y

Frédérick_Badie

r' D2 r F2 r' F2 r2 U2 r B2 l' B2 D2 r2 U2

(25,15)

Mirror[Rotation[Alg.1(v3)]]

Y

Frédérick_Badie

U2 l2 D2 B2 r' B2 l U2 l2 F2 l' F2 l D2 l'

(25,15)

Inverse[Rotation[Alg.1(v3)]]

Y

Frédérick_Badie

U2 r2 D2 B2 l B2 r' U2 r2 F2 r F2 r' D2 r

(25,15)

Mirror[Inverse[Rotation[Alg.1(v3)]]]

Y

Frédérick_Badie

r B2 r' U2 r U2 l2 B2 r' U2 l U2 F2 l2 F2

(25,15)

Alg.1(v4)

Y

Frédérick_Badie

l' B2 l U2 l' U2 r2 B2 l U2 r' U2 F2 r2 F2

(25,15)

Mirror[Alg.1(v4)]

Y

Frédérick_Badie

F2 l2 F2 U2 l' U2 r B2 l2 U2 r' U2 r B2 r'

(25,15)

Inverse[Alg.1(v4)]

Y

Frédérick_Badie

l D2 l' F2 l F2 r2 D2 l' F2 r F2 U2 r2 U2

(25,15)

Rotation[Alg.1(v4)]

Y

Frédérick_Badie

F2 r2 F2 U2 r U2 l' B2 r2 U2 l U2 l' B2 l

(25,15)

Mirror[Inverse[Alg.1(v4)]]

Y

Frédérick_Badie

r' D2 r F2 r' F2 l2 D2 r F2 l' F2 U2 l2 U2

(25,15)

Mirror[Rotation[Alg.1(v4)]]

Y

Frédérick_Badie

U2 r2 U2 F2 r' F2 l D2 r2 F2 l' F2 l D2 l'

(25,15)

Inverse[Rotation[Alg.1(v4)]]

Y

Frédérick_Badie

U2 l2 U2 F2 l F2 r' D2 l2 F2 r F2 r' D2 r

(25,15)

Mirror[Inverse[Rotation[Alg.1(v4)]]]

Y

Frédérick_Badie

r B2 l' B2 l U2 r2 F2 l' F2 r U2 F2 l2 F2

(25,15)

Alg.1(v5)

Y

Frédérick_Badie

l' B2 r B2 r' U2 l2 F2 r F2 l' U2 F2 r2 F2

(25,15)

Mirror[Alg.1(v5)]

Y

Frédérick_Badie

F2 l2 F2 U2 r' F2 l F2 r2 U2 l' B2 l B2 r'

(25,15)

Inverse[Alg.1(v5)]

Y

Frédérick_Badie

l D2 r' D2 r F2 l2 U2 r' U2 l F2 U2 r2 U2

(25,15)

Rotation[Alg.1(v5)]

Y

Frédérick_Badie

F2 r2 F2 U2 l F2 r' F2 l2 U2 r B2 r' B2 l

(25,15)

Mirror[Inverse[Alg.1(v5)]]

Y

Frédérick_Badie

r' D2 l D2 l' F2 r2 U2 l U2 r' F2 U2 l2 U2

(25,15)

Mirror[Rotation[Alg.1(v5)]]

Y

Frédérick_Badie

U2 r2 U2 F2 l' U2 r U2 l2 F2 r' D2 r D2 l'

(25,15)

Inverse[Rotation[Alg.1(v5)]]

Y

Frédérick_Badie

U2 l2 U2 F2 r U2 l' U2 r2 F2 l D2 l' D2 r

(25,15)

Mirror[Inverse[Rotation[Alg.1(v5)]]]

Y

Frédérick_Badie

Group 2: F2 move conjugation + rotation of Group 1

r2 U2 B2 r B2 r' B2 r2 U2 l U2 l' B2 r U2

(25,15)

Alg.2(v1)

Y

l2 U2 B2 l' B2 l B2 l2 U2 r' U2 r B2 l' U2

(25,15)

Mirror[Alg.2(v1)]

Y

U2 r' B2 l U2 l' U2 r2 B2 r B2 r' B2 U2 r2

(25,15)

Inverse[Alg.2(v1)]

Y

l2 F2 D2 l D2 l' D2 l2 F2 r F2 r' D2 l F2

(25,15)

Rotation[Alg.2(v1)]

Y

U2 l B2 r' U2 r U2 l2 B2 l' B2 l B2 U2 l2

(25,15)

Mirror[Inverse[Alg.2(v1)]]

Y

r2 F2 D2 r' D2 r D2 r2 F2 l' F2 l D2 r' F2

(25,15)

Mirror[Rotation[Alg.2(v1)]]

Y

F2 l' D2 r F2 r' F2 l2 D2 l D2 l' D2 F2 l2

(25,15)

Inverse[Rotation[Alg.2(v1)]]

Y

F2 r D2 l' F2 l F2 r2 D2 r' D2 r D2 F2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.2(v1)]]]

Y

r2 U2 B2 l U2 r' U2 l2 B2 l B2 r' B2 r U2

(25,15)

Alg.2(v2)

Y

l2 U2 B2 r' U2 l U2 r2 B2 r' B2 l B2 l' U2

(25,15)

Mirror[Alg.2(v2)]

Y

U2 r' B2 r B2 l' B2 l2 U2 r U2 l' B2 U2 r2

(25,15)

Inverse[Alg.2(v2)]

Y

l2 F2 D2 r F2 l' F2 r2 D2 r D2 l' D2 l F2

(25,15)

Rotation[Alg.2(v2)]

Y

U2 l B2 l' B2 r B2 r2 U2 l' U2 r B2 U2 l2

(25,15)

Mirror[Inverse[Alg.2(v2)]]

Y

r2 F2 D2 l' F2 r F2 l2 D2 l' D2 r D2 r' F2

(25,15)

Mirror[Rotation[Alg.2(v2)]]

Y

F2 l' D2 l D2 r' D2 r2 F2 l F2 r' D2 F2 l2

(25,15)

Inverse[Rotation[Alg.2(v2)]]

Y

F2 r D2 r' D2 l D2 l2 F2 r' F2 l D2 F2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.2(v2)]]]

Y

r2 D2 F2 l' F2 r U2 r2 B2 r' B2 r D2 r' U2

(25,15)

Alg.2(v3)

Y

l2 D2 F2 r F2 l' U2 l2 B2 l B2 l' D2 l U2

(25,15)

Mirror[Alg.2(v3)]

Y

U2 r D2 r' B2 r B2 r2 U2 r' F2 l F2 D2 r2

(25,15)

Inverse[Alg.2(v3)]

Y

l2 B2 U2 r' U2 l F2 l2 D2 l' D2 l B2 l' F2

(25,15)

Rotation[Alg.2(v3)]

Y

U2 l' D2 l B2 l' B2 l2 U2 l F2 r' F2 D2 l2

(25,15)

Mirror[Inverse[Alg.2(v3)]]

Y

r2 B2 U2 l U2 r' F2 r2 D2 r D2 r' B2 r F2

(25,15)

Mirror[Rotation[Alg.2(v3)]]

Y

F2 l B2 l' D2 l D2 l2 F2 l' U2 r U2 B2 l2

(25,15)

Inverse[Rotation[Alg.2(v3)]]

Y

F2 r' B2 r D2 r' D2 r2 F2 r U2 l' U2 B2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.2(v3)]]]

Y

r2 U2 B2 r B2 l' D2 r2 B2 l B2 l' D2 l U2

(25,15)

Alg.2(v4)

Y

l2 U2 B2 l' B2 r D2 l2 B2 r' B2 r D2 r' U2

(25,15)

Mirror[Alg.2(v4)]

Y

U2 l' D2 l B2 l' B2 r2 D2 l B2 r' B2 U2 r2

(25,15)

Inverse[Alg.2(v4)]

Y

l2 F2 D2 l D2 r' B2 l2 D2 r D2 r' B2 r F2

(25,15)

Rotation[Alg.2(v4)]

Y

U2 r D2 r' B2 r B2 l2 D2 r' B2 l B2 U2 l2

(25,15)

Mirror[Inverse[Alg.2(v4)]]

Y

r2 F2 D2 r' D2 l B2 r2 D2 l' D2 l B2 l' F2

(25,15)

Mirror[Rotation[Alg.2(v4)]]

Y

F2 r' B2 r D2 r' D2 l2 B2 r D2 l' D2 F2 l2

(25,15)

Inverse[Rotation[Alg.2(v4)]]

Y

F2 l B2 l' D2 l D2 r2 B2 l' D2 r D2 F2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.2(v4)]]]

Y

r2 U2 B2 l U2 r' U2 l2 B2 r D2 r' D2 l U2

(25,15)

Alg.2(v5)

Y

l2 U2 B2 r' U2 l U2 r2 B2 l' D2 l D2 r' U2

(25,15)

Mirror[Alg.2(v5)]

Y

U2 l' D2 r D2 r' B2 l2 U2 r U2 l' B2 U2 r2

(25,15)

Inverse[Alg.2(v5)]

Y

l2 F2 D2 r F2 l' F2 r2 D2 l B2 l' B2 r F2

(25,15)

Rotation[Alg.2(v5)]

Y

U2 r D2 l' D2 l B2 r2 U2 l' U2 r B2 U2 l2

(25,15)

Mirror[Inverse[Alg.2(v5)]]

Y

r2 F2 D2 l' F2 r F2 l2 D2 r' B2 r B2 l' F2

(25,15)

Mirror[Rotation[Alg.2(v5)]]

Y

F2 r' B2 l B2 l' D2 r2 F2 l F2 r' D2 F2 l2

(25,15)

Inverse[Rotation[Alg.2(v5)]]

Y

F2 l B2 r' B2 r D2 l2 F2 r' F2 l D2 F2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.2(v5)]]]

Y

Group 3: cyclic shift of Group 1

r B2 r' U2 l B2 l2 B2 U2 l U2 l' U2 l2 B2

(25,15)

Alg.3(v1)

Y

l' B2 l U2 r' B2 r2 B2 U2 r' U2 r U2 r2 B2

(25,15)

Mirror[Alg.3(v1)]

Y

B2 l2 U2 l U2 l' U2 B2 l2 B2 l' U2 r B2 r'

(25,15)

Inverse[Alg.3(v1)]

Y

l D2 l' F2 r D2 r2 D2 F2 r F2 r' F2 r2 D2

(25,15)

Rotation[Alg.3(v1)]

Y

B2 r2 U2 r' U2 r U2 B2 r2 B2 r U2 l' B2 l

(25,15)

Mirror[Inverse[Alg.3(v1)]]

Y

r' D2 r F2 l' D2 l2 D2 F2 l' F2 l F2 l2 D2

(25,15)

Mirror[Rotation[Alg.3(v1)]]

Y

D2 r2 F2 r F2 r' F2 D2 r2 D2 r' F2 l D2 l'

(25,15)

Inverse[Rotation[Alg.3(v1)]]

Y

D2 l2 F2 l' F2 l F2 D2 l2 D2 l F2 r' D2 r

(25,15)

Mirror[Inverse[Rotation[Alg.3(v1)]]]

Y

r B2 r' U2 r D2 r2 U2 F2 l F2 r' D2 r2 B2

(25,15)

Alg.3(v2)

Y

l' B2 l U2 l' D2 l2 U2 F2 r' F2 l D2 l2 B2

(25,15)

Mirror[Alg.3(v2)]

Y

B2 r2 D2 r F2 l' F2 U2 r2 D2 r' U2 r B2 r'

(25,15)

Inverse[Alg.3(v2)]

Y

l D2 l' F2 l B2 l2 F2 U2 r U2 l' B2 l2 D2

(25,15)

Rotation[Alg.3(v2)]

Y

B2 l2 D2 l' F2 r F2 U2 l2 D2 l U2 l' B2 l

(25,15)

Mirror[Inverse[Alg.3(v2)]]

Y

r' D2 r F2 r' B2 r2 F2 U2 l' U2 r B2 r2 D2

(25,15)

Mirror[Rotation[Alg.3(v2)]]

Y

D2 l2 B2 l U2 r' U2 F2 l2 B2 l' F2 l D2 l'

(25,15)

Inverse[Rotation[Alg.3(v2)]]

Y

D2 r2 B2 r' U2 l U2 F2 r2 B2 r F2 r' D2 r

(25,15)

Mirror[Inverse[Rotation[Alg.3(v2)]]]

Y

r B2 r' U2 r D2 l2 D2 B2 l B2 r' U2 l2 B2

(25,15)

Alg.3(v3)

Y

l' B2 l U2 l' D2 r2 D2 B2 r' B2 l U2 r2 B2

(25,15)

Mirror[Alg.3(v3)]

Y

B2 l2 U2 r B2 l' B2 D2 l2 D2 r' U2 r B2 r'

(25,15)

Inverse[Alg.3(v3)]

Y

l D2 l' F2 l B2 r2 B2 D2 r D2 l' F2 r2 D2

(25,15)

Rotation[Alg.3(v3)]

Y

B2 r2 U2 l' B2 r B2 D2 r2 D2 l U2 l' B2 l

(25,15)

Mirror[Inverse[Alg.3(v3)]]

Y

r' D2 r F2 r' B2 l2 B2 D2 l' D2 r F2 l2 D2

(25,15)

Mirror[Rotation[Alg.3(v3)]]

Y

D2 r2 F2 l D2 r' D2 B2 r2 B2 l' F2 l D2 l'

(25,15)

Inverse[Rotation[Alg.3(v3)]]

Y

D2 l2 F2 r' D2 l D2 B2 l2 B2 r F2 r' D2 r

(25,15)

Mirror[Inverse[Rotation[Alg.3(v3)]]]

Y

r B2 l' B2 l D2 l2 D2 B2 r D2 l' D2 r2 B2

(25,15)

Alg.3(v4)

Y

l' B2 r B2 r' D2 r2 D2 B2 l' D2 r D2 l2 B2

(25,15)

Mirror[Alg.3(v4)]

Y

B2 r2 D2 l D2 r' B2 D2 l2 D2 l' B2 l B2 r'

(25,15)

Inverse[Alg.3(v4)]

Y

l D2 r' D2 r B2 r2 B2 D2 l B2 r' B2 l2 D2

(25,15)

Rotation[Alg.3(v4)]

Y

B2 l2 D2 r' D2 l B2 D2 r2 D2 r B2 r' B2 l

(25,15)

Mirror[Inverse[Alg.3(v4)]]

Y

r' D2 l D2 l' B2 l2 B2 D2 r' B2 l B2 r2 D2

(25,15)

Mirror[Rotation[Alg.3(v4)]]

Y

D2 l2 B2 r B2 l' D2 B2 r2 B2 r' D2 r D2 l'

(25,15)

Inverse[Rotation[Alg.3(v4)]]

Y

D2 r2 B2 l' B2 r D2 B2 l2 B2 l D2 l' D2 r

(25,15)

Mirror[Inverse[Rotation[Alg.3(v4)]]]

Y

r' U2 r U2 l' D2 r2 D2 B2 l' D2 r D2 l2 B2

(25,15)

Alg.3(v5)

Y

l U2 l' U2 r D2 l2 D2 B2 r D2 l' D2 r2 B2

(25,15)

Mirror[Alg.3(v5)]

Y

B2 l2 D2 r' D2 l B2 D2 r2 D2 l U2 r' U2 r

(25,15)

Inverse[Alg.3(v5)]

Y

l' F2 l F2 r' B2 l2 B2 D2 r' B2 l B2 r2 D2

(25,15)

Rotation[Alg.3(v5)]

Y

B2 r2 D2 l D2 r' B2 D2 l2 D2 r' U2 l U2 l'

(25,15)

Mirror[Inverse[Alg.3(v5)]]

Y

r F2 r' F2 l B2 r2 B2 D2 l B2 r' B2 l2 D2

(25,15)

Mirror[Rotation[Alg.3(v5)]]

Y

D2 r2 B2 l' B2 r D2 B2 l2 B2 r F2 l' F2 l

(25,15)

Inverse[Rotation[Alg.3(v5)]]

Y

D2 l2 B2 r B2 l' D2 B2 r2 B2 l' F2 r F2 r'

(25,15)

Mirror[Inverse[Rotation[Alg.3(v5)]]]

Y

Group 4: B2 move conjugation of Group 3

r2 U2 r' U2 r U2 B2 r2 B2 r U2 l' B2 l B2

(25,15)

Alg.4(v1)

Y

l2 U2 l U2 l' U2 B2 l2 B2 l' U2 r B2 r' B2

(25,15)

Mirror[Alg.4(v1)]

Y

B2 l' B2 l U2 r' B2 r2 B2 U2 r' U2 r U2 r2

(25,15)

Inverse[Alg.4(v1)]

Y

l2 F2 l' F2 l F2 D2 l2 D2 l F2 r' D2 r D2

(25,15)

Rotation[Alg.4(v1)]

Y

B2 r B2 r' U2 l B2 l2 B2 U2 l U2 l' U2 l2

(25,15)

Mirror[Inverse[Alg.4(v1)]]

Y

r2 F2 r F2 r' F2 D2 r2 D2 r' F2 l D2 l' D2

(25,15)

Mirror[Rotation[Alg.4(v1)]]

Y

D2 r' D2 r F2 l' D2 l2 D2 F2 l' F2 l F2 l2

(25,15)

Inverse[Rotation[Alg.4(v1)]]

Y

D2 l D2 l' F2 r D2 r2 D2 F2 r F2 r' F2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.4(v1)]]]

Y

r2 D2 r F2 l' F2 U2 r2 D2 r' U2 r B2 r' B2

(25,15)

Alg.4(v2)

Y

l2 D2 l' F2 r F2 U2 l2 D2 l U2 l' B2 l B2

(25,15)

Mirror[Alg.4(v2)]

Y

B2 r B2 r' U2 r D2 r2 U2 F2 l F2 r' D2 r2

(25,15)

Inverse[Alg.4(v2)]

Y

l2 B2 l U2 r' U2 F2 l2 B2 l' F2 l D2 l' D2

(25,15)

Rotation[Alg.4(v2)]

Y

B2 l' B2 l U2 l' D2 l2 U2 F2 r' F2 l D2 l2

(25,15)

Mirror[Inverse[Alg.4(v2)]]

Y

r2 B2 r' U2 l U2 F2 r2 B2 r F2 r' D2 r D2

(25,15)

Mirror[Rotation[Alg.4(v2)]]

Y

D2 l D2 l' F2 l B2 l2 F2 U2 r U2 l' B2 l2

(25,15)

Inverse[Rotation[Alg.4(v2)]]

Y

D2 r' D2 r F2 r' B2 r2 F2 U2 l' U2 r B2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.4(v2)]]]

Y

r2 U2 l' B2 r B2 D2 r2 D2 l U2 l' B2 l B2

(25,15)

Alg.4(v3)

Y

l2 U2 r B2 l' B2 D2 l2 D2 r' U2 r B2 r' B2

(25,15)

Mirror[Alg.4(v3)]

Y

B2 l' B2 l U2 l' D2 r2 D2 B2 r' B2 l U2 r2

(25,15)

Inverse[Alg.4(v3)]

Y

l2 F2 r' D2 l D2 B2 l2 B2 r F2 r' D2 r D2

(25,15)

Rotation[Alg.4(v3)]

Y

B2 r B2 r' U2 r D2 l2 D2 B2 l B2 r' U2 l2

(25,15)

Mirror[Inverse[Alg.4(v3)]]

Y

r2 F2 l D2 r' D2 B2 r2 B2 l' F2 l D2 l' D2

(25,15)

Mirror[Rotation[Alg.4(v3)]]

Y

D2 r' D2 r F2 r' B2 l2 B2 D2 l' D2 r F2 l2

(25,15)

Inverse[Rotation[Alg.4(v3)]]

Y

D2 l D2 l' F2 l B2 r2 B2 D2 r D2 l' F2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.4(v3)]]]

Y

r2 D2 l D2 r' B2 D2 l2 D2 l' B2 l B2 r' B2

(25,15)

Alg.4(v4)

Y

l2 D2 r' D2 l B2 D2 r2 D2 r B2 r' B2 l B2

(25,15)

Mirror[Alg.4(v4)]

Y

B2 r B2 l' B2 l D2 l2 D2 B2 r D2 l' D2 r2

(25,15)

Inverse[Alg.4(v4)]

Y

l2 B2 r B2 l' D2 B2 r2 B2 r' D2 r D2 l' D2

(25,15)

Rotation[Alg.4(v4)]

Y

B2 l' B2 r B2 r' D2 r2 D2 B2 l' D2 r D2 l2

(25,15)

Mirror[Inverse[Alg.4(v4)]]

Y

r2 B2 l' B2 r D2 B2 l2 B2 l D2 l' D2 r D2

(25,15)

Mirror[Rotation[Alg.4(v4)]]

Y

D2 l D2 r' D2 r B2 r2 B2 D2 l B2 r' B2 l2

(25,15)

Inverse[Rotation[Alg.4(v4)]]

Y

D2 r' D2 l D2 l' B2 l2 B2 D2 r' B2 l B2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.4(v4)]]]

Y

r2 D2 l D2 r' B2 D2 l2 D2 r' U2 l U2 l' B2

(25,15)

Alg.4(v5)

Y

l2 D2 r' D2 l B2 D2 r2 D2 l U2 r' U2 r B2

(25,15)

Mirror[Alg.4(v5)]

Y

B2 l U2 l' U2 r D2 l2 D2 B2 r D2 l' D2 r2

(25,15)

Inverse[Alg.4(v5)]

Y

l2 B2 r B2 l' D2 B2 r2 B2 l' F2 r F2 r' D2

(25,15)

Rotation[Alg.4(v5)]

Y

B2 r' U2 r U2 l' D2 r2 D2 B2 l' D2 r D2 l2

(25,15)

Mirror[Inverse[Alg.4(v5)]]

Y

r2 B2 l' B2 r D2 B2 l2 B2 r F2 l' F2 l D2

(25,15)

Mirror[Rotation[Alg.4(v5)]]

Y

D2 r F2 r' F2 l B2 r2 B2 D2 l B2 r' B2 l2

(25,15)

Inverse[Rotation[Alg.4(v5)]]

Y

D2 l' F2 l F2 r' B2 l2 B2 D2 r' B2 l B2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.4(v5)]]]

Y

Symmetrical algorithms

Group 1

• The first and last turns of each of these algorithms can be made wide turns to still achieve a single dedge flip.

r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 r2

(25,15)

Alg.5(v1)

Y

l2 B2 U2 r' U2 l U2 l' U2 F2 l' F2 r B2 l2

(25,15)

Mirror[Alg.5(v1)]

Y

r2 B2 l F2 r' F2 U2 r' U2 r U2 l' U2 B2 r2

(25,15)

Inverse[Alg.5(v1)]

Y

l2 D2 F2 r F2 l' F2 l F2 U2 l U2 r' D2 l2

(25,15)

Rotation[Alg.5(v1)]

Y

l2 B2 r' F2 l F2 U2 l U2 l' U2 r U2 B2 l2

(25,15)

Mirror[Inverse[Alg.5(v1)]]

Y

r2 D2 F2 l' F2 r F2 r' F2 U2 r' U2 l D2 r2

(25,15)

Mirror[Rotation[Alg.5(v1)]]

Y

l2 D2 r U2 l' U2 F2 l' F2 l F2 r' F2 D2 l2

(25,15)

Inverse[Rotation[Alg.5(v1)]]

Y

r2 D2 l' U2 r U2 F2 r F2 r' F2 l F2 D2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.5(v1)]]]

Y

r2 B2 U2 l U2 l' B2 r B2 U2 r U2 r' B2 r2

(25,15)

Alg.5(v2)

Y

l2 B2 U2 r' U2 r B2 l' B2 U2 l' U2 l B2 l2

(25,15)

Mirror[Alg.5(v2)]

Y

r2 B2 r U2 r' U2 B2 r' B2 l U2 l' U2 B2 r2

(25,15)

Inverse[Alg.5(v2)]

Y

l2 D2 F2 r F2 r' D2 l D2 F2 l F2 l' D2 l2

(25,15)

Rotation[Alg.5(v2)]

Y

l2 B2 l' U2 l U2 B2 l B2 r' U2 r U2 B2 l2

(25,15)

Mirror[Inverse[Alg.5(v2)]]

Y

r2 D2 F2 l' F2 l D2 r' D2 F2 r' F2 r D2 r2

(25,15)

Mirror[Rotation[Alg.5(v2)]]

Y

l2 D2 l F2 l' F2 D2 l' D2 r F2 r' F2 D2 l2

(25,15)

Inverse[Rotation[Alg.5(v2)]]

Y

r2 D2 r' F2 r F2 D2 r D2 l' F2 l F2 D2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.5(v2)]]]

Y

r2 F2 D2 r' D2 r F2 l' F2 D2 l' D2 l F2 r2

(25,15)

Alg.5(v3)

Y

l2 F2 D2 l D2 l' F2 r F2 D2 r D2 r' F2 l2

(25,15)

Mirror[Alg.5(v3)]

Y

r2 F2 l' D2 l D2 F2 l F2 r' D2 r D2 F2 r2

(25,15)

Inverse[Alg.5(v3)]

Y

l2 U2 B2 l' B2 l U2 r' U2 B2 r' B2 r U2 l2

(25,15)

Rotation[Alg.5(v3)]

Y

l2 F2 r D2 r' D2 F2 r' F2 l D2 l' D2 F2 l2

(25,15)

Mirror[Inverse[Alg.5(v3)]]

Y

r2 U2 B2 r B2 r' U2 l U2 B2 l B2 l' U2 r2

(25,15)

Mirror[Rotation[Alg.5(v3)]]

Y

l2 U2 r' B2 r B2 U2 r U2 l' B2 l B2 U2 l2

(25,15)

Inverse[Rotation[Alg.5(v3)]]

Y

r2 U2 l B2 l' B2 U2 l' U2 r B2 r' B2 U2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.5(v3)]]]

Y

r2 F2 D2 r' D2 l D2 l' D2 B2 l' B2 r F2 r2

(25,15)

Alg.5(v4)

Y

l2 F2 D2 l D2 r' D2 r D2 B2 r B2 l' F2 l2

(25,15)

Mirror[Alg.5(v4)]

Y

r2 F2 r' B2 l B2 D2 l D2 l' D2 r D2 F2 r2

(25,15)

Inverse[Alg.5(v4)]

Y

l2 U2 B2 l' B2 r B2 r' B2 D2 r' D2 l U2 l2

(25,15)

Rotation[Alg.5(v4)]

Y

l2 F2 l B2 r' B2 D2 r' D2 r D2 l' D2 F2 l2

(25,15)

Mirror[Inverse[Alg.5(v4)]]

Y

r2 U2 B2 r B2 l' B2 l B2 D2 l D2 r' U2 r2

(25,15)

Mirror[Rotation[Alg.5(v4)]]

Y

l2 U2 l' D2 r D2 B2 r B2 r' B2 l B2 U2 l2

(25,15)

Inverse[Rotation[Alg.5(v4)]]

Y

r2 U2 r D2 l' D2 B2 l' B2 l B2 r' B2 U2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.5(v4)]]]

Y

r2 B2 U2 r B2 r' B2 l U2 F2 r F2 l' B2 r2

(25,15)

Alg.5(v5)

Y

l2 B2 U2 l' B2 l B2 r' U2 F2 l' F2 r B2 l2

(25,15)

Mirror[Alg.5(v5)]

Y

r2 B2 l F2 r' F2 U2 l' B2 r B2 r' U2 B2 r2

(25,15)

Inverse[Alg.5(v5)]

Y

l2 D2 F2 l D2 l' D2 r F2 U2 l U2 r' D2 l2

(25,15)

Rotation[Alg.5(v5)]

Y

l2 B2 r' F2 l F2 U2 r B2 l' B2 l U2 B2 l2

(25,15)

Mirror[Inverse[Alg.5(v5)]]

Y

r2 D2 F2 r' D2 r D2 l' F2 U2 r' U2 l D2 r2

(25,15)

Mirror[Rotation[Alg.5(v5)]]

Y

l2 D2 r U2 l' U2 F2 r' D2 l D2 l' F2 D2 l2

(25,15)

Inverse[Rotation[Alg.5(v5)]]

Y

r2 D2 l' U2 r U2 F2 l D2 r' D2 r F2 D2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.5(v5)]]]

Y

r2 F2 D2 l' F2 l F2 r' D2 B2 l' B2 r F2 r2

(25,15)

Alg.5(v6)

Y

l2 F2 D2 r F2 r' F2 l D2 B2 r B2 l' F2 l2

(25,15)

Mirror[Alg.5(v6)]

Y

r2 F2 r' B2 l B2 D2 r F2 l' F2 l D2 F2 r2

(25,15)

Inverse[Alg.5(v6)]

Y

l2 U2 B2 r' U2 r U2 l' B2 D2 r' D2 l U2 l2

(25,15)

Rotation[Alg.5(v6)]

Y

l2 F2 l B2 r' B2 D2 l' F2 r F2 r' D2 F2 l2

(25,15)

Mirror[Inverse[Alg.5(v6)]]

Y

r2 U2 B2 l U2 l' U2 r B2 D2 l D2 r' U2 r2

(25,15)

Mirror[Rotation[Alg.5(v6)]]

Y

l2 U2 l' D2 r D2 B2 l U2 r' U2 r B2 U2 l2

(25,15)

Inverse[Rotation[Alg.5(v6)]]

Y

r2 U2 r D2 l' D2 B2 r' U2 l U2 l' B2 U2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.5(v6)]]]

Y

r2 B2 U2 r B2 l' D2 r D2 B2 l U2 r' B2 r2

(25,15)

Alg.5(v7)

Y

l2 B2 U2 l' B2 r D2 l' D2 B2 r' U2 l B2 l2

(25,15)

Mirror[Alg.5(v7)]

Y

r2 B2 r U2 l' B2 D2 r' D2 l B2 r' U2 B2 r2

(25,15)

Inverse[Alg.5(v7)]

Y

l2 D2 F2 l D2 r' B2 l B2 D2 r F2 l' D2 l2

(25,15)

Rotation[Alg.5(v7)]

Y

l2 B2 l' U2 r B2 D2 l D2 r' B2 l U2 B2 l2

(25,15)

Mirror[Inverse[Alg.5(v7)]]

Y

r2 D2 F2 r' D2 l B2 r' B2 D2 l' F2 r D2 r2

(25,15)

Mirror[Rotation[Alg.5(v7)]]

Y

l2 D2 l F2 r' D2 B2 l' B2 r D2 l' F2 D2 l2

(25,15)

Inverse[Rotation[Alg.5(v7)]]

Y

r2 D2 r' F2 l D2 B2 r B2 l' D2 r F2 D2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.5(v7)]]]

Y

r2 F2 D2 l' F2 r U2 l' U2 F2 r' D2 l F2 r2

(25,15)

Alg.5(v8)

Y

l2 F2 D2 r F2 l' U2 r U2 F2 l D2 r' F2 l2

(25,15)

Mirror[Alg.5(v8)]

Y

r2 F2 l' D2 r F2 U2 l U2 r' F2 l D2 F2 r2

(25,15)

Inverse[Alg.5(v8)]

Y

l2 U2 B2 r' U2 l F2 r' F2 U2 l' B2 r U2 l2

(25,15)

Rotation[Alg.5(v8)]

Y

l2 F2 r D2 l' F2 U2 r' U2 l F2 r' D2 F2 l2

(25,15)

Mirror[Inverse[Alg.5(v8)]]

Y

r2 U2 B2 l U2 r' F2 l F2 U2 r B2 l' U2 r2

(25,15)

Mirror[Rotation[Alg.5(v8)]]

Y

l2 U2 r' B2 l U2 F2 r F2 l' U2 r B2 U2 l2

(25,15)

Inverse[Rotation[Alg.5(v8)]]

Y

r2 U2 l B2 r' U2 F2 l' F2 r U2 l' B2 U2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.5(v8)]]]

Y

Group 2: cyclic shift and re-conjugation of Group 1

• The first and last turns of each of these algorithms can be made wide turns to still achieve a single dedge flip.

r2 B2 U2 l' U2 r U2 r' U2 B2 r' B2 l B2 r2

(25,15)

Alg.6(v1)

Y

l2 B2 U2 r U2 l' U2 l U2 B2 l B2 r' B2 l2

(25,15)

Mirror[Alg.6(v1)]

Y

r2 B2 l' B2 r B2 U2 r U2 r' U2 l U2 B2 r2

(25,15)

Inverse[Alg.6(v1)]

Y

l2 D2 F2 r' F2 l F2 l' F2 D2 l' D2 r D2 l2

(25,15)

Rotation[Alg.6(v1)]

Y

l2 B2 r B2 l' B2 U2 l' U2 l U2 r' U2 B2 l2

(25,15)

Mirror[Inverse[Alg.6(v1)]]

Y

r2 D2 F2 l F2 r' F2 r F2 D2 r D2 l' D2 r2

(25,15)

Mirror[Rotation[Alg.6(v1)]]

Y

l2 D2 r' D2 l D2 F2 l F2 l' F2 r F2 D2 l2

(25,15)

Inverse[Rotation[Alg.6(v1)]]

Y

r2 D2 l D2 r' D2 F2 r' F2 r F2 l' F2 D2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.6(v1)]]]

Y

r2 B2 U2 l' U2 l F2 r' F2 U2 r' U2 r B2 r2

(25,15)

Alg.6(v2)

Y

l2 B2 U2 r U2 r' F2 l F2 U2 l U2 l' B2 l2

(25,15)

Mirror[Alg.6(v2)]

Y

r2 B2 r' U2 r U2 F2 r F2 l' U2 l U2 B2 r2

(25,15)

Inverse[Alg.6(v2)]

Y

l2 D2 F2 r' F2 r U2 l' U2 F2 l' F2 l D2 l2

(25,15)

Rotation[Alg.6(v2)]

Y

l2 B2 l U2 l' U2 F2 l' F2 r U2 r' U2 B2 l2

(25,15)

Mirror[Inverse[Alg.6(v2)]]

Y

r2 D2 F2 l F2 l' U2 r U2 F2 r F2 r' D2 r2

(25,15)

Mirror[Rotation[Alg.6(v2)]]

Y

l2 D2 l' F2 l F2 U2 l U2 r' F2 r F2 D2 l2

(25,15)

Inverse[Rotation[Alg.6(v2)]]

Y

r2 D2 r F2 r' F2 U2 r' U2 l F2 l' F2 D2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.6(v2)]]]

Y

r2 F2 D2 r D2 r' B2 l B2 D2 l D2 l' F2 r2

(25,15)

Alg.6(v3)

Y

l2 F2 D2 l' D2 l B2 r' B2 D2 r' D2 r F2 l2

(25,15)

Mirror[Alg.6(v3)]

Y

r2 F2 l D2 l' D2 B2 l' B2 r D2 r' D2 F2 r2

(25,15)

Inverse[Alg.6(v3)]

Y

l2 U2 B2 l B2 l' D2 r D2 B2 r B2 r' U2 l2

(25,15)

Rotation[Alg.6(v3)]

Y

l2 F2 r' D2 r D2 B2 r B2 l' D2 l D2 F2 l2

(25,15)

Mirror[Inverse[Alg.6(v3)]]

Y

r2 U2 B2 r' B2 r D2 l' D2 B2 l' B2 l U2 r2

(25,15)

Mirror[Rotation[Alg.6(v3)]]

Y

l2 U2 r B2 r' B2 D2 r' D2 l B2 l' B2 U2 l2

(25,15)

Inverse[Rotation[Alg.6(v3)]]

Y

r2 U2 l' B2 l B2 D2 l D2 r' B2 r B2 U2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.6(v3)]]]

Y

r2 F2 D2 r D2 l' D2 l D2 F2 l F2 r' F2 r2

(25,15)

Alg.6(v4)

Y

l2 F2 D2 l' D2 r D2 r' D2 F2 r' F2 l F2 l2

(25,15)

Mirror[Alg.6(v4)]

Y

r2 F2 r F2 l' F2 D2 l' D2 l D2 r' D2 F2 r2

(25,15)

Inverse[Alg.6(v4)]

Y

l2 U2 B2 l B2 r' B2 r B2 U2 r U2 l' U2 l2

(25,15)

Rotation[Alg.6(v4)]

Y

l2 F2 l' F2 r F2 D2 r D2 r' D2 l D2 F2 l2

(25,15)

Mirror[Inverse[Alg.6(v4)]]

Y

r2 U2 B2 r' B2 l B2 l' B2 U2 l' U2 r U2 r2

(25,15)

Mirror[Rotation[Alg.6(v4)]]

Y

l2 U2 l U2 r' U2 B2 r' B2 r B2 l' B2 U2 l2

(25,15)

Inverse[Rotation[Alg.6(v4)]]

Y

r2 U2 r' U2 l U2 B2 l B2 l' B2 r B2 U2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.6(v4)]]]

Y

r2 B2 U2 r' F2 r F2 l' U2 B2 r' B2 l B2 r2

(25,15)

Alg.6(v5)

Y

l2 B2 U2 l F2 l' F2 r U2 B2 l B2 r' B2 l2

(25,15)

Mirror[Alg.6(v5)]

Y

r2 B2 l' B2 r B2 U2 l F2 r' F2 r U2 B2 r2

(25,15)

Inverse[Alg.6(v5)]

Y

l2 D2 F2 l' U2 l U2 r' F2 D2 l' D2 r D2 l2

(25,15)

Rotation[Alg.6(v5)]

Y

l2 B2 r B2 l' B2 U2 r' F2 l F2 l' U2 B2 l2

(25,15)

Mirror[Inverse[Alg.6(v5)]]

Y

r2 D2 F2 r U2 r' U2 l F2 D2 r D2 l' D2 r2

(25,15)

Mirror[Rotation[Alg.6(v5)]]

Y

l2 D2 r' D2 l D2 F2 r U2 l' U2 l F2 D2 l2

(25,15)

Inverse[Rotation[Alg.6(v5)]]

Y

r2 D2 l D2 r' D2 F2 l' U2 r U2 r' F2 D2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.6(v5)]]]

Y

r2 B2 U2 r' F2 l D2 r' D2 F2 l' U2 r B2 r2

(25,15)

Alg.6(v6)

Y

l2 B2 U2 l F2 r' D2 l D2 F2 r U2 l' B2 l2

(25,15)

Mirror[Alg.6(v6)]

Y

r2 B2 r' U2 l F2 D2 r D2 l' F2 r U2 B2 r2

(25,15)

Inverse[Alg.6(v6)]

Y

l2 D2 F2 l' U2 r B2 l' B2 U2 r' F2 l D2 l2

(25,15)

Rotation[Alg.6(v6)]

Y

l2 B2 l U2 r' F2 D2 l' D2 r F2 l' U2 B2 l2

(25,15)

Mirror[Inverse[Alg.6(v6)]]

Y

r2 D2 F2 r U2 l' B2 r B2 U2 l F2 r' D2 r2

(25,15)

Mirror[Rotation[Alg.6(v6)]]

Y

l2 D2 l' F2 r U2 B2 l B2 r' U2 l F2 D2 l2

(25,15)

Inverse[Rotation[Alg.6(v6)]]

Y

r2 D2 r F2 l' U2 B2 r' B2 l U2 r' F2 D2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.6(v6)]]]

Y

r2 F2 D2 l B2 r' U2 l U2 B2 r D2 l' F2 r2

(25,15)

Alg.6(v7)

Y

l2 F2 D2 r' B2 l U2 r' U2 B2 l' D2 r F2 l2

(25,15)

Mirror[Alg.6(v7)]

Y

r2 F2 l D2 r' B2 U2 l' U2 r B2 l' D2 F2 r2

(25,15)

Inverse[Alg.6(v7)]

Y

l2 U2 B2 r D2 l' F2 r F2 D2 l B2 r' U2 l2

(25,15)

Rotation[Alg.6(v7)]

Y

l2 F2 r' D2 l B2 U2 r U2 l' B2 r D2 F2 l2

(25,15)

Mirror[Inverse[Alg.6(v7)]]

Y

r2 U2 B2 l' D2 r F2 l' F2 D2 r' B2 l U2 r2

(25,15)

Mirror[Rotation[Alg.6(v7)]]

Y

l2 U2 r B2 l' D2 F2 r' F2 l D2 r' B2 U2 l2

(25,15)

Inverse[Rotation[Alg.6(v7)]]

Y

r2 U2 l' B2 r D2 F2 l F2 r' D2 l B2 U2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.6(v7)]]]

Y

r2 F2 D2 l B2 l' B2 r D2 F2 l F2 r' F2 r2

(25,15)

Alg.6(v8)

Y

l2 F2 D2 r' B2 r B2 l' D2 F2 r' F2 l F2 l2

(25,15)

Mirror[Alg.6(v8)]

Y

r2 F2 r F2 l' F2 D2 r' B2 l B2 l' D2 F2 r2

(25,15)

Inverse[Alg.6(v8)]

Y

l2 U2 B2 r D2 r' D2 l B2 U2 r U2 l' U2 l2

(25,15)

Rotation[Alg.6(v8)]

Y

l2 F2 l' F2 r F2 D2 l B2 r' B2 r D2 F2 l2

(25,15)

Mirror[Inverse[Alg.6(v8)]]

Y

r2 U2 B2 l' D2 l D2 r' B2 U2 l' U2 r U2 r2

(25,15)

Mirror[Rotation[Alg.6(v8)]]

Y

l2 U2 l U2 r' U2 B2 l' D2 r D2 r' B2 U2 l2

(25,15)

Inverse[Rotation[Alg.6(v8)]]

Y

r2 U2 r' U2 l U2 B2 r D2 l' D2 l B2 U2 r2

(25,15)

Mirror[Inverse[Rotation[Alg.6(v8)]]]

Y

Shortest single slice quarter turn algorithms

• The current upperbound in the single slice quarter turn metric for the single dedge flip is 23. This section contains over 100 algorithms with this quantity of moves.
The majority of algorithms in this section, like the (25,15) solutions, were found by using the 3x3x3 Classic Setup in Cube Explorer. For this particular set, a search up through depth 18 was performed. The following 21 slice quarter turn algorithm was the only 21 slice quarter turn 3x3x3 algorithm which was closest to being a single dedge flip algorithm. Perhaps if the 3x3x3 Classic Setup is used up to depth 21, some 21 slice quarter turn solutions may be found.

r' F l' F2 r F r' F2 D F' D l D' F D' l F2 r

(21,18)

N

Christopher Mowla

Update
On March 28, 2018, Ben Whitmore found two 21 single slice quarter turn algorithms with his ksolve++ program. (They also happen to be supercube safe.)

r' U' u r U' r U r u' r' u r U r U' r Uw' r' U2 //Safe

(20,19)

N

Ben Whitmore

r' Uw' r U' r U r u r' u' r U r U' r u U' r' U2 //Safe

(20,19)

N

Ben Whitmore

However, unlike all of the 23 single slice quarter turn algorithms presented in this section (which can be applied to all big cube sizes), these two algorithms only work on the 4x4x4.
• Like the (25,15) solutions, these solutions have been grouped next to their transformations. To save space, we label each group's algorithm as v1,v2,v3, and so forth.
• However, unlike the (25,15) solutions, this is by no means the complete set of all possible unique (let alone all inverse, mirror, and rotation transformations of the unique solutions) 23 single slice quarter turn solutions. (There were 36 "unique" (25,15) solutions from which all 288 algorithms were derived from.) Perhaps in the future we will have efficient optimal solvers in the single slice quarter turn metric which we can use to do a brute force search.

x Rw2 U2 Rw U2 l' U2 Rw' F2 U' l' U F2 U' l U' Rw2 x'

(23,16)

v1

Y

Christopher Mowla

x Rw2 U2 Lw F2 l' F2 Lw' F2 U' l' U F2 U' l U' Rw2 x'

(23,16)

v2

Y

Christopher Mowla

x Rw2 U2 Rw U2 l' U2 Rw' F2 U l' U' F2 U l U Rw2 x'

(23,16)

v1

N

Christopher Mowla

x Rw2 U2 Lw F2 l' F2 Lw' F2 U l' U' F2 U l U Rw2 x'

(23,16)

v2

N

Christopher Mowla

x' Rw2 U2 Lw' U2 r U2 Lw F2 U r U' F2 U r' U Rw2 x

(23,16)

cmowlaparity

Y

Christopher Mowla

x' Rw2 U2 Rw' F2 r F2 Rw F2 U r U' F2 U r' U Rw2 x

(23,16)

cmowlaparity(v2)

Y

Christopher Mowla

x' Rw2 U2 Lw' U2 r U2 Lw F2 U' r U F2 U' r' U' Rw2 x

(23,16)

v1

N

Christopher Mowla

x' Rw2 U2 Rw' F2 r F2 Rw F2 U' r U F2 U' r' U' Rw2 x

(23,16)

v2

N

Christopher Mowla

Rw' U2 B r' B' U2 F2 Lw' B2 r' B2 Lw B r F2 B' Rw

(22,17)

v1

Y

Christopher Mowla

Rw' U2 B r' B' U2 F2 Rw' U2 r' U2 Rw B r F2 B' Rw

(22,17)

v2

Y

Christopher Mowla

Rw' U2 B' r' B U2 F2 Lw' B2 r' B2 Lw B' r F2 B Rw

(22,17)

v1

N

Christopher Mowla

Rw' U2 B' r' B U2 F2 Rw' U2 r' U2 Rw B' r F2 B Rw

(22,17)

v2

N

Christopher Mowla

r' U' r' U B2 U' r U' r' U2 r' B2 l U2 l' U2 r2

(23,17)

Y

Christopher Mowla

r' U r' U' B2 U r U r' U2 r' B2 l U2 l' U2 r2

(23,17)

N

Christopher Mowla

r B l B' D2 B l' B l B2 r F2 l' F2 l D2 r2

(23,17)

v1

Y

Christopher Mowla

r B l B' D2 B l' B l B2 l D2 l' D2 r D2 r2

(23,17)

v2

Y

Christopher Mowla

r B' l B D2 B' l' B' l B2 r F2 l' F2 l D2 r2

(23,17)

v1

N

Christopher Mowla

r B' l B D2 B' l' B' l B2 l D2 l' D2 r D2 r2

(23,17)

v2

N

Christopher Mowla

Rw D B2 D' l D B2 U2 Rw D2 l D2 Rw' D' l' U2 Rw'

(23,17)

v1

Y

Christopher Mowla

Rw D B2 D' l D B2 U2 Lw B2 l B2 Lw' D' l' U2 Rw'

(23,17)

v2

Y

Christopher Mowla

Rw D' B2 D l D' B2 U2 Rw D2 l D2 Rw' D l' U2 Rw'

(23,17)

v1

N

Christopher Mowla

Rw D' B2 D l D' B2 U2 Lw B2 l B2 Lw' D l' U2 Rw'

(23,17)

v2

N

Christopher Mowla

Rw' F2 D2 B l B' D2 B l' B' Rw' B2 l B2 Rw F2 Rw

(23,17)

v1

Y

Christopher Mowla

Rw' F2 D2 B l B' D2 B l' B' Lw' D2 l D2 Lw F2 Rw

(23,17)

v2

Y

Christopher Mowla

Rw' F2 D2 B' l B D2 B' l' B Rw' B2 l B2 Rw F2 Rw

(23,17)

v1

N

Christopher Mowla

Rw' F2 D2 B' l B D2 B' l' B Lw' D2 l D2 Lw F2 Rw

(23,17)

v2

N

Christopher Mowla

r2 D2 r D2 l' D l D' F2 D l' D l D2 l F2 r

(23,17)

Y

Christopher Mowla

r2 D2 r D2 l' D' l D F2 D' l' D' l D2 l F2 r

(23,17)

N

Christopher Mowla

Rw' B' U2 B r' B' U2 F2 Lw' B2 r' B2 Lw B r F2 Rw

(23,17)

v1

Y

Christopher Mowla

Rw' B' U2 B r' B' U2 F2 Rw' U2 r' U2 Rw B r F2 Rw

(23,17)

v2

Y

Christopher Mowla

Rw' B U2 B' r' B U2 F2 Lw' B2 r' B2 Lw B' r F2 Rw

(23,17)

v1

N

Christopher Mowla

Rw' B U2 B' r' B U2 F2 Rw' U2 r' U2 Rw B' r F2 Rw

(23,17)

v2

N

Christopher Mowla

l' r' D' r D' B2 D r' D' B2 l' D2 l' F2 l F2 l2

(23,17)

v1

Y

Christopher Mowla

l r U' r U' F2 U r' U' F2 l' U2 l' B2 l B2 r2

(23,17)

v2

Y

Christopher Mowla

l' r' D r D B2 D' r' D B2 l' D2 l' F2 l F2 l2

(23,17)

v1

N

Christopher Mowla

l r U r U F2 U' r' U F2 l' U2 l' B2 l B2 r2

(23,17)

v2

N

Christopher Mowla

Rw U2 F2 D' r' D F2 D' r D Lw D2 r' D2 Lw' U2 Rw'

(23,17)

v1

Y

Christopher Mowla

Rw U2 F2 D' r' D F2 D' r D Rw F2 r' F2 Rw' U2 Rw'

(23,17)

v2

Y

Christopher Mowla

Rw U2 F2 D r' D' F2 D r D' Lw D2 r' D2 Lw' U2 Rw'

(23,17)

v1

N

Christopher Mowla

Rw U2 F2 D r' D' F2 D r D' Rw F2 r' F2 Rw' U2 Rw'

(23,17)

v2

N

Christopher Mowla

r2 B' r B' U2 B r' B' U2 l' B2 Lw' D2 l D2 Lw r

(23,17)

v1

Y

Christopher Mowla

l2 F' r F' D2 F r' F' D2 l' F2 l' U2 l U2 l' r'

(23,17)

v2

Y

Christopher Mowla

r2 B r B U2 B' r' B U2 l' B2 Lw' D2 l D2 Lw r

(23,17)

v1

N

Christopher Mowla

l2 F r F D2 F' r' F D2 l' F2 l' U2 l U2 l' r'

(23,17)

v2

N

Christopher Mowla

r2 U2 l' U2 l F' r' F U2 F' r F' r' F2 r' U2 r'

(23,17)

v1

Y

Christopher Mowla

r2 U2 r' F2 l D' r' D F2 D' r D' r' D2 l' U2 r'

(23,17)

v2

Y

Christopher Mowla

r2 U2 l' U2 l F r' F' U2 F r F r' F2 r' U2 r'

(23,17)

v1

N

Christopher Mowla

r2 U2 r' F2 l D r' D' F2 D r D r' D2 l' U2 r'

(23,17)

v2

N

Christopher Mowla

r2 B2 D2 r D b' D F2 D' b D r' B2 l r F2 r

(23,17)

Y

Christopher Mowla

r2 B2 D2 r D' f' D' F2 D f D' r' B2 l r F2 r

(23,17)

N

Christopher Mowla

r2 F2 U2 l' U' f U' B2 U f' U' r' B2 r l F2 r

(23,17)

Y

Christopher Mowla

r2 F2 U2 l' U b U B2 U' b' U r' B2 r l F2 r

(23,17)

N

Christopher Mowla

r2 D f D' B2 D f' D r D2 F2 r' B2 l r F2 r

(23,17)

Y

Christopher Mowla

r2 D' b D B2 D' b' D' r D2 F2 r' B2 l r F2 r

(23,17)

N

Christopher Mowla

r2 U' b' U F2 U' b U' l' U2 B2 r' B2 r l F2 r

(23,17)

Y

Christopher Mowla

r2 U f' U' F2 U f U l' U2 B2 r' B2 r l F2 r

(23,17)

N

Christopher Mowla

r2 D2 r D b' D F2 D' b D r' B2 l B2 r F2 r

(23,17)

v1

Y

Christopher Mowla

r2 D2 r D b' D F2 D' b D l' D2 l D2 l F2 r

(23,17)

v2

Y

Christopher Mowla

r2 D2 r D' f' D' F2 D f D' r' B2 l B2 r F2 r

(23,17)

v1

N

Christopher Mowla

r2 D2 r D' f' D' F2 D f D' l' D2 l D2 l F2 r

(23,17)

v2

N

Christopher Mowla

r2 F2 r' B2 l D f D' B2 D f' D r D2 r F2 r

(23,17)

v1

Y

Christopher Mowla

r2 F2 l' D2 l F u F' D2 F u' F r F2 l F2 r

(23,17)

v2

Y

Christopher Mowla

r2 B2 r U2 r' B' d' B U2 B' d B' r' U2 r' U2 r'

(23,17)

v3

Y

Christopher Mowla

r2 F2 r' B2 l D' b D B2 D' b' D' r D2 r F2 r

(23,17)

v1

N

Christopher Mowla

r2 F2 l' D2 l F' d F D2 F' d' F' r F2 l F2 r

(23,17)

v2

N

Christopher Mowla

r2 B2 r U2 r' B u' B' U2 B u B r' U2 r' U2 r'

(23,17)

v3

N

Christopher Mowla

Rw2 B r' U R' U r U' R U' r B U2 r U2 r' B2 Rw2

(23,18)

N

Christopher Mowla

Lw' B' r' B D2 B' r B' r' B2 r' D2 l B2 l' B2 r Lw

(23,18)

Y

Christopher Mowla

Lw' B r' B' D2 B r B r' B2 r' D2 l B2 l' B2 r Lw

(23,18)

N

Christopher Mowla

r Lw F2 l' F2 l D' r' D F2 D' r D' r' D2 r' F2 Lw'

(23,18)

v1

Y

Christopher Mowla

Rw l F2 l' F2 r F' r' F U2 F' r F' r' F2 r' U2 Rw'

(23,18)

v2

Y

Christopher Mowla

r Lw F2 r' D2 l B' r' B D2 B' r B' r' B2 l' F2 Lw'

(23,18)

v3

Y

Christopher Mowla

r Lw F2 l' F2 l D r' D' F2 D r D r' D2 r' F2 Lw'

(23,18)

v1

N

Christopher Mowla

Rw l F2 l' F2 r F r' F' U2 F r F r' F2 r' U2 Rw'

(23,18)

v2

N

Christopher Mowla

r Lw F2 r' D2 l B r' B' D2 B r B r' B2 l' F2 Lw'

(23,18)

v3

N

Christopher Mowla

Rw' U' r' U B2 U' r U' r' U2 l' D2 r D2 r' B2 Rw l

(23,18)

v1

Y

Christopher Mowla

Rw' U' r' U B2 U' r U' r' U2 r' B2 r B2 l' B2 Rw l

(23,18)

v2

Y

Christopher Mowla

Rw' U r' U' B2 U r U r' U2 l' D2 r D2 r' B2 Rw l

(23,18)

v1

N

Christopher Mowla

Rw' U r' U' B2 U r U r' U2 r' B2 r B2 l' B2 Rw l

(23,18)

v2

N

Christopher Mowla

Rw' F2 D b' D l D2 l' D' b D Rw' B2 l B2 Rw F2 Rw

(23,18)

v1

Y

Christopher Mowla

Rw' F2 D b' D l D2 l' D' b D Lw' D2 l D2 Lw F2 Rw

(23,18)

v2

Y

Christopher Mowla

Rw' F2 D' f' D' l D2 l' D f D' Rw' B2 l B2 Rw F2 Rw

(23,18)

v1

N

Christopher Mowla

Rw' F2 D' f' D' l D2 l' D f D' Lw' D2 l D2 Lw F2 Rw

(23,18)

v2

N

Christopher Mowla

r2 B' D2 r D b' D F2 D' b D r' B2 l B' r F2 r

(23,18)

Y

Christopher Mowla

r2 B D2 r D' f' D' F2 D f D' r' B2 l B r F2 r

(23,18)

N

Christopher Mowla

r2 B D2 r D b' D F2 D' b D r' B2 l B r F2 r

(23,18)

Y

Christopher Mowla

r2 B' D2 r D' f' D' F2 D f D' r' B2 l B' r F2 r

(23,18)

N

Christopher Mowla

r' Lw' B' d' B U2 B' d B' l' B2 D2 r' D2 r l U2 Lw

(23,18)

Y

Christopher Mowla

r' Lw' B u' B' U2 B u B l' B2 D2 r' D2 r l U2 Lw

(23,18)

N

Christopher Mowla

Rw l D2 F2 l' F' d F' U2 F d' F' l D2 r' l' U2 Rw'

(23,18)

Y

Christopher Mowla

Rw l D2 F2 l' F u F U2 F' u' F l D2 r' l' U2 Rw'

(23,18)

N

Christopher Mowla

Rw l F' u' F D2 F' u F' l' F2 U2 l D2 r' l' U2 Rw'

(23,18)

Y

Christopher Mowla

Rw l F d' F' D2 F d F l' F2 U2 l D2 r' l' U2 Rw'

(23,18)

N

Christopher Mowla

Rw l F2 l' F' d F' U2 F d' F' Lw D2 r' D2 Lw' U2 Rw'

(23,18)

v1

Y

Christopher Mowla

Rw l F2 l' F' d F' U2 F d' F' Rw F2 r' F2 Rw' U2 Rw'

(23,18)

v2

Y

Christopher Mowla

Rw l F2 l' F u F U2 F' u' F Lw D2 r' D2 Lw' U2 Rw'

(23,18)

v1

N

Christopher Mowla

Rw l F2 l' F u F U2 F' u' F Rw F2 r' F2 Rw' U2 Rw'

(23,18)

v2

N

Christopher Mowla

r' Lw' U2 B2 l' B' u B' D2 B u' B' r' D2 r l U2 Lw

(23,18)

Y

Christopher Mowla

r' Lw' U2 B2 l' B d B D2 B' d' B r' D2 r l U2 Lw

(23,18)

N

Christopher Mowla

Rw l U2 r F2 r' U' b' U F2 U' b U' l' U2 r' U2 Rw'

(23,18)

v1

Y

Christopher Mowla

r Lw U2 r F2 r' U' b' U F2 U' b U' r' F2 r' F2 Lw'

(23,18)

v2

Y

Christopher Mowla

Rw l U2 r F2 r' U f' U' F2 U f U l' U2 r' U2 Rw'

(23,18)

v1

N

Christopher Mowla

r Lw U2 r F2 r' U f' U' F2 U f U r' F2 r' F2 Lw'

(23,18)

v2

N

Christopher Mowla

Rw U2 F' d F' r' F2 r F d' F' Lw D2 r' D2 Lw' U2 Rw'

(23,18)

v1

Y

Christopher Mowla

Rw U2 F' d F' r' F2 r F d' F' Rw F2 r' F2 Rw' U2 Rw'

(23,18)

v2

Y

Christopher Mowla

Rw U2 F u F r' F2 r F' u' F Lw D2 r' D2 Lw' U2 Rw'

(23,18)

v1

N

Christopher Mowla

Rw U2 F u F r' F2 r F' u' F Rw F2 r' F2 Rw' U2 Rw'

(23,18)

v2

N

Christopher Mowla




Shortest block quarter turn move algorithms

• All of the algorithms in this section are wide turn based (they require – not just possess for easier execution – wide turns to create a single dedge flip).
• Most of these algorithms affect the centers of the supercube differently than the previously shown algorithms.
Despite that the shortest algorithm in BQTM listed is 18, it could be possible that a 17 BQTM exists, as below is a 16 BQTM algorithm which is nearly "pure". (Found in late January of 2018.)

Rw Dw' Rw U' Rw Bw' r' Bw Rw' U R' Dw Rw' Bw' r' Bw

(16,16)

N

Christopher Mowla

UPDATE: On June 8, 2025, Tom Rokicki completed a depth 17 (half turn move) search (on an optimal 4x4x4 solver) and confirmed that the (18,18) solution listed in this section is optimal for the quarter turn metric for this case.


Symmetrical algorithms

Rw' E' Lw2 Uw' l Uw Lw2 Dw' L2 Uw' l Uw Lw' L' Dw E Rw

(20,17)

N

Christopher Mowla

y' Rw Uw2 x' U D r y Uw2 Rw2 Uw r Uw' Rw2 Dw r' u D' x Uw2 Rw' y

(22,17)

N

Christopher Mowla

Rw' 3Uw' u' Lw' r' Uw' r' Uw 3Lw r' Uw Lw' Uw' r' Uw Lw E' Rw

(18,18)

N

Christopher Mowla

y' Rw Uw' L Uw' y R Uw' r Uw Rw' Dw Rw2 Uw' r Uw Rw2 L' Uw Rw' y

(19,18)

Challenger

N

Christopher Mowla

y' Rw' Uw L' Uw y' R' Uw r' Uw' Rw Dw' Rw2 Uw r' Uw' Rw2 L Uw' Rw y

(19,18)

Reverter

N

Christopher Mowla

z Dw' M D Lw' Uw' r' Uw Lw Uw' Lw2' Bw' r' Bw Rw' R' u y' M' Uw x2 z'

(19,18)

Holy Grail

N

Christopher Mowla

f' U2 r U u' r' U r' Uw' r' Uw' u' r' U r' U r f

(19,18)

N

Ben Whitmore

(Rw' E) Uw' r' Uw' Fw Uw Rw' Fw' r' Fw Rw r Uw' Fw' Uw2 (E' Rw)

(19,18)

N

Christopher Mowla

(Rw' E) Uw' r' Uw' r' Uw r Bw r Bw' Uw' Bw r' Bw' Uw2 (E' Rw)

(19,18)

N

Christopher Mowla

(Rw' E) Uw' r' Uw' r' Uw r2 Bw' Uw' Bw r' Bw' Uw Bw Uw (E' Rw)

(19,18)

N

Christopher Mowla

(Rw B' r) Bw' Dw Bw l' Dw' l' Dw l2 Bw' Dw' Bw l' (r' B Rw')

(19,18)

N

Christopher Mowla

(Lw' u) F' Lw' b' Lw Fw Lw' Bw2 Uw' b' Uw Bw' B' Lw f' (u' Lw)

(19,18)

N

Christopher Mowla

Uw' Fw2 L' R' u Lw Rw Uw2 Rw d Lw' Bw2 Lw d' 3Rw Rw R Fw2 Uw

(23,18)

N

Christopher Mowla

(Rw' E) Uw b' Uw r Uw r' Uw' b Uw r' Uw' r Uw r Uw (E' Rw)

(19,19)

N

Christopher Mowla

(Rw' E) Uw' r' Uw Lw' Uw' Fw Rw' Fw' r' Fw Rw r Fw' Uw Lw (E' Rw)

(19,19)

N

Christopher Mowla

(Rw' U l') Uw Fw' r Fw r Fw' r' Uw' b' Uw Fw Uw' b (l U' Rw)

(19,19)

N

Christopher Mowla

(Rw' U l') Uw Fw' r Fw Uw' Fw' Rw Uw r Uw' Rw' r' Fw (l U' Rw)

(19,19)

N

Christopher Mowla

(Rw' E) Bw r' F Lw' Fw r Fw' Lw F' Bw' Lw Bw r Bw' Lw' (E' Rw)

(19,19)

N

Christopher Mowla

(Rw' E) r' F Lw' Fw r Fw' Lw F' Bw' Lw Bw r Bw' Lw' Bw (E' Rw)

(19,19)

N

Christopher Mowla

(Rw' E) L Uw' Lw' r' Uw' r' Uw r' R Bw Rw' Uw' r' Uw l (E' Rw)

(19,19)

N

Christopher Mowla

(Rw' E) Uw' r' Uw Rw' Bw' Rw Fw' Lw' f' Lw Fw f Rw' Bw Rw (E' Rw)

(19,19)

N

Christopher Mowla

(Rw' E') L Dw' Lw' l' Fw' l' Fw l' L Dw Lw' Fw' l' Fw l (E Rw)

(19,19)

N

Christopher Mowla

r' Uw U r' Uw' r Uw r Uw r' Uw2 r' Uw r Uw r Uw u r

(20,19)

N

Ben Whitmore

r' Uw' u' r' Uw' r' Uw' r Uw2 r Uw' r' Uw' r' Uw r Uw' U' r

(20,19)

N

Ben Whitmore

Dw' F' z Rw2 B2 x' l' Dw Uw Rw2 Dw' l' Dw Rw2 Dw' l u' U Rw2 D Rw z'

(24,19)

N

Christopher Mowla

Uw Bw' R' r u' R' U' r U R u Rw r Bw Rw' Uw' r Uw Rw Uw'

(20,20)

N

Christopher Mowla

Uw Fw' D r' Dw b Dw b' Dw' r Dw b' Dw' b Dw b Dw2 D' Fw Uw'

(21,20)

N

Christopher Mowla

Fw' U' l u2 Lw Bw' Lw' u Lw Uw Bw u' Bw' Uw' Bw Lw' u l' U Fw

(21,20)

N

Christopher Mowla

(Rw U L F' U) l' Dw Uw Rw2 Dw' l' Dw Rw2 Dw' l Uw' (U' F L' U' Rw')

(23,20)

N

Christopher Mowla

Bw' R' u Rw' r' Uw Bw' Uw' r Uw Rw Bw r' Bw' Rw' Bw Uw' Rw u' R Bw

(21,21)

N

Christopher Mowla

Bw' R' u Rw' l' Dw Bw' Dw' l Dw Lw Bw l' Bw' Lw' Bw Dw' Rw u' R Bw

(21,21)

N

Christopher Mowla

x' Rw' Fw' r' Uw' Rw' b z Rw' u R U r' U' R' u' Rw B r Fw' Rw Uw Fw Rw x

(22,22)

N

Christopher Mowla


Non-symmetrical algorithms

z Rw Uw' l Uw Rw' Uw' d' L d' Rw Dw l Dw' Rw' d L' d l' Uw z'

(19,19)

N

Christopher Mowla

r' U r U r U' u r' E r' E' r' Uw' r U r U r' U2

(20,19)

N

Ben Whitmore

r' U r U r Uw' r' E' r' E r' U' u r U r U r' U2

(20,19)

N

Ben Whitmore

r' U' r U' r Uw r' E r' E' r' U u' r U' r U' r' U2

(20,19)

N

Ben Whitmore

r' U' r U' r U u' r' E' r' E r' Uw r U' r U' r' U2

(20,19)

N

Ben Whitmore

r' Uw Rw Bw' r F' R' r u' R' U' r' U R u r' R F Bw Rw' Uw'

(21,21)

N

Christopher Mowla

x2 Rw' Fw Rw' B Rw' Uw U l U' r U l' U' Uw' Rw B' R Fw' Rw Uw r Uw' x2

(22,22)

N

Christopher Mowla

z' Fw' Uw r' Uw' Fw Uw r' Fw' U L u l Uw' r' Uw l' u' L' U' Fw r2 Uw' z

(23,22)

N

Christopher Mowla

z' Fw' Uw r' Uw' Fw Uw r' Fw' f' U L U' f r' f' U L' U' f Fw r2 Uw' z

(23,22)

N

Christopher Mowla

z' Fw' Uw r' Uw' Fw Uw r' Fw' f' L Fw R' F' r' F R Fw' L' f Fw r2 Uw' z

(23,22)

N

Christopher Mowla

z' f' Uw Rw' Fw' r' Fw Rw Uw' f r' Uw Rw' Fw' U L U' r' U L' U' Fw Rw Uw' r2 z

(25,24)

5x5x5 Alg

N

Christopher Mowla




Longer/faster algorithms

• The following list of algorithms are all unique solutions generated with the 3x3x3 Classic Setup which use 2 different face turns.
There is only one unique algorithm which just uses the face turn U2. (If its inverse and mirror transformations are also considered, then there are four such algorithms.)

r U2 r U2 r' U2 r U2 l' U2 r U2 r' U2 l r2 U2 r'

(27,18)

Alg.1

Y

r U2 r2 l' U2 r U2 r' U2 l U2 r' U2 r U2 r' U2 r'

(27,18)

Inverse[Alg.1(v1)]

Y

l' U2 l' U2 l U2 l' U2 r U2 l' U2 l U2 r' l2 U2 l

(27,18)

Mirror[Alg.1(v1)]

Y

l' U2 l2 r U2 l' U2 l U2 r' U2 l U2 l' U2 l U2 l

(27,18)

Inverse[Mirror[Alg.1(v1)]]

Y

It is common speedcubing practice to execute the inner slice turns of (l r2) as (M r'). Similar adjustments (such as (l' r) to (M')) can be done to several algorithms on this page.

r U2 r U2 r' U2 r U2 l' U2 r U2 r' U2 (M r') U2 r'

(26,18)

Alg.1 (v2)

N

However, the majority of algorithms on this page are presented without this "simplification" to show the "full nxnxn direct translation version" of the algorithm and also to show a version whose inner slice turns can be converted to wide turns to still preserve the first 3 layers (F3L).




• Algorithms with moves such as (Lw Rw') or (Lw' Rw) reduce to the cube rotations x' and x, respectively, when all slice turns are substituted with wide turns.
• All solutions ending in B2 may be conjugated with B2 to create a handful more of "unique" algorithms.
• All other algorithms can be modified by inverses, mirrors, cube rotations, or a combination of any of these to create every possible algorithm which uses 2 faces which are generated from the 3x3x3 Classic Setup. (Clearly this was done for the 15 BHTM solutions.)
• Even though it is very likely that algorithms related by transformations are listed next to each other, no official sorting by transformations was done.
• Since these algorithms use either U and F faces or U and B faces, all algorithms which affect one or the other have been grouped together for convenience of personal preference.

r' U2 r U2 l F2 l' r' F2 r' U2 r' U2 F2 r2 F2

(25,16)

Y

r' U2 r U2 l' U2 l r U2 r U2 r' U2 F2 r2 F2

(25,16)

Y

r' U2 l F2 l' F2 l r F2 r F2 l' U2 F2 r2 F2

(25,16)

Y

r B2 r' U2 l B2 l2 B2 U2 r B2 l' B2 l' r' B2

(25,16)

Y

l' r' B2 r' B2 l U2 B2 r2 B2 r U2 l' B2 l B2

(25,16)

Y

r' U2 r U2 r U2 r2 F2 l r2 F2 l' U2 F2 r2 F2

(27,16)

Y

r' U2 l F2 r F2 l2 U2 l r2 U2 r' U2 F2 r2 F2

(27,16)

Y

r' U2 r' U2 r2 U2 r U2 r2 B2 r U2 r U2 r2 B2

(27,16)

Y

r' U2 l' B2 r2 B2 l U2 r2 B2 r U2 r U2 r2 B2

(27,16)

Y

r U2 r U2 r' U2 l F2 l' F2 r F2 r' F2 r' U2 r'

(25,17)

Y

r U2 r U2 r' U2 r U2 l' U2 l F2 r' F2 r' U2 r'

(25,17)

Y

r U2 l F2 l' U2 r U2 r' F2 r F2 r' F2 r' U2 r'

(25,17)

Y

r U2 l F2 r' F2 r F2 l' F2 r F2 r' F2 r' U2 r'

(25,17)

Y

r' F2 l' F2 r U2 r' F2 r F2 l' F2 l F2 l U2 l

(25,17)

Y

r' F2 l' F2 r U2 l' U2 r U2 r' F2 l F2 l U2 l

(25,17)

Y

l r U2 F2 l' F2 r F2 r' F2 U2 r' U2 l U2 l' r'

(25,17)

Y

l r U2 r F2 r' F2 U2 r' U2 l F2 l' F2 U2 l' r'

(25,17)

Y

r' U2 r' U2 l F2 r' U2 r' U2 r2 F2 r' U2 l' U2 r2

(27,17)

Y

r' U2 r U2 r' F2 l r F2 l' r2 U2 r' U2 F2 r2 F2

(27,17)

Y

r' U2 r U2 r U2 l' r' U2 l r2 U2 r' U2 F2 r2 F2

(27,17)

Y

r' U2 l F2 l' F2 r2 U2 l F2 r' F2 l' r F2 r2 F2

(27,17)

Y

r' U2 l F2 r F2 l' r' F2 l r2 F2 l' U2 F2 r2 F2

(27,17)

Y

r' U2 l F2 l2 r U2 l r U2 l F2 l' U2 F2 r2 F2

(27,17)

Y

r' U2 l F2 r' U2 r2 U2 F2 r' F2 r F2 r2 U2 l' r

(27,17)

Y

r2 U2 r' U2 l F2 U2 r2 U2 r F2 l' U2 l U2 l' r

(27,17)

Y

r2 U2 r' F2 r F2 l' U2 l F2 r' U2 r' U2 r2 F2 r

(27,17)

Y

r2 U2 l' U2 l F2 r' F2 r F2 r' U2 r' U2 r2 F2 r

(27,17)

Y

r2 U2 l' U2 r U2 r' U2 l F2 r' U2 r' U2 r2 F2 r

(27,17)

Y

r2 F2 r F2 l2 r U2 F2 l' F2 l F2 r' F2 U2 l r

(27,17)

Y

r2 F2 r' U2 l F2 l' U2 r U2 r' F2 r' U2 r2 F2 r

(27,17)

Y

r2 F2 r' U2 l F2 r' F2 r F2 l' F2 r' U2 r2 F2 r

(27,17)

Y

r2 F2 r' U2 r U2 r' U2 l F2 l' F2 r' U2 r2 F2 r

(27,17)

Y

r2 F2 r' U2 r U2 r' U2 r U2 l' U2 r U2 l2 F2 l'

(27,17)

Y

l r2 F2 r F2 r F2 l' r' F2 r' U2 r' U2 F2 r2 F2

(27,17)

Y

r B2 r' U2 r U2 l2 B2 l' B2 l B2 U2 l2 U2 l r'

(27,17)

Y

r B2 l' B2 r B2 r2 U2 l' U2 r B2 U2 l2 U2 l r'

(27,17)

Y

r' U2 r U2 r' B2 r2 B2 U2 r' U2 l' r2 B2 l r B2

(27,17)

Y

r' U2 r U2 r' B2 r2 B2 U2 l' B2 r B2 l2 U2 l' r

(27,17)

Y

r2 U2 r B2 r U2 r2 B2 r B2 r U2 l' B2 l U2 r

(27,17)

Y

r2 U2 r' U2 r U2 B2 r2 B2 r U2 r' U2 l U2 l' r

(27,17)

Y

r2 U2 r' U2 r' B2 r2 U2 r' U2 l' r' B2 r B2 l B2

(27,17)

Y

r2 U2 r' U2 r' B2 l' r' B2 r' B2 r2 B2 r B2 l B2

(27,17)

Y

r2 U2 l' U2 r U2 r' U2 r U2 r' B2 r' B2 r2 U2 l

(27,17)

Y

r2 U2 l' B2 r B2 l r' B2 r2 B2 r U2 l' B2 l B2

(27,17)

Y

r2 B2 U2 l' U2 r U2 r' U2 B2 l r2 U2 l U2 l' r'

(27,17)

Y

r2 B2 U2 l' U2 r U2 r' U2 l r' U2 r' U2 r B2 r2

(27,17)

Y

r2 B2 r U2 l' B2 l r' B2 r' B2 r B2 r' U2 B2 r2

(27,17)

Y

r2 B2 r U2 r' B2 l' B2 l U2 l' B2 l U2 r2 U2 r'

(27,17)

Y

r2 B2 r U2 r' U2 B2 r' B2 r B2 l' B2 l r' B2 r2

(27,17)

Y

r2 B2 r U2 r' U2 l r' U2 l' B2 r B2 r' U2 B2 r2

(27,17)

Y

r2 B2 r U2 r' U2 l r' U2 r' U2 r U2 l' U2 B2 r2

(27,17)

Y

r2 B2 l' r B2 l' B2 l U2 r' U2 B2 r' B2 l B2 r2

(27,17)

Y

r U r U' F2 U r' U' l F2 l' F2 U2 r' U2 r2 F2 r

(25,18)

Y

r2 U2 l U2 r' U2 r F2 r F r' F U2 F' r F l' r2

(25,18)

Y

r2 F2 r F2 l2 r U2 r' F2 U' l' U F2 U' l U' l r

(25,18)

Y

r' F2 l' F2 l F2 l' U' l' U F2 U' l U F2 l2 F2 r

(25,18)

Y

r U2 r U2 r' U2 r B r B' U2 B r' B' U2 r2 U2 r'

(25,18)

Y

r2 U2 l' U2 r U' r' U B2 U' r U' r' U2 r' B2 l r2

(25,18)

Y

r2 B2 r U2 r' U2 B r' B' l U2 l' U2 l' B' l B' r2

(25,18)

Y

r2 B2 r U2 r' U2 B r' B' r B2 l' B2 r' B' l B' r2

(25,18)

Y

r2 B' r B' U2 B r' B' U2 l' B2 l r2 U2 l U2 l' r'

(25,18)

Y

(r2 U2)2 (r U2)2 r' U2 B2 U' r' U B2 U' r U'

(27,18)

N

Christopher Mowla

(r2 U2)2 (r U2)2 r' U2 B2 U r' U' B2 U r U

(27,18)

N

Christopher Mowla

r U2 r2 U2 r' U2 l F2 l' U2 l U2 r F2 r' U2 l' r'

(27,18)

Y

r U2 r U2 r' U2 r U2 r' F2 r F2 r' F2 l r2 F2 l'

(27,18)

Y

r U2 r U2 r' U2 r U2 r' F2 r F2 l' U2 l r2 U2 r'

(27,18)

Y

r U2 l F2 r' F2 l' r2 U2 l' U2 l F2 r' F2 r' U2 r'

(27,18)

Y

r U2 l F2 l' U2 r U2 l' U2 r U2 l r2 F2 r' U2 r'

(27,18)

Y

r' U2 r U2 r U2 r2 F2 r' U2 r' U2 F2 l' r' U2 l r'

(27,18)

Y

r' U2 r U2 r' F2 r2 U2 r U2 r' U2 l' r U2 l r F2

(27,18)

Y

r' U2 r U2 r' F2 l r F2 r F2 r' F2 l' r F2 r2 F2

(27,18)

Y

r' U2 r U2 l' U2 l r U2 l F2 r' F2 l' r F2 r2 F2

(27,18)

Y

r' U2 l F2 r' U2 r2 U2 F2 l' U2 r U2 l r U2 l' r

(27,18)

Y

r' U2 l F2 l' F2 r2 U2 l F2 l' U2 l' r U2 l r F2

(27,18)

Y

r' F2 r2 U2 r U2 r F2 r' F2 r F2 r' F2 l F2 l' r'

(27,18)

Y

r' F2 l' F2 r U2 l' U2 r U2 l' U2 l U2 l2 r' U2 l

(27,18)

Y

r' F2 l' F2 r U2 l' U2 l F2 l' F2 r U2 l2 r' U2 l

(27,18)

Y

r' F2 l' F2 r U2 l' U2 l F2 l' F2 l F2 l2 r' F2 r

(27,18)

Y

r' F2 l' F2 l F2 l' F2 r F2 l' F2 r U2 l2 r' U2 l

(27,18)

Y

r' F2 l' F2 l F2 l' F2 r F2 l' F2 l F2 l2 r' F2 r

(27,18)

Y

r' F2 l' F2 l F2 l' F2 l' r2 U2 r' F2 l F2 l U2 l

(27,18)

Y

r' F2 l2 F2 r U2 r' F2 r U2 r' U2 l' F2 l U2 l r

(27,18)

Y

r2 U2 r' F2 r F2 r' F2 r F2 r' U2 r' U2 l' r' U2 l

(27,18)

Y

r2 U2 r' F2 r F2 r' F2 r F2 r' U2 r F2 l r F2 l'

(27,18)

Y

r2 F2 U2 l' U2 l F' r' F U2 F' r F r2 U2 r2 F2 r

(27,18)

Y

r2 F2 r' U2 r U2 r' U2 r U2 r' F2 r' U2 l' r' U2 l

(27,18)

Y

r2 F2 r' U2 r U2 r' U2 r U2 r' F2 r F2 l r F2 l'

(27,18)

Y

r2 F2 l' r F2 l' U2 l U2 l' U2 F2 l' F2 l U2 l r

(27,18)

Y

l r' U2 l' U2 l F2 r' U2 r2 U2 F2 r' F2 r F2 l' r'

(27,18)

Y

r U2 r U2 l' B2 r B2 r' U2 r U2 r' U2 l r2 U2 r'

(27,18)

Y

r U2 r U2 l' B2 r B2 l' B2 l U2 r' U2 r' B2 l r2

(27,18)

Y

r U2 r U2 l' B2 l U2 l' U2 r U2 r' U2 r' B2 l r2

(27,18)

Y

r U2 l2 B2 l' B2 l' U2 l U2 l' U2 r B2 r' B2 l' r'

(27,18)

Y

r U2 l2 B2 l' B2 l' U2 r B2 l' B2 l B2 r' B2 l' r'

(27,18)

Y

r U2 l2 B2 l' B2 l' U2 r B2 r' U2 l U2 l' B2 l' r'

(27,18)

Y

r U2 l2 B2 l' U2 l' B2 l B2 r' U2 r B2 l' U2 l' r'

(27,18)

Y

r U2 l2 B2 l' U2 r' U2 l U2 l' U2 r B2 l' U2 l' r'

(27,18)

Y

r U2 l2 B2 l' U2 r' U2 r B2 l' B2 l B2 l' U2 l' r'

(27,18)

Y

r U2 l2 B2 r' B2 l B2 r' B2 r B2 r' B2 r U2 l r

(27,18)

Y

r B2 r' U2 r U2 l' r' U2 l' U2 r B2 U2 l2 U2 l r'

(27,18)

Y

r B2 r' U2 l B2 l2 B2 U2 r B2 r' U2 l' r' U2 l' r

(27,18)

Y

r B2 l B2 r' U2 l B2 l B2 l2 U2 l B2 r B2 l' r'

(27,18)

Y

r B2 l' B2 r B2 l' r' B2 l' B2 l B2 U2 l2 U2 l r'

(27,18)

Y

r B2 l' B2 r' U2 l r U2 l B2 l B2 U2 l2 U2 l r'

(27,18)

Y

r' U2 r U2 r' B2 l r U2 l' r U2 r' U2 r U2 r2 B2

(27,18)

Y

r' U2 r U2 r' B2 r2 B2 l' r B2 r' B2 r B2 l r B2

(27,18)

Y

r' U2 r U2 r' B2 r2 B2 U2 r' U2 r' B2 l' r' B2 l r

(27,18)

Y

r' U2 r U2 r' B2 r2 B2 U2 r' U2 r U2 l r U2 l' r

(27,18)

Y

r' U2 r' U2 l' r' B2 r B2 l r B2 r U2 r U2 r2 B2

(27,18)

Y

r' U2 r' U2 r U2 r' B2 r' B2 r2 U2 r' B2 l' B2 l r

(27,18)

Y

r' U2 r U2 r U2 l' r' U2 r' B2 r' B2 U2 r2 U2 l r'

(27,18)

Y

l r' U2 l' U2 r U2 r' B2 r2 B2 U2 l' B2 r B2 l r

(27,18)

Y

r2 U2 l' B2 l U2 l r' U2 l' r' B2 r U2 l' B2 l B2

(27,18)

Y

r2 B2 r U2 r' U2 B2 r' B2 r B2 r' U2 l r' U2 l' r'

(27,18)

Y

Rw U2 Rw2 U L U r U' L' U2 L U r' U' L' U Rw' U2 r U2 Rw' U2 Rw'

(29,23)

Y

Christopher Mowla

Rw U2 Rw2 U L' U r U' L U2 L' U r' U' L U Rw' U2 r U2 Rw' U2 Rw'

(29,23)

N

Christopher Mowla

2-Gen and "Semi 2-gen" algorithms

• 2 Gen (short for "2 generator") algorithms are algorithms which only use two different types of turns, such as r and U or Rw and U.
• There are five types (move sets) of 2-gen parity algorithms which can exist on the 4x4x4.
• Several 2-gen algorithms in this section were found using Kåre Krig's Ksolve program or Ben Whitmore's ksolve++ program.
• Although all 2-gen algorithms in this section are move optimal in their respective move sets (with the exception of the algorithms in the move set <Rw, 3Uw>, whose optimal solution is unknown but of which Tom Rokicki established a lowerbound of 50 and an upperbound of 87 half turn moves in 2018), here is a video tutorial by Christopher Mowla which shows how to create longer (but more humanly comprehensible) algorithms for all five types of 2-gen 4x4x4 single dedge flip algorithms by hand which are about double the length of optimal solutions in their respective move sets (on average).
  • The two exceptions to this general length trend (of being double move optimal in length) are algorithms for 'Type 3' (which Christopher later found an algorithm for which is about double the move optimal length, after he uploaded the video series) and algorithms for 'Type 5'.
    • No humanly comprehensible algorithms close to optimal in length for the 'Type 5' 2-gen move set have been found; however, there are short repeated sequences which achieve a single dedge flip on the 4x4x4 in this move set at the end of the following list of algorithms.
  • All five types of 2-gen cases have been named according to which of the five types they are labelled as in this video series.


2-Gen

r U2 r' U2 r U2 r U r U r U2 r2 U2 r U2 r U r' U' r2 U2

(31,22)

Type 1

N

Lucas Garron

r U2 r' U2 r U2 r U' r U' r U2 r2 U2 r U2 r U' r' U r2 U2

(31,22)

Type 1

N

Lucas Garron

r2 U r' U' r U2 r U2 r2 U2 r U' r U' r U2 r U2 r' U2 r U2

(31,22)

Type 1

N

Lucas Garron

r2 U' r' U r U2 r U2 r2 U2 r U r U r U2 r U2 r' U2 r U2

(31,22)

Type 1

N

Lucas Garron

y' Rw' U' Rw U' Rw U Rw2 U' Rw2 U2 Rw U2 Rw' U2 Rw U2 Rw U' Rw U Rw' U' Rw U2 Rw' U2 y

(34,26)

Type 4

Y

Bruce Norskog

y' Rw' U' Rw U' Rw U' Rw2 U2 Rw2 U' Rw U2 Rw' U2 Rw U2 Rw U Rw U Rw' U' Rw U2 Rw' U2 y

(34,26)

Type 4

Y

Bruce Norskog

y Rw' U2 Rw U' Rw' U Rw U Rw U2 Rw U2 Rw' U2 Rw U' Rw2 U2 Rw2 U' Rw U' Rw U' Rw' U2 y'

(34,26)

Type 4

Y

Bruce Norskog

y Rw' U2 Rw U' Rw' U Rw U' Rw U2 Rw U2 Rw' U2 Rw U2 Rw2 U' Rw2 U Rw U' Rw U' Rw' U2 y'

(34,26)

Type 4

Y

Bruce Norskog

r2 Uw2 r' Uw r Uw2 r Uw r Uw r Uw' r' Uw' r Uw r Uw r' Uw2 r2 Uw2 r' Uw' r Uw2 r2

(35,27)

Type 3

Y

Ben Whitmore

r 3Uw r2 3Uw2 r2 3Uw2 r 3Uw' r 3Uw r 3Uw r' 3Uw' r' 3Uw' r' 3Uw r' 3Uw2 r2 3Uw2 r2 3Uw' r 3Uw2 r2 3Uw2 r2

(40,29)

Type 2

N

Ben Whitmore

r2 3Uw r 3Uw2 r 3Uw2 r' 3Uw r 3Uw r' 3Uw2 r 3Uw2 r2 3Uw2 r' 3Uw2 r' 3Uw' r' 3Uw' r 3Uw2 r2 3Uw2 r' 3Uw' r2

(40,29)

Type 2 (5x5x5 Alg)

N

Ben Whitmore

y Rw U Rw2 U' Rw U' Rw' U Rw U' Rw2 U2 Rw' U2 Rw U2 Rw' U Rw2 U' Rw U Rw' U' Rw U2 Rw U2 Rw2 U Rw' U' Rw' y'

(42,33)

Type 4 (5x5x5 Alg)

Y

Ben Whitmore

(Rw' 3Uw Rw 3Uw' Rw' 3Uw' Rw 3Uw Rw 3Uw' Rw' 3Uw Rw 3Uw Rw2 3Uw' Rw 3Uw Rw 3Uw Rw' 3Uw Rw 3Uw2 Rw 3Uw Rw 3Uw2 Rw' 3Uw Rw)3

(98,87)

Type 5

Y

Tom Rokicki

(3Uw Rw' 3Uw2 Rw2 3Uw2 Rw 3Uw' Rw 3Uw2 Rw 3Uw2 Rw' 3Uw2 Rw' 3Uw' Rw' 3Uw2 Rw 3Uw Rw' 3Uw2 Rw 3Uw' Rw2 3Uw2 Rw' 3Uw Rw2 3Uw Rw2 3Uw' Rw2 3Uw Rw2)3

(144,102)

Type 5

Y

xyzzy

z' (3Uw' Rw2 3Uw2 Rw2 3Uw2 Rw 3Uw2 Rw' 3Uw2 Rw)15 z

(240,150)

Type 5

Y

Christopher Mowla

(3Uw2 Rw 3Uw Rw 3Uw2 Rw' 3Uw' Rw 3Uw Rw)315

(3780,3150)

Type 5

Y

Tom Rokicki

z (3Uw' Rw2 3Uw Rw 3Uw' Rw' 3Uw2 Rw)585 z'

(5850,4680)

Type 5

Y

Christopher Mowla

Semi 2-Gen

Rw U2 Rw U Rw U2 Rw U2 Rw' U2 Rw' U r' U' Rw U2 Rw U2 Rw' U2 Rw' U r2 U2 Rw' U2 Rw'

(37,27)

N

Christopher Mowla

Rw U2 Rw U' Rw U2 Rw U2 Rw' U2 Rw' U' r' U Rw U2 Rw U2 Rw' U2 Rw' U' r2 U2 Rw' U2 Rw'

(37,27)

N

Christopher Mowla

4x4x4 supercube safe algorithms

• All of the algorithms in this section "prove" that one can just switch two 4x4x4 supercube wing edges without affecting its centers.
This is not true for the nxnxn supercube in general, but there is a mathematically quantifiable minimum number of center pieces which must be switched along with two wing edges, for example, of the 5x5x5 and larger nxnxn supercubes.

r' U' u r U' r U r u' r' u r U r U' r Uw' r' U2 //Safe

(20,19)

N

Ben Whitmore

r' Uw' r U' r U r u r' u' r U r U' r u U' r' U2 //Safe

(20,19)

N

Ben Whitmore

r' U2 r' U' r U r U r' D' f2 r f2 r' D U2 r' U r U r U r //Safe

(27,23)

N

Rw2 U2 r' U2 r' U' l r u2 l' r' U l r u2 x U2 l' U2 r U2 r' U2 Rw2 x' //Safe

(33,23)

N

Per Kristen Fredlund

r2 U (r2 U2)2 (r' U)2 r' (U2 r2)2 U' r U' (r U2)3 r U' r U2 r2 //Safe

(42,29)

N

Ben Whitmore

(Rw' U')(U' r2 U Rw (r2 u r2 u') U' r' U Rw')3 (U Rw) //Safe

(43,36)

N

Christopher Mowla




Three dedge flips

In the last layer



u2 f' U f2 Lw2 U2 f U2 B2 Lw2 f2 U f u2 Lw2 B2 Lw2

(29,17)

N

Clément Gallet

R2 b' U' (L2 f2)2 U b U f' U2 (f M2)2 U R2

(27,18)

N

Clément Gallet

R2 D' l D2 r' B2 r' D2 l2 D2 F2 r' B2 r F2 r2 D' R2

(29,18)

N

Walter Randelshofer

y M U r B2 l' D2 l' B2 r2 B2 U2 l' D2 l U2 l2 B2 U' M' y'

(29,19)

N

y M U r2 D2 r' U2 r D2 B2 l2 B2 r U2 r B2 l' B2 U' M' y'

(29,19)

N

M2' U' x' r2 U2 M r' U2 r F2 U2 r2 U2 M' r x U2 r U2 r' U' M2'

(31,20)

N

y M U l r F2 l B u' B D2 B' u B r' D2 B2 l2 F2 r' U' M' y'

(27,21)

N

Christopher Mowla

y M U l r F2 l B' d' B' D2 B d B' r' D2 B2 l2 F2 r' U' M' y'

(27,21)

N

Christopher Mowla

y M U r' D2 l' r' B2 D2 l' B u B' D2 B u' B l D2 r2 U' M' y'

(27,21)

N

Christopher Mowla

y M U r' D2 l' r' B2 D2 l' B' d B D2 B' d' B' l D2 r2 U' M' y'

(27,21)

N

Christopher Mowla

y M U l' r' U2 l' D' f D' B2 D f' D' r B2 D2 l2 U2 r U' M' y'

(27,21)

N

Christopher Mowla

y M U l' r' U2 l' D b D B2 D' b' D r B2 D2 l2 U2 r U' M' y'

(27,21)

N

Christopher Mowla

Rw U Rw U2 Rw U2 Rw U Rw' U2 r' U2 Rw' U Rw U2 Rw U2 Rw U Rw

(27,21)

v1

N

Kåre Krig

Rw' U' Rw' U2 Rw' U2 Rw' U' Rw U2 r U2 Rw U' Rw' U2 Rw' U2 Rw' U' Rw'

(27,21)

v2

N

Kåre Krig

Rw U' R' U' r U' R U' r2 U R' U r' U R U r U2 r' U2 Rw' r'

(25,22)

N

Ben Whitmore

Uw2 F2 Lw' U' L U Lw U2 Lw' U' L' U' l' U2 Lw' F2 l F2 Lw' U2 Lw' l' F2 Uw2

(33,24)

N

Christopher Mowla

Uw2 B2 Lw U L' U' Lw' U2 Lw U L U l U2 Lw B2 l' B2 Lw U2 Lw l B2 Uw2

(33,24)

N

Christopher Mowla

R2 f' U2 b L2 b' L2 f2 U2 f U2 f' U2 L2 b2 U' S2 L2 S L2 U L2 S' R2

(39,24)

N

André Boulouard

M2' U' l U2 r' F2 r D2 F2 r F2 r' Dw2 l r U' l' r' d2 l r U l M F2 l U' M2' //Safe

(38,28)

N

Per Kristen Fredlund

M2' U' r' U2 l F2 l' D2 F2 l' F2 l Dw2 l' r' U l r d2 l' r' U' r' M F2 r' U' M2' //Safe

(38,28)

N

Per Kristen Fredlund

L R U2 L' R' U' L R U2 L2 x' U B2 r2 U2 r D2 r' U2 x' U2 l2 U2 r' B2 r' U2 x l U' M' U' //Safe

(42,29)

N

François Courtès

y Rw U Rw2 U2 Rw2 U Rw' U Rw2 U Rw2 U2 Rw U' Rw' U' Rw U' Rw U' Rw U' Rw' U2 Rw' U2 Rw U2 Rw2 U2 Rw' y'

(42,31)

Y

Ben Whitmore

y2 3Uw' r2 3Uw2 r 3Uw' r' 3Uw2 r2 3Uw2 r' 3Uw2 r 3Uw2 r' 3Uw2 r2 3Uw2 r2 3Uw2 r2 3Uw2 r 3Uw2 r' 3Uw2 r2 3Uw2 r 3Uw' r' 3Uw2 r2 3Uw y2

(53,33)

N

Ben Whitmore

In the M layer



r B2 l' D2 l' B2 r2 B2 U2 l' D2 l U2 l2 B2

(25,15)

N

l' B2 r D2 r B2 l2 B2 U2 r D2 r' U2 r2 B2

(25,15)

N

B2 l2 U2 l' D2 l U2 B2 r2 B2 l D2 l B2 r'

(25,15)

N

l D2 r' B2 r' D2 l2 D2 F2 r' B2 r F2 r2 D2

(25,15)

N

B2 r2 U2 r D2 r' U2 B2 l2 B2 r' D2 r' B2 l

(25,15)

N

r' D2 l B2 l D2 r2 D2 F2 l B2 l' F2 l2 D2

(25,15)

N

D2 r2 F2 r' B2 r F2 D2 l2 D2 r B2 r D2 l'

(25,15)

N

D2 l2 F2 l B2 l' F2 D2 r2 D2 l' B2 l' D2 r

(25,15)

N

r2 D2 r' U2 r D2 B2 l2 B2 r U2 r B2 l' B2

(25,15)

N

l2 D2 l U2 l' D2 B2 r2 B2 l' U2 l' B2 r B2

(25,15)

N

B2 l B2 r' U2 r' B2 l2 B2 D2 r' U2 r D2 r2

(25,15)

N

l2 B2 l' F2 l B2 D2 r2 D2 l F2 l D2 r' D2

(25,15)

N

B2 r' B2 l U2 l B2 r2 B2 D2 l U2 l' D2 l2

(25,15)

N

r2 B2 r F2 r' B2 D2 l2 D2 r' F2 r' D2 l D2

(25,15)

N

D2 r D2 l' F2 l' D2 r2 D2 B2 l' F2 l B2 l2

(25,15)

N

D2 l' D2 r F2 r D2 l2 D2 B2 r F2 r' B2 r2

(25,15)

N

l r F2 l B u' B D2 B' u B r' D2 B2 l2 F2 r'

(23,17)

N

Christopher Mowla

l r F2 l B' d' B' D2 B d B' r' D2 B2 l2 F2 r'

(23,17)

N

Christopher Mowla

r' D2 l' r' B2 D2 l' B u B' D2 B u' B l D2 r2

(23,17)

N

Christopher Mowla

r' D2 l' r' B2 D2 l' B' d B D2 B' d' B' l D2 r2

(23,17)

N

Christopher Mowla

l' r' U2 l' D' f D' B2 D f' D' r B2 D2 l2 U2 r

(23,17)

N

Christopher Mowla

l' r' U2 l' D b D B2 D' b' D r B2 D2 l2 U2 r

(23,17)

N

Christopher Mowla

r F2 r2 F2 U2 D l D' F2 D l' D' r F2 r' U2 r2

(24,17)

N

Christopher Mowla

r F2 r2 F2 U2 D' l D F2 D' l' D r F2 r' U2 r2

(24,17)

N

Christopher Mowla

r F2 r2 F2 U2 D' l D F2 D' l' D r F2 r' U2 r2

(24,17)

N

Christopher Mowla

r F2 r2 F2 U2 D l D' F2 D l' D' r F2 r' U2 r2

(24,17)

N

Christopher Mowla

r' D2 l2 D2 F2 l' D b D' F2 D b' D M l D2 r2

(24,17)

N

Christopher Mowla

r' D2 l2 D2 F2 l' D' f D F2 D' f' D' M l D2 r2

(24,17)

N

Christopher Mowla

r2 U2 M' r U2 r' B2 U2 r2 U2 M r' B2 r' B2 r B2

(27,17)

N

r2 F2 r U2 r U2 F2 r2 U2 F2 r' U2 r U2 F2 r U2

(29,17)

N

r2 U2 F2 r U2 F2 r U2 r2 U2 F2 r U2 r F2 r' U2

(29,17)

N

(Rw' U' F' R' F R2 U' Lw' E' R2 E) r' (E' R2 E Lw U R2 F' R F U Rw) (r U2 r' U2) (U2 r2 U2 r2 U2 r2 U2)

(41,31)

N

Christopher Mowla

(Rw' U' F' R' F R2 U') r (U R2 F' R F U Rw) (D' U' F2 U D) (r' F2 r F2) (F2 r2 F2 r2 F2 r2 F2) (D' U' F2 U D)

(43,33)

N

Christopher Mowla

In miscellaneous portions of the cube

• There are many possible 3 dedge flip cases outside of the confinement of a single layer (e.g., the last layer) and the M layer. All algorithms are labeled/named according to which dedges they flip.

F2 D L' (b L2 f' D2 f' L2 b2 L2 U2 f' D2 f U2 f2 L2) L D' F2

(31,20)

(ur)(rf)(fu)

N

Walter Randelshofer

U L' u2 L2 F2 d F2 d' F2 d2 L2 u L2 u' F2 u L' U L E L2 E2 L U2

(36,24)

(ur)(rf)(fu)

N

André Boulouard

Rw U2 Rw U2 Rw U R U Rw U2 Rw' U' R' U' Rw' R' U R U Rw U2 Rw' U' R' U' Rw U2 Rw' U2 Rw'

(36,30)

(ur)(rf)(fu)

Y

Christopher Mowla

F (D' l D2 r' B2 r' D2 l2 D2 F2 r' B2 r F2 r2 D') F'

(27,18)

(ub)(rd)(fl)

N

Walter Randelshofer

R F (l' B2 r D2 r B2 l2 B2 U2 r D2 r' U2 r2 B2) F' R'

(27,19)

(ub)(rd)(fl)

N

Walter Randelshofer

E' Rw' B2 l U2 l' U2 r2 B2 r B2 r' B2 U2 r2 U2 E' R2 F2 E R' E R F2 R'

(37,24)

(ub)(rd)(fl)

N

Walter Randelshofer

F2 (D' l D2 r' B2 r' D2 l2 D2 F2 r' B2 r F2 r2 D') F2

(29,18)

(ub)(rd)(fd)

N

Walter Randelshofer

(Rw' U' F' R' F R2 U') r (U R2 F' R F U Rw)(D' F2 D)(r' F2 r F2)(F2 r2 F2 r2 F2 r2 F2)(D' F2 D)

(39,29)

(ub)(rd)(fd)

N

Christopher Mowla

Five dedge flips

• There are many possible 5 dedge flip cases. All algorithms are labeled/named according to which dedges they flip.

R2 U R2 f D2 f' R2 f R2 b2 D2 b' D2 b D2 R2 f2 U' S2 R2 S'

(34,21)

(ur)(uf)(ul)(ub)(dr)

N

Walter Randelshofer

L2 S' R2 U R2 f D2 f' R2 f R2 b2 D2 b' D2 b D2 R2 f2 U' S2 M2 x2

(36,22)

(ur)(uf)(ul)(ub)(dr)

N

André Boulouard

Rw U Rw2 U Rw' U' Rw U' Rw' U' Rw U' Rw U Rw' U Rw' U' Rw U2 Rw2 U2 Rw U Rw' U2 Rw U' Rw2 U Rw2 U2 Rw U2 Rw' U' Rw2 U' Rw2 U Rw'

(52,41)

(ur)(uf)(ul)(ub)(dr)

Y

Ben Whitmore

Rw U Rw U' Rw2 U2 Rw' U' Rw U Rw U Rw U Rw' U2 Rw' U2 Rw U2 Rw U' Rw' U' Rw2 U Rw' U Rw' U Rw2 U2 Rw' U2 Rw U2 Rw' U2 Rw2 U2 Rw

(54,41)

(ur)(uf)(ul)(ub)(fr)

Y

Ben Whitmore

Seven dedge flips

• There are many possible 7 dedge flip cases. All algorithms are labeled/named according to which dedges they flip.

D' M' F2 L2 F2 U' D' r' B2 l U2 r' B2 r2 B2 U2 r' U2 r U2 Rw2 B2 U'

(36,23)

(uf)(ul)(ub)(dr)(df)(dl)(db)

N

André Boulouard

u R2 d' F2 d' R2 u2 R2 B2 d' F2 d B2 d2 L2 B2 M' B2 D M2 F2 M F2 D' x2

(39,24)

(ur)(uf)(ul)(ub)(fl)(lb)(br)

N

Walter Randelshofer

Nine dedge flips

• There are several possible 9 dedge flip cases. All algorithms are labeled/named according to which dedges they flip.

D' S L2 B2 d2 F2 d F2 d' F2 L2 d2 L2 d' F2 u L2 Uw' D' S' R2 F2 R2 U'

(38,24)

(ur)(uf)(ul)(ub)(rf)(dr)(df)(dl)(db)

N

André Boulouard

Eleven dedge flips



• There is only one case of the 11 dedge flip (when applying symmetry). All algorithms listed solve/generate the case shown in the image above.

Rw2 F2 l U2 x' l U2 l2 x' U2 r' U2 l D2 F2 l U2 L R E M S x

(30,20)

N

Herbert Kociemba &Christopher Mowla

r D2 x U2 l D2 r' D2 l2 F2 l U2 l x' U2 r2 F2 R2 E L' R U' D y x R2

(31,20)

N

Herbert Kociemba

Lw2 F2 M r' F2 r' U2 r2 B2 r D2 l' B2 D2 r' F2 L2 E' M S' x2

(31,20)

N

Herbert Kociemba &Christopher Mowla

Rw2 F2 l U2 r' x' U2 l2 F2 r D2 M l x U2 B2 r F2 R2 E M S x2

(31,20)

N

Herbert Kociemba &Christopher Mowla

l B2 U2 r D2 l' D2 r2 B2 r U2 r F2 l2 U2 L2 S' M' E' 3Rw2

(32,20)

N

Herbert Kociemba &Walter Randelshofer

r D2 F2 l B2 r' B2 l2 D2 l F2 l U2 r2 F2 R2 E M S 3Lw2

(32,20)

N

Herbert Kociemba &Walter Randelshofer

r B2 r' U2 r' B2 l2 B2 D2 r' U2 r D2 Rw2 S' U2 B2 M' U2 R2 E' F2 U2 x2

(38,23)

N

Christopher Mowla

r B2 r' U2 r' B2 l2 B2 D2 r' U2 r D2 Rw2 S' U2 B2 M' D2 L2 E' B2 D2 x2

(38,23)

N

Christopher Mowla

y x2 r U' F2 U l F2 r U2 r U' B2 U r D2 L' R' B2 L2 B' R2 B D2 B E F M y2 x

(35,26)

N

Christopher Mowla

R' D2 R' E2 S' Rw' U2 l F2 r' U2 r2 U2 F2 r' F2 r F2 Rw2 S' R S2 E' L' B2 R' S2 z2

(40,27)

N

André Boulouard

D2 L' S2 R' B2 L' E S2 R S' Rw2 F2 r F2 r' F2 D2 r2 D2 r' F2 l D2 Rw' S' E2 R' z2

(41,27)

N

André Boulouard

M D2 F2 Rw2 F2 r' F2 r F2 U2 r2 U2 r F2 l' U2 l F' E F' U2 F E' B' D' U' S L2 x y'

(41,28)

N

André Boulouard

M S r U2 r F2 r' D2 r E2 r U' f2 Rw2 U f2 U' Rw2 f2 D' B2 R2 B2 L2 D2 Fw2 U2 Fw2 y

(45,28)

N

Bruce Norskog

y d E M R D R' Bw f M d' M' Fw b M' F Uw M' R' S L F' U R D' R' E S2 R F' z'

(30,29)

N

Bruce Norskog &Christopher Mowla

Double Parity Cases Only Affecting Dedges

In the last layer

Opposite double parity



• This is a 4-cycle case. Manually take the inverse of these algorithms to solve its inverse case, as it and its inverse are not the same case.

r B2 r' U2 r U2 r' F2 Rw D2 l D2 Rw' (S2) z2

(21,14)

Alg.1

Y

l' B2 l U2 l' U2 l F2 Lw' D2 r' D2 Lw (S2) z2

(21,14)

Mirror[Alg.1]

Y

l' B2 l U2 l' U2 l F2 Rw' B2 r' B2 Rw (S2) z2

(21,14)

Alg.2

Y

r B2 r' U2 r U2 r' F2 Lw B2 l B2 Lw' (S2) z2

(21,14)

Mirror[Alg.2]

Y

(S2) Lw' U2 r U2 Lw B2 l' D2 l D2 l' F2 l z2

(21,14)

Alg.3

Y

(S2) Rw U2 l' U2 Rw' B2 r D2 r' D2 r F2 r' z2

(21,14)

Mirror[Alg.3]

Y

(S2) Rw' F2 r F2 Rw B2 l' D2 l D2 l' F2 l z2

(21,14)

Alg.4

Y

(S2) Lw F2 l' F2 Lw' B2 r D2 r' D2 r F2 r' z2

(21,14)

Mirror[Alg.4]

Y

r U2 r' (E2) F2 l F2 l' F2 r F2 r' D2 l y2

(21,14)

Alg.5

Y

Kenneth Gustavsson

l' U2 l (E2) F2 r' F2 r F2 l' F2 l D2 r' y2

(21,14)

Mirror[Alg.5]

Y

Kenneth Gustavsson

r U2 r' (E2) F2 l F2 r' D2 r D2 l' D2 l y2

(21,14)

Alg.6

Y

l' U2 l (E2) F2 r' F2 l D2 l' D2 r D2 r' y2

(21,14)

Mirror[Alg.6]

Y

r U2 r' (E2) F2 r U2 l' U2 l F2 r' D2 l y2

(21,14)

Alg.7

Y

l' U2 l (E2) F2 l' U2 r U2 r' F2 l D2 r' y2

(21,14)

Mirror[Alg.7]

Y

l' U2 l (E2) F2 r' F2 l D2 r' B2 r B2 l' y2

(21,14)

Alg.8

Y

r U2 r' (E2) F2 l F2 r' D2 l B2 l' B2 r y2

(21,14)

Mirror[Alg.8]

Y

l F2 r' (S2) D2 l D2 r' B2 l U2 l' U2 l z2

(21,14)

Alg.9

Y

r' F2 l (S2) D2 r' D2 l B2 r' U2 r U2 r' z2

(21,14)

Mirror[Alg.9]

Y

r D2 l' F2 l F2 r' F2 r F2 (E2) l' U2 l y2

(21,14)

Alg.10

Y

l' D2 r F2 r' F2 l F2 l' F2 (E2) r U2 r' y2

(21,14)

Mirror[Alg.10]

Y

r D2 r' D2 l D2 l' F2 r F2 (E2) l' U2 l y2

(21,14)

Alg.11

Y

l' D2 l D2 r' D2 r F2 l' F2 (E2) r U2 r' y2

(21,14)

Mirror[Alg.11]

Y

r D2 l' F2 r U2 r' U2 l F2 (E2) l' U2 l y2

(21,14)

Alg.12

Y

l' D2 r F2 l' U2 l U2 r' F2 (E2) r U2 r' y2

(21,14)

Mirror[Alg.12]

Y

l B2 r' B2 r D2 l' F2 r F2 (E2) l' U2 l y2

(21,14)

Alg.13

Y

r' B2 l B2 l' D2 r F2 l' F2 (E2) r U2 r' y2

(21,14)

Mirror[Alg.13]

Y

l' D2 l D2 l' F2 r U2 l' U2 (S2) r B2 l' z2

(21,14)

Alg.14

Y

r D2 r' D2 r F2 l' U2 r U2 (S2) l' B2 r z2

(21,14)

Mirror[Alg.14]

Y

r2 B2 r' U2 r' U2 x' U2 r' U2 r U2 r' U2 r2 U2 x

(25,15)

Y

Frédérick Badie

x r F2 Lw' F r' F' U2 F r F' U2 Lw x' E2 l' U2 l y2

(21,16)

N

Christopher Mowla

x r F2 Lw' F' r' F U2 F' r F U2 Lw x' E2 l' U2 l y2

(21,16)

Y

Christopher Mowla

Rw2 B2 U2 l r2 U2 r' U2 r U2 F2 r F2 l' B2 Rw2

(27,16)

Y

Alexander Ooms

Rw U2 r' U2 r2 U2 M' U2 r U2 r' U2 l U2 r' U2 Rw'

(26,17)

N

Jayden McNeill

Rw' U2 r' U2 l U2 Rw' U2 r U2 3Rw U2 r2 U2 r' U2 Rw

(26,17)

Y

r2 U2 3Rw' U2 r' U2 Rw U2 l' U2 Rw U2 r U2 Rw' U2 r

(26,17)

Y

Rw2 U2 r' U2 r' U2 r U2 M' U2 r U2 r' U2 M U2 Rw2

(27,17)

N

Thom Barlow

Rw2 U B2 U' r' U B2 U' Rw r U B2 U r' U' B2 U' Rw

(23,18)

v1

N

Christopher Mowla

Rw2 3Dw L2 3Dw' r' 3Dw L2 3Dw' Rw r 3Dw L2 3Dw l' 3Dw' L2 3Dw' Rw

(23,18)

v2

N

Christopher Mowla

r2 U' B2 U r' U' B2 U r2 U' B2 U' r' U B2 U r

(23,18)

v1

N

Christopher Mowla

r2 3Dw' R2 3Dw r' 3Dw' R2 3Dw r2 3Dw' R2 3Dw' l' 3Dw R2 3Dw r

(23,18)

v2

N

Christopher Mowla

Rw U2 Lw' U2 l U2 Lw U2 Rw' U2 l U2 l' U2 x U2 l2 U2 x' U2

(29,18)

Y

r' f R' f Uw f' Uw' Rw u f' Uw' R' Uw' r' Uw R f' U r

(19,19)

Alg.1

N

Ben Whitmore

U2 x' r' (F2 U' l' U)5 r x

(29,23)

Y

Christopher Mowla

U2 x' r' (F2 U' l U)5 r x

(29,23)

Y

Christopher Mowla

U2 x' r' (U' l' U F2)5 r x

(29,23)

Y

Christopher Mowla

U2 x' r' (U' l U F2)5 r x

(29,23)

Y

Christopher Mowla

(F d' S r F Rw2 f2 u' r2 Uw2 S') r (S Uw2 r2 u f2 Rw2 F' r' S' d F')

(31,23)

N

Christopher Mowla

(B Lw2 U' L' U r u2 b' r2 Bw2 E) r' (E' Bw2 r2 b u2 r' U' L U Lw2 B')

(31,23)

N

Christopher Mowla

r' U' R' U' r' U R U r U2 r' U' R' U' r2 U R U r U2 r U2 r' U2

(29,24)

N

Christopher Mowla

R U Rw U2 Rw U2 Rw' U' Rw2 U2 R2 Rw U' R' U' Rw U2 R' Rw2 U Rw2 U' R' U

(30,24)

N

xyzzy

U R' U' Rw2 U R' Rw2 U2 Rw U' R' U' R2 Rw U2 Rw2 U' Rw' U2 Rw U2 Rw U R

(30,24)

N

xyzzy

U Rw U2 Rw U2 Rw' U' Rw2 U2 R2 Rw U' R' U' Rw U2 R' Rw2 U Rw2 U' R' U R

(30,24)

N

xyzzy

R U R' U' Rw2 U R' Rw2 U2 Rw U' R' U' R2 Rw U2 Rw2 U' Rw' U2 Rw U2 Rw U

(31,24)

N

xyzzy

(z' Lw Uw2 U' Rw L2 U2 Lw' U' Lw R F2 U' R' U) r' (U' R U F2 R' Lw' U Lw U2 L2 Rw' U Uw2 Lw' z)

(33,29)

1 Slicer

N

Christopher Mowla

U' r U r2 U2 r U2 r2 U' r' U' r' U2 r2 U r2 U r U2 r' U r' U' r' U2 r2 U' r' U' r2 //Safe

(41,30)

N

Ben Whitmore

(z' Lw Uw2 U' Rw L2 U2 Lw' U' Rw R U2 3Lw U' R2 U) r' (U' R2 U 3Lw' U2 R' Rw' U Lw U2 L2 Rw' U Uw2 Lw' z)

(37,31)

1 Slicer(2)

N

Christopher Mowla

y Rw' U2 Rw U' Rw' U Rw' U' Rw2 U Rw2 U2 Rw' U2 Rw U2 Rw' U2 Rw' U Rw' U' Rw U' Rw U2 Rw U2 Rw2 U2 Rw U2 Rw2 y'

(46,33)

Y

Ben Whitmore

Adjacent double parity



• This is a 4-cycle case. Manually take the inverse of these algorithms to solve its inverse case, as it and its inverse are not the same case.

y' l' f U R D U L' u L D' U' R' U' f l f2 y

(17,16)

N

Tom Rokicki

y' l2 f l U' B' D' U' F u F' D U B U l f' y

(17,16)

N

Tom Rokicki

y' r b' U R D U L' d' L D' U' R' U' b' r' b2 y

(17,16)

N

Tom Rokicki

y' r2 b' r' U' B' D' U' F d' F' D U B U r' b y

(17,16)

N

Tom Rokicki

y' l f U2 L2 U' L' R' d L R U L2 U2 f l' f2 y

(21,16)

N

Tom Rokicki

y' l2 f' l U2 F2 U F B d F' B' U' F2 U2 l f y

(21,16)

N

Tom Rokicki

y' r2 b r' U2 F2 U F B u' F' B' U' F2 U2 r' b' y

(21,16)

N

Tom Rokicki

y' r' b' U2 L2 U' L' R' u' L R U L2 U2 b' r b2 y

(21,16)

N

Tom Rokicki

y' f2 l f' U2 L2 U L' R' d' L R U' L2 U2 f' l' y

(21,16)

N

Tom Rokicki

y' f' l' U2 F2 U' F B d' F' B' U F2 U2 l' f l2 y

(21,16)

N

Tom Rokicki

y' b r U2 F2 U' F B u F' B' U F2 U2 r b' r2 y

(21,16)

N

Tom Rokicki

y' b2 r' b U2 L2 U L' R' u L R U' L2 U2 b r y

(21,16)

N

Tom Rokicki

(y' r2 F Rw' F') r2 U2 (r' U2)3 (F Rw F' r2 y)

(23,16)

N

Christopher Mowla

(y Rw' z' L' U r F) r2 U2 (r' U2)3 (F' r' U' L z Rw y')

(23,18)

N

Christopher Mowla

Dw2 D F2 U l' F2 l F2 U l D' U2 r' D F2 D' r Dw2

(23,18)

N

Clément Gallet

Dw2 D F2 U l' F2 l F2 U l D U2 r' D' F2 D r d2

(23,18)

N

Clément Gallet

(x' r2 U' Rw U) M' U2 (r' U2)3 l r (U' Rw' U r2 x)

(24,18)

N

Christopher Mowla

u2 D' F2 U l' F2 l F2 U l D U2 r' D' F2 D r u2

(24,18)

N

Clément Gallet

R B y2 r U2 r' E2 F2 l F2 l' F2 r F2 r' D2 l B' R'

(25,18)

N

Kenneth Gustavsson

l2 U' B2 U l' U' l' B2 l U F2 U' B2 l B2 U F2 l'

(25,18)

N

Clément Gallet

l2 U' B2 U l' F2 U' Lw' B2 l B2 Lw U F2 U' B2 U l'

(25,18)

N

Clément Gallet

l2 U' B2 U l' F2 U' Rw' U2 l U2 Rw U F2 U' B2 U l'

(25,18)

N

Clément Gallet

R B' r2 B2 r B2 r' B2 r B2 U2 r U2 r B2 r2 B' R'

(27,18)

N

l2 F2 U' B2 U F2 l' F2 U' Lw' B2 l B2 Lw B2 U F2 l'

(27,18)

N

Clément Gallet

l2 F2 U' B2 U F2 l' F2 U' Rw' U2 l U2 Rw B2 U F2 l'

(27,18)

N

Clément Gallet

r2 B2 l2 U' r' B2 r U l U' B2 U l B2 D2 l' D2 r2

(27,18)

N

Clément Gallet

(x Rw F2 U' R Rw) U' (r U2)3 r2 U' (Rw' R' U F2 Rw' x')

(25,19)

N

Christopher Mowla

(y Rw2 U x') l U' L' U' l' U L U F2 l' F2 r U2 r' U2 (x U' Rw2 y')

(25,19)

N

Christopher Mowla

(r2 F') Rw' F' U2 F r F l D2 r' D2 F2 l' U2 F R (F r2)

(26,19)

N

Christopher Mowla

r' B L' B l' B' L B' l U l B2 Lw' F2 l' F2 Lw B2 U' M'

(24,20)

N

Christopher Mowla

(y Rw2 U x') l U' L' U' l' U L U M' U2 l' U2 l U2 r' U2 (x U' Rw2 y')

(26,20)

N

Christopher Mowla

(y Rw2 F L) F U2 l F2 D2 r D2 l' F r' F U2 F' r (L' F' Rw2 y')

(27,20)

N

Christopher Mowla

(z' Rw' U' Rw' l' U L') U' r2 (U2 r')3 U' (L U' l Rw U Rw z)

(25,21)

N

Christopher Mowla

(Lw2 B' R B l b2 u' l2 Uw2 S) l' (S' Uw2 l2 u b2 l' B' R' B Lw2)

(29,21)

N

Christopher Mowla

(y x' Lw2 F' L2 F r f2 u' r2 Uw2 S') r' (S Uw2 r2 u f2 r' F' L2 F Lw2 x y')

(31,21)

N

Christopher Mowla

(y Bw' R U b' R' Bw' Uw2 Rw' u2 l2 S') u (S l2 u2 Rw Uw2 Bw R b U' R' Bw y')

(29,23)

N

Christopher Mowla

Rw2 U2 R2 Rw U' R' U' Rw U2 R2 U' R Rw2 U2 Rw' U R U Rw U2 Rw2 U Rw U2

(32,24)

N

xyzzy

U2 Lw2 U2 Lw L2 U' L' U' Lw U2 L2 U' Lw2 L U2 Lw' U L U Lw U2 Lw2 U Lw

(32,24)

N

xyzzy

(z Rw' Uw2 U Lw' R2 U2 Rw U Lw' U2 F' R2 F) l' (F' R2 F U2 Lw U' Rw' U2 R2 Lw U' Uw2 Rw z')

(33,27)

1 Slicer(1)

N

Christopher Mowla

(Lw U Rw2 F Rw U' Rw U' F2 U Rw2 U' R' U') r (U R U Rw2 U' F2 U Rw' U Rw' F' Rw2 U' Lw')

(35,29)

1 Slicer(2)

N

Christopher Mowla

Uw Bw' l Bw Uw' Bw' l Uw2 R' Uw' L' B' L B U' L U l U' L' U B' L' B L Uw R Uw2 l2 Bw

(33,30)

N

Christopher Mowla

Uw Bw' l Bw Uw' Bw' l Uw2 R' Uw' L' U' B' L B L U l U' L' B' L' B U L Uw R Uw2 l2 Bw

(33,30)

N

Christopher Mowla

((Lw x) Uw2 Lw Rw Uw2 Rw2 U2 Rw U' Lw U2 F L2 U F' U') l' (U F U' L2 F' U2 Lw' U Rw' U2 Rw2 Uw2 Lw' Rw' Uw2 (Lw' x'))

(45,33)

5x5x5 1 Slicer

N

Christopher Mowla

Three flips



• Note that there are several possible dedge flip double parity cases for the last layer. This case is probably just the first of many to come on this page, as fast algorithms for other cases are quite possible to achieve.

u2 f' U' f2 Lw2 U2 f' U2 B2 Lw2 f2 U' f u2 Lw2 B2 Lw2

(29,17)

N

Clément Gallet

Rw' U2 r U2 Rw' x' U2 r' U' R' U' Rw' U2 Rw U R U' Rw R U2 x

(24,19)

N

Christopher Mowla

y M U r U2 r2 U2 r' U2 F2 r' F2 r2 U2 l F2 l' F2 r2 U' M' y'

(31,20)

N

(y' r' U2 M l U)(r' U' F' R' F R2 U') r' (U R2 F' R F U r) r U2 r' U2 (U' M' l' U2 r y)

(33,27)

N

Christopher Mowla

In the M layer



• Note that none of these algorithms affect the cube in the same manner as their inverses.

r U2 r' U2 r' D2 r D2 r' B2 r B2 r'

(19,13)

N

Christopher Mowla

r U2 r' U2 M' l' U2 r U2 l' U2 r U2 l'

(20,14)

N

Christopher Mowla

r U2 l' U2 l r2 U2 l U2 r' U2 l U2 l r2

(23,15)

reParity(v2)

N

reThinking the Cube

(r' U' F' R' F R2 U' Lw2 E L2 E') r' (E L2 E' Lw2 U R2 F' R F U r) (r U2 r' U2)

(35,26)

N

Christopher Mowla




OLL parity algorithms which don't preserve the last layer



• Some of the algorithms in this category are the most widely used speedsolving algorithms.
• None of these algorithms will result in a single dedge flip or the "pure form" of double parity if single slices are substituted for the wide turns.

Single parity

1 flip

• With the exception of Kåre Krig's 2 gen algorithms, all algorithms listed in this section can be derived from algorithms from the Longer/Faster Algorithms section by using 4x4x4 wide turn equivalences such as Rw = (Lw x), Lw2 = (Rw2 x2), etc.
• Specifically, these algorithms were derived from the 3x3x3 Classic Setup's solutions which are at most 18 single slice half turns. They only involve 2 face turns and thus there are additional transformations which are "legal" in wide turns, but "illegal" transformations to keep these algorithms pure dedge flips.
  • Therefore we have more 15 BHTM solutions which preserve F3L in wide turns than we have for pure dedge flips, for example.
  • Most of these algorithms need to be adjusted to work on the 5x5x5.
Interesting Fact:
The following algorithm cannot be used to solve any form of OLL parity if any of its wide turns are converted to single slice turns!

Rw2 F2 U2 Rw' F' Uw L' U2 L Uw' F' Rw' U2 Rw2 F2 Rw

(23,16)

Y

Christopher Mowla




Rw' U2 Rw U2 Rw U2 Rw2 F2 Rw' U2 Rw' U2 F2 Rw2 F2

(25,15)

Y

Rw' U2 Rw U2 Rw' F2 Rw2 U2 Rw U2 Rw' U2 F2 Rw2 F2

(25,15)

Y

Rw' U2 Lw F2 Rw' U2 Rw2 U2 F2 Lw' U2 Lw F2 Rw2 U2 x

(25,15)

Y

Rw' U2 Rw U2 Rw' B2 Rw2 B2 U2 Lw' B2 Lw U2 Rw2 B2

(25,15)

Y

Rw' U2 Rw U2 Rw' B2 Rw2 B2 U2 Rw' U2 Rw U2 Rw2 B2

(25,15)

Y

Rw' U2 Rw U2 Lw' U2 Rw2 B2 Lw U2 Lw' B2 U2 Rw2 U2 x'

(25,15)

Y

Rw2 U2 Rw' U2 Rw U2 B2 Rw2 B2 Rw U2 Rw' U2 Rw B2

(25,15)

Y

x' Rw2 B2 Lw B2 Rw' U2 B2 Lw2 B2 Lw' U2 Lw U2 Lw' B2

(25,15)

Y

Lw Rw U2 F2 Lw' F2 Rw F2 Rw' F2 U2 Rw' U2 Lw U2 Lw2 x'

(25,16)

Y

Lw Rw U2 F2 Lw' F2 Rw F2 Rw' F2 U2 Rw' U2 Lw U2 Rw2 x

(25,16)

Y

Rw' U2 Rw U2 Rw' B2 Rw2 B2 U2 Rw' U2 Rw U2 Lw Rw U2 x

(25,16)

Y

Rw' U2 Rw U2 Rw U2 Lw' Rw' U2 Rw' B2 Rw' B2 U2 Rw2 U2 x'

(25,16)

Y

Rw' U2 Rw U2 Lw Rw2 U2 Rw2 U2 F2 Lw' U2 Lw F2 Rw2 U2 x

(27,16)

Y

Rw' U2 Rw U2 Lw Rw2 U2 Rw2 U2 F2 Lw' U2 Rw U2 Lw2 F2 x2

(27,16)

Y

Rw' U2 Rw U2 Lw2 Rw B2 Lw2 U2 Rw U2 Lw' B2 U2 Rw2 U2 x'

(27,16)

Y

Rw' U2 Rw U2 Rw' B2 Rw2 B2 U2 Rw' U2 Rw' B2 Rw2 U2 Rw2

(27,16)

Y

Rw' U2 Rw U2 Rw U2 Lw2 B2 Lw Rw2 B2 Rw' B2 U2 Rw2 U2 x'

(27,16)

Y

Rw' U2 Rw' U2 Rw2 U2 Rw' B2 Rw2 U2 Lw' B2 Lw U2 Rw2 B2

(27,16)

Y

Rw' U2 Rw' U2 Rw2 U2 Rw' B2 Rw2 U2 Rw' U2 Rw U2 Rw2 B2

(27,16)

Y

Rw2 U2 B2 Rw B2 Rw B2 U2 Rw U2 Rw' U2 Rw' U2 Rw2 U2 B2

(27,16)

Y

Rw2 U2 Rw' U2 Rw' B2 Rw2 U2 Rw' U2 Rw2 U2 Rw U2 Lw U2 x

(27,16)

Y

x' Rw2 B2 Lw2 Rw' U2 Lw' U2 B2 Lw2 B2 Lw' U2 Lw U2 Lw' B2

(27,16)

Y

Rw' U' Rw' U B2 U' Rw U' Rw' U2 Rw' x' U2 Rw U2 Lw' U2 Rw2

(23,17)

Y

Christopher Mowla

Rw2 U2 Lw' U2 Rw U' Rw' U B2 U' Rw U' Rw' U2 Rw' x' U2 Rw'

(23,17)

Y

Christopher Mowla

Lw Rw U' Rw U' F2 U Rw' U' x U2 Rw' U2 Rw' U2 Lw U2 Rw2 x

(23,17)

Y

Christopher Mowla

Rw U2 x Rw U2 Rw U2 Rw' U2 Lw U2 Rw' U2 Rw U2 Rw' U2 Rw'

(25,17)

lucasparity

Y

Lucas Garron & Stefan Pochmann

Rw U2 Rw U2 Rw' U2 Rw U2 Lw' U2 Rw U2 Rw' U2 x' Rw' U2 Rw'

(25,17)

lucasparity

Y

Lucas Garron & Stefan Pochmann

Rw U2 Rw U2 Lw' B2 Lw U2 Lw' U2 Rw U2 Rw' U2 Rw' x' U2 Rw'

(25,17)

Y

Rw U2 Rw U2 Rw' U2 Rw U2 Rw' F2 Rw F2 Rw' F2 Rw' U2 Rw'

(25,17)

Y

Rw' U2 Rw U2 Rw U2 Lw' Rw' U2 Lw2 Rw F2 Lw' U2 F2 Rw2 F2

(27,17)

Y

Rw' U2 Rw' U2 Rw2 U2 Rw U2 Lw Rw U2 Rw F2 Rw F2 Rw2 U2 x

(27,17)

Y

Rw' U2 Rw' U2 Rw U2 Rw' B2 Rw' B2 Rw2 U2 Rw' B2 Lw' B2 Lw2 x

(27,17)

Y

Rw' U2 Rw' U2 Rw U2 Rw' B2 Rw' B2 Rw2 U2 Rw' B2 Rw' U2 Rw2

(27,17)

Y

Rw2 U2 Lw' U2 Rw U2 Lw' B2 Lw U2 Rw' B2 Rw' B2 Rw2 U2 Lw

(27,17)

Y

Lw Rw U2 Lw' Rw2 U2 Rw' U2 B2 Rw' B2 Lw U2 Lw' U2 B2 Rw2

(27,17)

Y

Rw U2 Rw U2 Rw' U2 Lw U Rw U' F2 U Rw' U' x U2 Rw2 U2 Rw'

(25,18)

Y

Christopher Mowla

R U r U R' U' Rw2 R U2 Rw2 U Rw U2 Rw' U2 Rw U' Rw2 U2 Rw'

(27,20)

N

Kåre Krig &xyzzy

Rw' U' Rw2 U' Rw' U Rw' U' Rw' U2 Rw U Rw U' Rw U Rw2 U Rw2 U2 Rw U2 Rw2 U2 Rw

(33,25)

Y

Kåre Krig

Rw' U2 Rw' U2 Rw U Rw' U Rw' U2 Rw2 U Rw2 U' Rw2 U' Rw' U' Rw U Rw' U2 Rw2 U2 Rw

(34,25)

Y

Kåre Krig

Rw' U2 Rw' U2 Rw U Rw' U Rw U' Rw2 U' Rw2 U2 Rw2 U' Rw U' Rw U Rw' U2 Rw2 U2 Rw

(34,25)

Y

Kåre Krig

Rw' U2 Rw' U2 Rw U Rw' U Rw U Rw2 U2 Rw2 U' Rw2 U Rw U' Rw U Rw' U2 Rw2 U2 Rw

(34,25)

Y

Kåre Krig

Rw' U Rw' U' Rw' U2 Rw2 U2 Rw2 U' Rw U2 Rw U2 Rw' U2 Rw' U' Rw2 U Rw2 U Rw' U2 Rw

(35,25)

Y

Kåre Krig

Rw' U Rw' U' Rw' U2 Rw2 U2 Rw2 U Rw U2 Rw U2 Rw' U2 Rw' U Rw2 U Rw2 U Rw' U2 Rw

(35,25)

Y

Kåre Krig

Rw' U Rw' U' Rw' U2 Rw2 U2 Rw2 U Rw U2 Rw U2 Rw' U2 Rw U' Rw2 U' Rw2 U' Rw U2 Rw

(35,25)

Y

Kåre Krig

Rw' U Rw' U' Rw U' Rw2 U2 Rw2 U2 Rw' U2 Rw U2 Rw' U2 Rw' U' Rw2 U Rw2 U Rw' U2 Rw

(35,25)

Y

Kåre Krig

Rw' U Rw' U' Rw U Rw2 U2 Rw2 U2 Rw' U2 Rw U2 Rw' U2 Rw' U Rw2 U Rw2 U Rw' U2 Rw

(35,25)

Y

Kåre Krig

Rw' U Rw' U' Rw U Rw2 U2 Rw2 U2 Rw' U2 Rw U2 Rw' U2 Rw U' Rw2 U' Rw2 U' Rw U2 Rw

(35,25)

Y

Kåre Krig




Three flip

One can make a 3 dedge flip algorithm from a 1 dedge flip algorithm by conjugating the 1 dedge flip algorithm with moves which change the way the last layer pieces are in the last layer while keeping all pieces affected by the particular 1 dedge flip algorithm in the last layer.

The shortest of such conjugations is (B' R'),

(B' R') Rw U2' x Rw U2' Rw U2' Rw' U2 Lw U2' Rw' U2 Rw U2' Rw' U2 Rw' (R B)

(27,21)

Y

Lucas Garron

but one can see longer conjugations which solve different OLL + OLL parity cases in this thread and in the Word document "OLLparity+OLL 3flip" from Michael Fung's old page, for example.

On the other hand, the algorithms provided in this section are algorithms which flip 3 dedges without using this conjugation technique

Rw F U2 F' Rw U2 Rw U2 Rw U' L' U2 L F 3Lw' U' R U' x' U F Rw

(25,21)

Y

Christopher Mowla

Rw U' R U2 R' U2 R' U' Rw U2 Rw R U2 Rw U' R' U2 R' U2 R U' Rw

(28,22)

Y

Kåre Krig

Rw U' R U2 R' U2 R' U' Rw U2 Rw R U2 Rw U' Rw' U2 R' U2 Rw U' Rw

(28,22)

Y

Kåre Krig

Rw U' Rw U2 R' U2 Rw' U' Rw U2 Rw R U2 Rw U' Rw' U2 R' U2 Rw U' Rw

(28,22)

Y

Kåre Krig

Double parity

1 flip

Rw' U2 Rw' U2 Lw U2 Rw' U2 Rw U2 x U2 Rw2 U2 Rw' U2 Rw

(25,16)

v1

Y

Lw U2 Lw U2 Rw' U2 Lw U2 Lw' U2 x U2 Lw2 U2 Lw U2 Lw'

(25,16)

v2

Y

Rw U2 Rw' U2 Rw U2 Rw U2 Lw' U2 Rw U2 Rw' U2 x' U2 Rw2

(25,16)

v1

Y

Lw' U2 Lw U2 Lw' U2 Lw' U2 Rw U2 Lw' U2 Lw U2 x' U2 Lw2

(25,16)

v2

Y

Rw2 U2 x' U2 Rw' U2 Rw U2 Lw' U2 Rw U2 Rw U2 Rw' U2 Rw

(25,16)

v1

Y

Lw2 U2 x' U2 Lw U2 Lw' U2 Rw U2 Lw' U2 Lw' U2 Lw U2 Lw'

(25,16)

v2

Y

Rw U2 Rw' U2 Rw2 U2 x U2 Rw U2 Rw' U2 Lw U2 Rw' U2 Rw'

(25,16)

v1

Y

Lw' U2 Lw U2 Lw2 U2 x U2 Lw' U2 Lw U2 Rw' U2 Lw U2 Lw

(25,16)

v2

Y

Rw U2 Rw U2 Rw U R Rw U2 R2 U Rw U Rw2 R U Rw U' Rw

(23,19)

v1

Y

Kåre Krig

Rw' U2 Rw' U2 Rw' U' R' Rw' U2 R2 U' Rw' U' Rw2 R' U' Rw' U Rw'

(23,19)

v2

Y

Kåre Krig

Rw' U2 Rw' U2 Rw R U Rw2 R' U2 Rw' R' U R U Rw' U2 Rw2 U' Rw2

(26,20)

Y

Kåre Krig

z' U2 Lw' R2 x U' F2 U Lw' U2 r' U2 Rw F' L2 F r2 U2 x' U' F2 U x U2 Lw' U2 z

(32,22)

N

Christopher Mowla

Rw' U' Rw' U' Rw U Rw U2 Rw2 U2 Rw U2 Rw U Rw' U Rw U Rw2 U2 Rw' U2 Rw U2 Rw'

(33,25)

Y

Kåre Krig

Rw' U Rw2 U Rw2 U2 Rw U2 Rw' U2 Rw U Rw U Rw U2 Rw' U2 Rw' U2 Rw2 U' Rw2 U Rw'

(35,25)

Y

Kåre Krig

Rw U2 Rw' U2 Rw U2 Rw2 U' Rw' U' Rw U' Rw' U2 Rw' U2 Rw2 U2 Rw' U' Rw' U Rw U Rw

(33,25)

Y

Ben Whitmore

Three flip

Rw U' Rw U' Rw2 U Rw' U2 Rw U2 Rw' U Rw' U' Rw U2 Rw U2 Rw U2 Rw U' Rw' U2 Rw' U2 Rw

(35,27)

Y

Kåre Krig




OLL parity algorithms which don't preserve F3L

• One advantage of having access to algorithms which don't preserve the first three layers of the 4x4x4 is that the shortest algorithms that this category of algorithms has to offer have fewer moves than algorithms which preserve the first three layers. (This is of interest to certain cubers.)
• Unlike algorithms which preserve the first three layers, which are at minimum 14 block half turns (opposite double parity) and 18 block quarter turns (one dedge flip), some algorithms in this section are just 13 block half turns, and some are just 15 block quarter turns.

Corner permuters



• Most of these algorithms are also just front right F3L slot destroyers.

Single Parity

r' U2 F' U2 F U2 r' U2 r' U2 r' F' U2 F r'

(21,15)

N

Christopher Mowla & Bruce Norskog

l U2 F' U2 F U2 l U2 l U2 l F' U2 F l

(21,15)

N

Christopher Mowla & Bruce Norskog

r' U R U2 R' U' r' U2 r' U2 r' U' R U2 R' U r'

(21,17)

N

Christopher Mowla & Bruce Norskog

l' U' R U2 R' U l' U2 l' U2 l' U R U2 R' U' l'

(21,17)

N

Christopher Mowla & Bruce Norskog

r2 S' r U' R U' r U2 r U2 r U' R' U' r S r2

(21,17)

N

Christopher Mowla

r2 S r U' R U' r U2 r U2 r U' R' U' r S' r2

(21,17)

N

Christopher Mowla

Rw U' R' U2 R U r U2 r U2 r U R' U2 R U' Rw R2

(22,18)

N

Kåre Krig

Rw U' R' U2 R U r U2 Rw R U2 r U R U2 R' U' Rw

(22,18)

N

Kåre Krig

Rw U' R U2 R' U r U2 Rw R U2 r U R' U2 R U' Rw

(22,18)

N

Kåre Krig

Rw R U' R2 U2 R2 U r U2 Rw U2 r U R U2 R' U' Rw

(24,18)

N

Kåre Krig

Rw U' R' U2 R U Rw R2 U2 r U2 Rw U R' U2 R U' Rw R2

(24,19)

N

Kåre Krig

Rw U' R' U2 R U Rw U2 r U2 Rw R2 U R' U2 R U' Rw R2

(24,19)

N

Kåre Krig

Rw U' R' U2 R U' R U2 r U2 Rw U2 r U R U2 R' U' Rw

(24,19)

N

Kåre Krig

(r' U' R U r)(r U2 r2 U2 r' U2 r U2 r' U2 r2 U2 r)(r U R' U' r')

(31,21)

N

Christopher Mowla

Front-right F3L slot destroyers



Single Parity

l U2 l' F U R U' Lw' D2 r D2 r U2 Rw' U2 Lw F

(22,17)

N

Christopher Mowla

Rw U2 Lw' U2 l D2 l D2 Rw' U' L U F r' U2 r F

(22,17)

N

Christopher Mowla

Rw U R' U' Rw2 R U2 Rw2 U Rw U2 Rw' U2 Rw U' Rw2 U2 Rw'

(24,18)

Y

Kåre Krig

Rw U R' U' Rw2 U2 Rw2 R U Rw U2 Rw' U2 Rw U' Rw2 U2 r'

(24,18)

N

Kåre Krig

Rw U2 R U R' U2 R U r U2 Rw U2 Rw U R U2 R' U' Rw

(24,19)

N

Kåre Krig

Double Parity

Rw U2 Rw U2 Rw U Rw U2 R' U Rw U Rw2 R U Rw U' Rw

(21,18)

Y

Kåre Krig

Rw U2 Rw U2 Rw' U' R' Rw2 U2 r U' R' U' Rw U2 r2 U Rw2

(24,18)

N

Kåre Krig

(r' U' R U r)(r2 F2 Uw2 r Uw2 F2 r2)(r U R' U' r')(Fw2 l' Uw2 l r Uw2 r' Fw2)

(31,23)

N

Christopher Mowla

Petrus



• These algorithms "just" scramble two adjacent faces' worth of pieces.


Single Parity

Rw B' U2 B Rw B2 Lw B2 Rw B D2 B' Rw

(17,13)

v1

Y

Bruce Norskog

x' Rw' U F2 U' Lw' U2 Rw' U2 Rw' F' U2 F Rw'

(17,13)

v2

Y

Bruce Norskog

x' Rw' U F2 U' Lw' U2 Rw' U2 Lw' U' B2 U Rw' x'

(17,13)

v3

Y

Bruce Norskog

x' Rw' U F2 U' Lw' U2 Rw' 3Dw2 Rw' U' F2 U Lw' x

(17,13)

v4

Y

Bruce Norskog

Rw' S' L2 S Rw' U2 Rw' U2 Rw' F U2 F' U' L' U Rw'

(20,16)

Y

Christopher Mowla

Rw' S R2 S' Rw' U2 Rw' U2 Rw' F' U2 F U R U' Rw'

(20,16)

Y

Christopher Mowla

Rw U2 Rw' U2 Rw D2 r D2 l' U L' U' B' l' U2 l x'

(21,16)

N

Christopher Mowla

Rw' U R' U2 R U' Rw' U2 Rw' U2 Rw' U' R U2 R' U Rw'

(21,17)

Y

Kåre Krig, Bruce Norskog, & Christopher Mowla

Rw U' R' U2 R U r U2 r U2 r U R' U2 R U' Rw

(21,17)

N

Kåre Krig

Rw U' R' U2 R U Rw U2 Rw U2 r U R U2 R' U' Rw

(21,17)

N

Kåre Krig

Rw' U' L' U2 L U Rw' U2 Rw' U2 Rw' U L' U2 L U' Rw'

(21,17)

Y

Bruce Norskog & Christopher Mowla

Rw U2 Lw' U2 l D2 l D2 Rw' U' L U F Rw' U2 Rw F

(22,17)

N

Christopher Mowla

Rw U' R2 U2 R2 U r U2 Rw U2 r U R U2 R' U' Rw

(23,17)

N

Kåre Krig

Rw U2 Rw2 U' Rw' U2 Rw U2 Rw' U Rw2 U2 Rw2 U R2 U' Rw'

(25,17)

Y

Kåre Krig & Christopher Mowla

Rw U2 Rw2 U Rw' U2 Rw U2 Rw' U' Rw2 U2 r2 U R2 U' Rw'

(25,17)

N

Kåre Krig & Christopher Mowla

Rw U' R2 U2 R2 U r U2 r U2 r U R2 U2 R2 U' Rw

(25,17)

N

Kåre Krig

Rw' U R U2 R' U' r' U2 Rw' U2 Rw' F' U' R' U' R F Rw'

(21,18)

N

Kåre Krig

Rw' U R U2 R' U' Rw' U2 Rw' U2 Rw' R' U' R U2 R' U Rw'

(22,18)

Y

Kåre Krig

Rw U' R' U2 R U Rw U2 r U2 Rw R2 U R' U2 R U' Rw

(23,18)

N

Kåre Krig

Rw U' R U2 R' U Rw U2 r U2 Rw R2 U R U2 R' U' Rw

(23,18)

N

Kåre Krig

Rw U' R U2 R' U Rw U2 Rw R2 U2 Rw U R' U2 R U' Rw

(23,18)

Y

Kåre Krig

Rw U R U' Rw2 R U2 Rw2 U' Rw U2 Rw' U2 Rw U Rw2 U2 Rw'

(24,18)

Y

Kåre Krig

Rw' U R U2 R' U' Rw' U2 Rw' U2 Rw' R2 U' R2 U2 R2 U Rw'

(24,18)

Y

Kåre Krig

Rw U R' U' Rw2 U2 Rw2 R U Rw U2 Rw' U2 Rw U' Rw2 U2 Rw'

(24,18)

Y

Kåre Krig

Rw U R' U Rw U2 r U2 r U' Rw' U R U R' U2 Rw U' Rw

(22,19)

N

Kåre Krig

Rw U R' U r U2 r U2 Rw U' Rw' U R U R' U2 Rw U' Rw

(22,19)

N

Kåre Krig

Rw U' Rw U2 R' U R U Rw' U' r U2 Rw U2 Rw U R' U Rw

(22,19)

N

Kåre Krig

Rw U' R' U2 R U Rw R U2 r U2 Rw R U R' U2 R U' Rw

(23,19)

N

Kåre Krig

Rw U' Rw U2 R' U R U Rw' U' Rw U2 r U2 Rw U R2 U Rw

(23,19)

N

Kåre Krig

Double Parity

Lw' U2 F2 Lw' U' F R' F Lw2 U2 Rw U2 Lw2 U' x'

(20,14)

v1

Y

Bruce Norskog

Lw' U2 F2 Lw' U' F 3Lw' U Lw2 x' U2 Rw U2 Lw2 U' x'

(20,14)

v2

Y

Bruce Norskog

Rw U2 F2 Rw U' F R' F Rw2 U2 Lw' U2 Rw2 U' x'

(20,14)

v1

Y

Bruce Norskog

Rw U2 F2 Rw U' F 3Lw' U Rw2 x' U2 Lw' U2 Rw2 U' x'

(20,14)

v2

Y

Bruce Norskog

r' U R U' r' U2 r U R' U r U2 r F2 Lw' U2 Lw

(21,17)

N

Christopher Mowla

Rw' U R U (Rw' U2)3 Rw2 U R' U' Rw2 U' R' U Rw'

(24,19)

Y

Stefan Pochmann

X-Cross preservers



Single Parity

Rw' U2 F U2 F' U2 Rw' U2 Rw' U2 Rw' F U2 F' Rw'

(21,15)

Y

Bruce Norskog & Christopher Mowla

Rw U D' R2 U' R2 D x' U2 Lw U2 Lw U2 Lw F' U2 F Lw

(22,16)

Y

Christopher Mowla

Rw F U2 F' Rw U2 Rw U2 Rw R U2 F U2 F' U2 r

(22,16)

N

Christopher Mowla

Rw' U R U2 R' U' Rw' U2 Rw' U2 Rw' F U' B' U' S Dw' z'

(20,17)

Y

Christopher Mowla

Lw U' L' R U2 L U Lw U2 Lw U2 Lw U' R' U2 R U r x'

(22,18)

N

Christopher Mowla

Lw U' L' R' U2 L U Lw U2 Lw U2 Lw U' R U2 R' U Lw R

(23,19)

Y

Christopher Mowla

Random pseudo-3x3x3 positions



• Bruce Norskog explained how he found the algorithms (the algorithms in this section attributed solely to him) in this post.
• To see a large list of (U,Rw,R) solutions that have been found by Kåre Krig, see this thread. (Several of the algorithms in that list are listed in this section already.)

Single Parity

Rw B U2 B' r B2 l B2 r B' D2 B Rw

(17,13)

N

Bruce Norskog

Rw U F2 U' Lw F2 r U2 Rw U B2 U' Rw

(17,13)

N

Bruce Norskog

Rw U' F2 U Lw F2 r U2 Rw U' B2 U Rw

(17,13)

N

Bruce Norskog

Uw M R D R' Bw f M d' M' Fw b M' F Uw

(15,15)

N

Bruce Norskog

Rw' S' L2 S Rw' U2 Rw' U2 Rw' F' U2 F U R U' Rw'

(20,16)

Y

Christopher Mowla

Rw' S R2 S' Rw' U2 Rw' U2 Rw' F U2 F' U' L' U Rw'

(20,16)

Y

Christopher Mowla

Rw U2 Rw' U2 Rw D2 Rw D2 Rw' F L' F' U' Rw' D2 Rw

(21,16)

Y

Christopher Mowla

r U' M' U M' U' M' U' M Uw2 r Uw2 U' M' U2 M' U M U2 M U M' U M U M' U' l'

(31,28)

N

Christopher Mowla

Double Parity

Rw F2 U2 Lw F U' R U' Rw2 B2 Rw' B2 Rw2

(19,13)

Y

Bruce Norskog

Rw F2 U2 Lw F' U L' U Rw2 B2 Rw' B2 Rw2

(19,13)

Y

Bruce Norskog

Rw F2 U2 Lw F U' (L' R) U' Rw2 B2 Rw' B2 Rw2

(19,13)

Y

Bruce Norskog

Rw F2 U2 Lw F' U (L' R) U Rw2 B2 Rw' B2 Rw2

(19,13)

Y

Bruce Norskog

Rw U2 Lw' U2 x' (Rw' U2 Lw U2)2 Lw'

(19,13)

reParity

Y

reThinking the Cube

Rw U' b M r' D' b D l' b2 Lw E' B' Rw

(15,14)

N

Bruce Norskog

Rw2 U2 Rw U2 Rw U2 Rw U2 D' Rw F' U2 F Rw'

(20,14)

Y

xyzzy

Rw2 U2 Rw' U2 Rw' U2 Rw' U2 D Rw' B U2 B' Rw

(20,14)

Y

xyzzy

Rw' U F2 U' Rw F2 B' Rw F2 Rw F2 Rw F2 Rw2

(20,14)

Y

xyzzy

Uw M F R' B u d l' u d F R' B M Uw

(15,15)

N

Bruce Norskog

Uw M F' L B' u d l' u d F' L B' M Uw

(15,15)

N

Bruce Norskog

Rw U2 Rw' U2 D' Rw D F2 Rw F2 Rw F2 Rw' D' Rw'

(20,15)

Y

xyzzy

Rw' U2 Rw' U2 Rw' U2 Rw2 B R D' Rw B' U2 B Rw'

(20,15)

Y

xyzzy

Rw2 U2 Rw U2 Rw U2 Rw U2 D L' Rw F U2 F' Rw'

(21,15)

Y

xyzzy

Rw2 U2 Rw U2 Rw U2 Rw D L' U2 Rw F U2 F' Rw'

(21,15)

Y

xyzzy

Rw2 U2 Rw' F2 Rw2 U F R' L U F' Rw U2 F2 Rw

(21,15)

Y

xyzzy

Rw2 U2 Rw' F2 Rw2 U' F R' L U' F Rw U2 F2 Rw

(21,15)

Y

xyzzy

Rw U2 F2 Rw F2 Rw2 U L2 B' L B U Rw' F2 Rw2

(22,15)

Y

xyzzy

Rw U2 F2 Rw F2 Rw2 U B' L' B L2 U Rw' F2 Rw2

(22,15)

Y

xyzzy

Rw U2 F2 Rw F2 Rw2 U' R2 B R' B' U' Rw' F2 Rw2

(22,15)

Y

xyzzy

Rw U2 F2 Rw F2 Rw2 U' B R B' R2 U' Rw' F2 Rw2

(22,15)

Y

xyzzy

Rw D' B2 D Rw' D' Rw' U2 Rw' B2 U2 Rw U2 B2 Rw2

(22,15)

Y

xyzzy

Rw2 U2 B2 Rw B2 U2 Rw' B2 Rw' F' Rw' F U2 F' Rw

(22,15)

Y

xyzzy

Rw2 F2 Rw' D L2 B L' B' D Rw2 F2 Rw F2 D2 Rw

(22,15)

Y

xyzzy

Rw2 F2 Rw' D B L B' L2 D Rw2 F2 Rw F2 D2 Rw

(22,15)

Y

xyzzy

Rw2 F2 Rw' D2 B2 Rw' D B R' D F2 B' Rw2 B2 Rw'

(22,15)

Y

xyzzy

Rw2 F2 Rw' D2 B2 Rw' D' B' L D' F2 B Rw2 B2 Rw'

(22,15)

Y

xyzzy

Rw2 F2 Rw' D2 B2 Rw' F D' L F' D' F2 Rw2 B2 Rw'

(22,15)

Y

xyzzy

Rw2 F2 Rw' D2 B2 Rw' F' D R' F D F2 Rw2 B2 Rw'

(22,15)

Y

xyzzy

Rw2 F2 Rw' D' R2 B' R B D' Rw2 F2 Rw F2 D2 Rw

(22,15)

Y

xyzzy

Rw2 F2 Rw' D' B' R' B R2 D' Rw2 F2 Rw F2 D2 Rw

(22,15)

Y

xyzzy

r' U M' U' M' U M' U M Uw2 l' Uw2 M' U M U' M' U M' U M U2 M' U2 r

(29,25)

N

Christopher Mowla




Non Dedge-Preserving Last Layer 2-Cycle Cases

• This section contains all of the 2-cycle cases that can occur in the last layer besides the one dedge flip.
• These are additional cases which arise in the K4 Method and other direct solving methods.

In opposite dedges

Opposite



Symmetrical algorithms

r U2 r U2 F2 r F2 l' U2 l U2 r2

(19,12)

Y

r U2 l F2 D2 r D2 l' F2 r U2 r2

(19,12)

Y

r' U2 l' B2 D2 r' D2 l B2 r' U2 r2

(19,12)

Y

r' U2 r' U2 B2 r' B2 l U2 l' U2 r2

(19,12)

Y

l F2 r F2 D2 l B2 r' B2 r D2 l2

(19,12)

Y

l F2 r F2 D2 r D2 r' D2 l D2 l2

(19,12)

Y

l' B2 r' B2 D2 l' F2 r F2 r' D2 l2

(19,12)

Y

l' B2 r' B2 D2 r' D2 r D2 l' D2 l2

(19,12)

Y

r U2 r U2 M' U2 r U2 r' U2 l U2 r2

(20,13)

N

Rw' U2 r' U2 3Rw' U2 r' U2 r U2 l' U2 Rw2

(20,13)

N

z Rw' U l' Uw' Rw Uw l' Fw' l' Fw l2 Uw' Rw' u Rw z'

(16,15)

N

Christopher Mowla

l F2 r F2 l D f' D r D2 r' D' f D l2

(19,15)

Y

Christopher Mowla

b u Bw' Rw' f' Rw Bw Rw' f2 Uw' f' Uw f' Rw u' b'

(17,16)

N

Christopher Mowla

z Rw U' l Uw Rw' 3Uw' l Bw l Bw' l' d' l' Uw Rw u' Rw' z'

(17,17)

N

Christopher Mowla

Dw' Lw U b' Uw' Fw Uw b' Lw' b' Lw b2 Uw' Fw' u Lw' Dw

(18,17)

N

Christopher Mowla

Dw' b U' Lw d' Lw' Uw Lw d' Fw' d' Fw d2 Lw' u' b' Dw

(18,17)

N

Christopher Mowla

Rw' S B d' Bw' Uw Bw d' Lw' d' Lw d2 Bw' Uw' b S' Rw

(18,17)

N

Christopher Mowla

Rw' u' B d' Bw' Uw Bw d' Lw' d' Lw d2 Bw' Uw' b u Rw

(18,17)

N

Christopher Mowla

Rw' f U' Lw d' Lw' Uw Lw d' Fw' d' Fw d2 Lw' u' f' Rw

(18,17)

N

Christopher Mowla

Dw' B Fw Uw Fw' Dw' d' Lw' d' Lw d' D Fw Uw' Fw' d' B' Dw

(18,18)

N

Christopher Mowla

Dw' B d' Bw' Uw Bw F d' Lw' d' Lw d F' d Bw' Uw' b Dw

(18,18)

N

Christopher Mowla

Dw' Lw U b' Uw' Fw l' Fw' l' Fw l Uw b Uw' Fw' u Lw' Dw

(18,18)

N

Christopher Mowla

Uw' Bw L d' Lw' Uw f' Uw' f' Uw f Lw d Lw' Uw' l Bw' Uw

(18,18)

N

Christopher Mowla

Dw R' S Lw Fw Lw' b Lw Fw' Lw' b Uw b Uw' f' b' R Dw'

(18,18)

N

Christopher Mowla

Dw' R S' Lw Fw Lw' b Lw Fw' Lw' b Uw b Uw' f b R' Dw

(18,18)

N

Christopher Mowla

Rw' d f Lw Fw Lw' b Lw Fw' Lw' b Uw b Uw' S' b d' Rw

(18,18)

N

Christopher Mowla

Bw' b' M Fw Rw' u Rw u Rw' u' Fw' l' Fw Rw Fw' r b Bw

(18,18)

N

Christopher Mowla

(y' L U') (r' F U2 F')4 r' (U L' y)

(25,21)

Y

Floyd Newberry

(y' L U') (r F U2 F')4 r (U L' y)

(25,21)

Y

Floyd Newberry

Non-symmetrical algorithms

Rw U2 r U2 Rw U2 Rw2 F r F' Rw2 U2 Rw2 F r' F'

(23,16)

N

Christopher Mowla

l' B2 l D r' D' l' B2 l D l r F2 r' F2 l' D'

(21,17)

Y

Christopher Mowla

r' U2 r D r' D' r' U2 l' B2 r' B2 l r D r D'

(21,17)

Y

Christopher Mowla

z Rw Uw' l Dw Bw' D' L' d Rw Dw l Dw' Rw' d' L d' l' Uw z'

(18,18)

N

Christopher Mowla

z Rw' Uw l' Dw' Fw D L d' Rw' Dw' l' Dw Rw d L' d l Uw' z'

(18,18)

N

Christopher Mowla

Dw' Fw l' Fw' Uw Fw l' Uw' u' R u l' u' R' Uw u l2 Fw' y'

(19,18)

N

Christopher Mowla

l U l U' l' U' l U l U l' D' b2 l b2 l' U2 D l' U //Safe

(23,20)

N

r' U' R U r' U R' U' r U R U2 R' U' r' U2 r' U2 r' U2 r'

(25,21)

N

Christopher Mowla

U2 l' U2 r U2 r' U2 x' U2 r' U2 r' U' l r u2 l' r' U l r u2 x //Safe

(29,21)

N

Per Kristen Fredlund

Diagonal



• Note: The mirror case can also arise as well. One can simply apply the mirror image of this case's algorithms to get algorithms for its mirror case.
• All 36 unique (25,14) solutions are listed. To obtain all possible (25,14) solutions for the top/last layer, also take y2 cube rotations of all 36 and then also consider their inverses. (This gives a total of 144 algorithms.)
It is interesting to note that due to the following algorithm's symmetry,

(Rw2 F2 U2) r (U2 F2 Rw2) (F2 l' U2 l r U2 r' F2)

(25,15)

N

Christopher Mowla

the bold inner slice r turn simply can be inverted to achieve the mirror case.

(Rw2 F2 U2) (r') (U2 F2 Rw2) (F2 l' U2 l r U2 r' F2)

(25,15)

N

Christopher Mowla

In fact, the combination of the first algorithm above with the inverse of the second,
(Rw2 F2 U2 r U2 F2 Rw2 F2 l' U2 l r U2 r' F2)(F2 r U2 r' l' U2 l F2 Rw2 F2 U2 (r) U2 F2 Rw2)

cancels to:
Rw2 F2 U2 r (r) U2 F2 Rw2 = (Rw2 F2 U2) r2 (U2 F2 Rw2) = Stefan Pochmann's nxnxn cube PLL parity algorithm.

This is interesting because this 2-cycle case can be thought of as "half" of the opposite parity case. This should make perfect sense, because, as mentioned earlier, the opposite dedge swap PLL parity case is a "2 2-cycle". This case is a "1 2-cycle" or a single 2-cycle.


Symmetrical algorithms

Uw Lw' Uw' l' Uw Lw Fw' Lw2 Uw' l' Uw Lw' L' Fw Uw'

(16,15)

N

Christopher Mowla

Uw Lw' Uw' l' Uw Lw2 Dw' Lw Uw' l' Uw L' Dw Lw' Uw'

(16,15)

N

Christopher Mowla

Dw Fw Dw' f' Rw Fw l Fw' Rw' F' l' Bw Rw Bw' l Dw'

(16,16)

N

Christopher Mowla

Uw F Rw' Fw' l' Fw Lw Dw' Rw2 Bw' l' Bw Rw' L' f Uw'

(17,16)

N

Christopher Mowla

Fw U' Uw' r' Uw Lw Uw' r2 Fw' r' Fw r' Uw Lw' U Fw'

(17,16)

N

Christopher Mowla

Uw b Rw' Fw' Rw b' L Fw' Lw b Lw' Fw L' Rw' Fw Rw Uw'

(17,17)

N

Christopher Mowla

Dw M Uw Rw' d' Fw Dw b Dw' Fw' 3Uw' l' Uw Rw Uw' r Dw'

(17,17)

N

Christopher Mowla

Lw' U2 l U2 l U2 r' U2 l U2 l' U2 M' U2 l2 U2 Lw

(26,17)

N

F (l' F' U2 F)4 l' F'

(23,19)

Y

Floyd Newberry

F (l F' U2 F)4 l F'

(23,19)

Y

Floyd Newberry

x' U2 r' U' r U r U r' D' f2 r f2 r' D U2 r' U r U r U x //Safe

(25,21)

N

Rw2 U2 r' U R U r U2 r' U' R' U' r2 U R U r U2 r' U' R' U Rw2

(29,23)

N

Christopher Mowla

Lw' U2 l U2 l U l' r' u2 l r U' l' r' u2 x U2 r U2 l' U2 l U2 Lw x' //Safe

(31,23)

N

Per Kristen Fredlund

Rw' U' Rw U2 Rw U2 Rw' U2 Rw' U' r U Rw U2 Rw U2 Rw' U2 Rw' U' r2 U2 Rw

(31,23)

N

Christopher Mowla

Rw' U Rw U2 Rw U2 Rw' U2 Rw' U r U' Rw U2 Rw U2 Rw' U2 Rw' U r2 U2 Rw

(31,23)

N

Christopher Mowla


Non-symmetrical algorithms

l' S2 U2 l U2 l' U2 r U2 r' F2 l B2 r z2

(21,14)

Y

Kenneth Gustavsson

l2 B2 U2 l U2 l2 B2 l U2 l2 U2 B2 l B2

(25,14)

Y

Nicholas Ho

r2 D2 r D2 F2 l2 F2 l' F2 D2 l2 D2 r F2

(25,14)

Y

r2 F2 U2 l' U2 l2 F2 l' U2 r2 U2 F2 r F2

(25,14)

Y

r2 F2 U2 l F2 D2 r2 D2 l' F2 r2 U2 r' F2

(25,14)

Y

r2 F2 U2 l F2 D2 r2 D2 r F2 l2 U2 l F2

(25,14)

Y

r2 F2 U2 r U2 r2 F2 r U2 r2 U2 F2 r F2

(25,14)

Y

r2 U2 r U2 F2 r2 F2 r' F2 U2 r2 U2 r' F2

(25,14)

Y

l2 B2 U2 r' U2 r2 B2 r' U2 l2 U2 B2 l B2

(25,14)

Y

l2 B2 U2 r B2 D2 l2 D2 l B2 r2 U2 r B2

(25,14)

Y

l2 B2 U2 r B2 D2 l2 D2 r' B2 l2 U2 l' B2

(25,14)

Y

l2 D2 l D2 B2 r2 B2 r' B2 D2 r2 D2 l B2

(25,14)

Y

l2 U2 l U2 B2 l2 B2 l' B2 U2 l2 U2 l' B2

(25,14)

Y

r2 U2 r' U2 F2 l2 F2 r' F2 D2 l2 D2 l' F2

(25,14)

Y

r2 B2 D2 l' D2 l2 B2 r' U2 r2 U2 B2 l' B2

(25,14)

Y

l2 U2 l' U2 B2 r2 B2 l' B2 D2 r2 D2 r' B2

(25,14)

Y

l2 F2 D2 r' D2 r2 F2 l' U2 l2 U2 F2 r' F2

(25,14)

Y

r2 U2 r B2 l2 D2 l D2 B2 r2 U2 F2 r' F2

(25,14)

Y

l2 U2 l F2 r2 D2 r D2 F2 l2 U2 B2 l' B2

(25,14)

Y

r2 U2 l' B2 r2 D2 r' D2 B2 r2 U2 F2 r' F2

(25,14)

Y

l2 U2 r' F2 l2 D2 l' D2 F2 l2 U2 B2 l' B2

(25,14)

Y

l2 F2 D2 l' F2 D2 l2 D2 r F2 r2 D2 r F2

(25,14)

Y

r2 B2 D2 r' B2 D2 r2 D2 l B2 l2 D2 l B2

(25,14)

Y

r2 B2 D2 r D2 r2 B2 l U2 r2 U2 B2 l' B2

(25,14)

Y

l2 F2 D2 l' F2 D2 l2 D2 l' F2 l2 D2 l' F2

(25,14)

Y

r2 B2 D2 r' B2 D2 r2 D2 r' B2 r2 D2 r' B2

(25,14)

Y

l2 F2 D2 l D2 l2 F2 r U2 l2 U2 F2 r' F2

(25,14)

Y

r2 U2 l B2 r2 U2 l U2 B2 r2 U2 F2 r F2

(25,14)

Y

r2 U2 r' B2 l2 U2 r' U2 B2 r2 U2 F2 r F2

(25,14)

Y

l2 U2 l' F2 r2 U2 l' U2 F2 l2 U2 B2 l B2

(25,14)

Y

l2 U2 r F2 l2 U2 r U2 F2 l2 U2 B2 l B2

(25,14)

Y

r2 F2 D2 l' D2 l2 F2 r U2 r2 U2 F2 l F2

(25,14)

Y

r2 F2 D2 r' B2 U2 r2 U2 l' B2 r2 D2 l' F2

(25,14)

Y

r2 F2 D2 r' B2 U2 r2 U2 r B2 l2 D2 r F2

(25,14)

Y

r2 B2 U2 r U2 r2 B2 r' U2 r2 U2 B2 r' B2

(25,14)

Y

Nicholas Ho

r2 B2 U2 l F2 U2 r2 U2 r' F2 r2 U2 l' B2

(25,14)

Y

r2 B2 U2 l F2 U2 r2 U2 l F2 l2 U2 r B2

(25,14)

Y

Rw2 F2 U2 r U2 x U2 Rw2 U2 r' U2 l r U2 l' U2 x'

(25,15)

N

Christopher Mowla

Rw Fw r' Fw' Rw' Dw R' U Rw' Fw r' Fw' Rw U' Rw Dw'

(16,16)

N

Christopher Mowla

l' U2 l U2 l U2 r' U2 l U2 l' U2 F2 l2 F2 r

(25,16)

Y

l' F2 D2 r' D2 F' l2 U2 l2 F' l F l2 U2 l2 F'

(25,16)

N

Christopher Mowla

Rw' Uw' r Uw Rw Bw' F' R' r Uw' r Uw r' R F r' Bw

(17,17)

N

Christopher Mowla

Rw' Uw' r Uw Rw Bw' R' F' r Uw' r Uw r' F R r' Bw

(17,17)

N

Christopher Mowla

x' r U2 r U u' r' U r' U' r' u r u' r' U' r' U r' Uw x //Safe

(20,19)

N

Ben Whitmore

x' l' Uw Rw' Uw' l Uw Fw Uw B' l2 B r' B' l2 B Uw' Fw' Rw Uw' x

(21,19)

N

Christopher Mowla

x l U2 l' U2 l U2 l2 U' l' U l U2 l U2 l2 U2 l U l U l U2 x'

(31,22)

N

Lucas Garron

x l U2 l' U2 l U2 l2 U l' U' l U2 l U2 l2 U2 l U' l U' l U2 x'

(31,22)

N

Lucas Garron

Rw U Rw' U' Rw U2 Rw' U2 Rw' U' Rw U' Rw U Rw2 U' Rw2 U2 Rw U2 Rw' U2 Rw U2 Rw U'

(34,26)

Y

Bruce Norskog

Rw U Rw' U' Rw U2 Rw' U2 Rw' U' Rw U' Rw U' Rw2 U2 Rw2 U' Rw U2 Rw' U2 Rw U2 Rw U

(34,26)

Y

Bruce Norskog

In adjacent dedges

Close unoriented



Symmetrical algorithms

l2' U Lw U2 r' U2 Rw' U2 x U' r' U x' U2 x U' M' l'

(20,15)

N

Christopher Mowla

x' l2 F U2 l U2 r' U2 r U2 F2 r F2 l' F' l2 x

(23,15)

N

reThinking the Cube

r U2 r U2 F' L F' r F L' F l' U2 l U2 r2

(21,16)

N

Christopher Mowla

x' l2 F U2 l U2 r' U2 r U2 3Rw U2 r U2 Rw' F' l2 x

(24,16)

N

Christopher Mowla

Bw l' Uw' r' Uw Lw Uw' Rw' r' Fw' r' Fw r' R Uw L' Bw'

(17,17)

N

Christopher Mowla

r2 U2 l U2 r' U2 r U2 F' L F' r F L' F l' r2

(23,17)

N

Christopher Mowla

l2 U M U2 l U2 r' U2 r U2 M' U2 r U2 l' U' l2

(25,17)

N

Christopher Mowla

z Uw' Lw M' Bw l Bw' Rw' Bw Lw l Uw l Uw' L' l Bw' M x Uw z'

(18,18)

N

Christopher Mowla

y' Lw u' Bw R r' Bw' r' Bw r R' Dw r Dw' Bw' Dw r' U' y Lw' y

(18,18)

N

Christopher Mowla

y Fw' M Lw Fw r Fw' Lw' Fw Rw r Uw r Uw' r R' Fw' M' Fw y'

(18,18)

N

Christopher Mowla

l U l U l' U' l' U' l D f2 l' f2 l U2 D' l U' l' U' l2 //Safe

(25,21)

N

(L U2)(r' F U2 F')4 r' (U2 L')

(27,21)

N

Floyd Newberry

(L U2)(r F U2 F')4 r (U2 L')

(27,21)

N

Floyd Newberry

x' l' Rw' U2 Rw U' L' U' r' U L U Rw' U2 Rw U' L' U' r2 U L U l x

(25,22)

N

Christopher Mowla

Non-symmetrical algorithms

y' r' Uw r U2 r U u' r' U r' U' r' u r u' r' U' r' U y //Safe

(20,19)

N

Ben Whitmore

y' r' u' U r U2 r Uw r' U r' U' r' u' r u r' U' r' U y //Safe

(20,19)

N

Ben Whitmore

y' r' U' r' u r u' r' U' r' U r' Uw r U2 r U u' r' U y //Safe

(20,19)

N

Ben Whitmore

y' r' U' r' u' r u r' U' r' U r' u' U r U2 r Uw r' U y //Safe

(20,19)

N

Ben Whitmore

l Fw' Lw Dw l' Uw2 U L2 u' L' Rw' Uw' l Uw L Rw u L2 u Bw' Uw x

(23,21)

N

Christopher Mowla

l' Fw Rw Fw' Lw L B2 L B l Uw' l' Uw l' B' L' B2 L2 Fw Rw' Fw'

(24,21)

N

Christopher Mowla

z' Fw' Uw r' Uw' Fw Uw r' Fw2 L Fw U' R2 U r' U' R2 U Fw' L' Fw2 r2 Uw' z

(27,22)

N

Christopher Mowla

l' F2 l F2 U' Rw U2 l U2 Rw' B' M U' Lw2 F2 r2 F2 Lw2 U M' B U

(31,22)

N

Christopher Mowla

Far unoriented



Symmetrical algorithms

l2' U' Lw U2 r' U2 Rw' U2 x U r' U' x' U2 x U M' l'

(20,15)

N

Christopher Mowla

(x Rw2 F') U2 l U2 r' U2 r U2 F2 r F2 l' (F Rw2 x')

(23,15)

N

Christopher Mowla

f' R' r u Rw' Dw' Rw u2 Fw u Fw' u Rw' Dw R2 f

(18,16)

N

Christopher Mowla

r F f' u' Fw Dw Fw' u2 Rw' u' Rw u' Fw Dw' F2 r'

(18,16)

N

Christopher Mowla

(f' R') Lw' u Lw Uw2 Rw' U Lw' u Lw Uw' Rw Uw2 (R f)

(18,16)

N

Christopher Mowla

(f' R') Uw2 Rw' u Rw Uw2 Lw' U Rw' u Rw Uw' Lw (R f)

(18,16)

N

Christopher Mowla

(f' R') Uw2 Rw' Uw Lw' u' Lw U' Rw Uw2 Lw' u' Lw (R f)

(18,16)

N

Christopher Mowla

(f' R') Lw' Uw Rw' u' Rw U' Lw Uw2 Rw' u' Rw Uw2 (R f)

(18,16)

N

Christopher Mowla

r U2 r U2 F R' F r F' R F' l' U2 l U2 r2

(21,16)

N

Christopher Mowla

(x Rw2 F') U2 l U2 r' U2 r U2 3Rw U2 r U2 Rw' (F Rw2 x')

(24,16)

N

Christopher Mowla

r d' Fw Rw Bw' l Bw Rw' F Uw Bw r Bw' Uw' F' Fw' r'

(17,17)

N

Christopher Mowla

r d Fw F Uw Fw d' Fw' Uw' F' Rw Bw' l' Bw Rw' Fw' r'

(17,17)

N

Christopher Mowla

f' d Lw' Uw' Lw d' Lw' Uw R' Fw' Uw' f' Uw Fw Lw R f

(17,17)

N

Christopher Mowla

f' Lw' Uw' Lw d' Lw' Uw R' Fw' Uw' f' Uw Fw Lw R d f

(17,17)

N

Christopher Mowla

(f' R') Lw' Uw' Lw d' Lw' Uw Fw' Uw' f' Uw Fw Lw d (R f)

(17,17)

N

Christopher Mowla

(f' R') d Lw' Uw' Lw d' Lw' Uw Fw' Uw' f' Uw Fw Lw (R f)

(17,17)

N

Christopher Mowla

(Rw r) Bw' Dw f' Dw' l' Uw l Bw d Bw' Uw' Bw d' (r' Rw')

(17,17)

N

Christopher Mowla

(Rw r) Bw' Dw f' Dw' Bw Dw Bw' Uw' f' Uw Bw f Dw' (r' Rw')

(17,17)

N

Christopher Mowla

(Lw r) Uw' Fw l' Fw' Uw Fw Lw' Uw' l' Uw Lw l Fw' (Lw' r')

(17,17)

N

Christopher Mowla

(Rw r z) l Fw' Uw Fw l' Fw' l' Uw' l Uw l Uw' Fw (z' Rw' r')

(17,17)

N

Christopher Mowla

(Bw' f') Uw Lw' d Lw b Rw' b' Uw' l' Uw Rw Uw' l (f Bw)

(17,17)

N

Christopher Mowla

(Bw' f') Uw Lw' d Lw Uw' Lw' Uw Rw d Rw' d' Uw' Lw (f Bw)

(17,17)

N

Christopher Mowla

(x' r E') r' Uw Lw Uw' r Uw Lw' Bw Uw l Uw' Bw' Uw' (E r' x)

(17,17)

N

Christopher Mowla

(x' r E') r Uw Bw Uw l' Uw' Bw' Lw Uw' r' Uw Lw' Uw' (E r' x)

(17,17)

N

Christopher Mowla

Rw' l' x' U2 l U2 r' U2 r U2 x U R U r U' R' U' Rw

(21,17)

N

Christopher Mowla

r2 U2 l U2 r' U2 r U2 F R' F r F' R F' l' r2

(23,17)

N

Christopher Mowla

R (l' B U2 B')4 l' R'

(23,19)

N

Floyd Newberry

R (l B U2 B')4 l R'

(23,19)

N

Floyd Newberry

Rw' U R U r2 U' R' U' Rw' U2 Rw U R U r U' R' U' Rw' U2 Rw2

(25,21)

N

Christopher Mowla

l U' l U' l' U l' U l D' b2 l' b2 l U2 D l U l' U l2 //Safe

(25,21)

N

Rw r U' R' U r' U' R U2 R' U r U2 r U2 r U2 r2 U R U' Rw'

(27,22)

N

Christopher Mowla

Rw2 R U2 Rw U r U' Rw' U2 Rw' U2 Rw U2 Rw U r2 U' Rw' U2 Rw' U2 Rw' R'

(30,23)

N

Christopher Mowla


Non-symmetrical algorithms

x' U2 l' U2 l U2 r' U2 r B2 r' F D2 F' l' F D2 F' r B2 x

(27,19)

N

Christopher Mowla

z' Fw' Uw r' Uw' Fw Uw r' Fw2 U L U' Fw r' Fw' U L' U' Fw2 r2 Uw' z

(23,20)

N

Christopher Mowla

Lw Uw' l Uw Rw' B D' d B Fw Dw b Dw' Fw' B' D d' B' Uw' l' Uw x

(21,21)

N

Christopher Mowla

z' Fw' Uw r' Uw' Fw Uw r' Fw2 L Fw U' R U r' U' R' U Fw' L' Fw2 r2 Uw' z

(25,22)

N

Christopher Mowla

Oriented



• Note: The mirror case can also arise as well. One can simply apply the mirror image of this case's algorithms to get algorithms for its mirror case.


Symmetrical algorithms

Rw2 Uw' Bw r' Bw' Uw Bw Rw' Uw' r' Uw Rw r Bw' Rw2

(17,15)

N

Christopher Mowla

(y' Fw2) l Uw' Bw Uw l' Uw' l' Bw' l Bw l Bw' Uw (Fw2 y)

(17,15)

N

Christopher Mowla

(y' z' Fw2) Rw' Dw l' Dw' Rw Dw Lw' Bw' l' Bw Lw l Dw' (Fw2 z y)

(17,15)

N

Christopher Mowla

Rw2 f Uw' Bw Dw r' Dw' r' Bw' r Bw r Bw' Uw Rw2

(17,15)

N

Christopher Mowla

(y' z' Fw' l') Uw' Bw2 Uw' r Uw Bw2 Uw2 r Uw r' Uw2 (l Fw z y)

(19,15)

1st 15 Mover

N

Christopher Mowla

Uw Rw' Uw' r' Uw Rw Bw' Rw' r' Uw' r' Uw R r' Bw Uw'

(16,16)

N

Christopher Mowla

(y' x' Uw') r' Uw Rw Uw' l Uw Rw' Fw Uw r Uw' Fw' Uw' M' (Uw x y)

(16,16)

N

Christopher Mowla

y' Bw' R Dw r' Dw' Rw' Bw Rw' R' Dw r' Dw' Rw2 Fw' u Fw y

(17,16)

N

Christopher Mowla

(y' Bw') Uw2 r' Dw l' Dw' r Uw' Lw2 Uw' r' Uw Lw2 Uw' r (Bw y)

(19,16)

N

Christopher Mowla

y' l' r' U Rw U2 l' U2 Lw' U2 x' U' l' U F2 U' M' l' y

(20,16)

N

Christopher Mowla

x l' U L U l' U' L' U' F2 l' F2 r U2 r' U2 l2 x'

(21,16)

N

Christopher Mowla

(L' B') l' U2 l' U2 x U2 l' U2 x' r U2 r' U2 l2 (B L)

(23,16)

N

(x Rw E') Uw Rw Uw' l Uw Rw' Fw Uw r Uw' Fw' Uw' l' (E Rw' x')

(17,17)

N

Christopher Mowla

(x Rw E') Uw Fw Uw r' Uw' Fw' Rw Uw' l' Uw Rw' Uw' l (E Rw' x')

(17,17)

N

Christopher Mowla

(y' Bw' r') l' Bw Rw Bw' l Bw Rw' Uw Bw r Bw' Uw' Bw' (r Bw y)

(17,17)

N

Christopher Mowla

(y' Bw' R) l Dw Bw Dw r' Dw' Bw' Rw Dw' l' Dw Rw' Dw' (R' Bw y)

(17,17)

N

Christopher Mowla

(y' Bw') l' R Dw Rw Dw' l Dw Rw' Bw Dw r Dw' Bw' Dw' R' (Bw y)

(17,17)

N

Christopher Mowla

(y' Bw' M) Bw Rw Bw' l Bw Rw' Uw Bw r Bw' Uw' Bw' l' (M' Bw y)

(17,17)

N

Christopher Mowla

(z Dw) Fw Rw Fw' l Fw Rw' B Dw Fw r Fw' Dw' B' Fw' l' (Dw' z')

(17,17)

N

Christopher Mowla

(z Dw) B l' Fw Rw Fw' l Fw Rw' Dw Fw r Fw' Dw' Fw' B' (Dw' z')

(17,17)

N

Christopher Mowla

(z Dw) B l Fw Dw Fw r' Fw' Dw' Rw Fw' l' Fw Rw' Fw' B' (Dw' z')

(17,17)

N

Christopher Mowla

x l' U L U l' U' L' U' x U2 l' U2 l U2 r' U2 l r x2

(21,17)

N

Christopher Mowla

x l' U L U l' U' L' U' M' U2 l' U2 l U2 r' U2 l2 x'

(22,17)

N

Christopher Mowla

x l U2 l U2 M U L U l U' L' U' l' U2 r U2 l2 x'

(22,17)

N

Christopher Mowla

Rw U2 l U2 3Lw U L' U l U' L U' l' U2 r U2 Lw2 x'

(22,17)

N

Christopher Mowla

y' l U F2 U' l U F2 l F L F l F' L' F' l' U' l2 y

(21,18)

N

Christopher Mowla

x Lw' L' U L' U l' U' L U' M' U2 l' U2 l U2 r' U2 Lw2 x'

(23,18)

N

Christopher Mowla

(x L2)(r F U2 F')4 r (L2 x')

(25,19)

N

Floyd Newberry

(x L2)(r' F U2 F')4 r' (L2 x')

(25,19)

N

Floyd Newberry

y' r U r' U' r' u' r u r' U' r' U r' u' U r U2 r Uw r2 y //Safe

(21,20)

N

Ben Whitmore

r' U' r U' r' D b2 r b2 r' U2 D' r' U' r U' r U r' U r2 //Safe

(25,21)

N

y' x Lw' U' R' U' l2 U R U Lw' U2 Lw U' R' U' l U R U Lw' U2 Lw2 x' y

(25,21)

N

Christopher Mowla

x l2 U2 Lw U l U' Lw' U2 Lw' U2 Lw U2 Lw U l2 U' Lw' U2 Lw' U2 Lw' L2 x'

(30,22)

N

Christopher Mowla

(y Rw' U2 r') Rw2 U2 Rw U2 Rw U r U' Rw' U2 Rw' U2 Rw U2 Rw U r2 U' Rw' U2 Rw (r U2 Rw y')

(37,27)

N

Christopher Mowla


Non-symmetrical algorithms

Lw' Dw B2 Rw2 Fw r Fw' Rw2 B2 r' Dw' Lw Dw r Dw'

(19,15)

N

Christopher Mowla

r' Uw Lw Uw' r2 Fw2 Lw' Fw r' Fw' Lw Fw2 r' Uw Lw' Uw'

(19,16)

N

Christopher Mowla

y' Uw' Fw l' Fw' Uw Fw l' Uw2 R Uw l' Uw' R' Uw2 l2 Fw' y

(19,16)

N

Christopher Mowla

z x2 Rw' Uw l' Uw' Rw Uw l' Fw2 R Fw l' Fw' R' Fw2 l2 Uw' x2 z'

(19,16)

N

Christopher Mowla

z Uw' Rw Uw2 l' Uw r' Uw' l Uw2 R' Uw Bw2 Uw' r' Uw Bw2 z'

(20,16)

N

Christopher Mowla

r' Uw Rw Bw' r F' R' r Uw' r' Uw r' R F Bw Rw' Uw'

(17,17)

N

Christopher Mowla

r' Uw Rw Bw' r R' F' r Uw' r' Uw r' F R Bw Rw' Uw'

(17,17)

N

Christopher Mowla

y' Rw' Uw' r Uw Rw Bw' F' Rw R Uw' r Uw Rw' R' F r' Bw y

(17,17)

N

Christopher Mowla

y' Rw' Uw' r Uw Rw Bw' R2 F' r Uw' r Uw r' F Rw' R' Bw y

(18,17)

N

Christopher Mowla

x' f' Uw Rw' Fw' r' Fw Rw Uw' f r' Uw Rw' Fw' r' Fw Rw Uw' r2 x

(19,18)

N

Christopher Mowla

z' r' Uw Bw Uw' Bw L Bw r Bw' L' Bw' Uw Bw' Lw Uw' r Uw Lw' Uw' z

(19,19)

N

Christopher Mowla

r Uw' Lw Uw r' Uw' Fw' Uw' B Rw2 B' l B Rw2 B' Uw Fw Lw' Uw

(21,19)

N

Christopher Mowla

Rw2 U Rw' U Rw' U Rw U2 Rw U2 Rw' U Rw U' Rw' U' Rw' U2 Rw' U2 Rw U2 Rw' U Rw2 U2

(34,26)

Y

Bruce Norskog

Rw' U Rw' U' Rw2 U2 Rw' U2 Rw' U Rw U' Rw' U Rw2 U' Rw U2 Rw' U2 Rw U2 Rw2 U Rw' U' Rw U

(36,28)

5x5x5 Alg

Y

Ben Whitmore




Non Dedge-Preserving Last Layer 4-Cycle Cases in Two Dedges

• These are all of the other 4-cycles that can occur in two dedges in the last layer besides double parity and adjacent double parity.
• These are additional cases which arise in the K4 Method and other direct solving methods.

In opposite dedges

Checkerboard



• This is a 4-cycle case. Manually take the inverse of these algorithms to solve its inverse case, as it and its inverse are not the same case.

Rw2 U2 r E2 r' E2 r U2 Rw2

(15,9)

v1

N

Tom Rokicki & Ed Trice

Lw2 U2 l E2 l' E2 l U2 Lw2

(15,9)

v2

N

Tom Rokicki & Ed Trice

l f l U2 S' l S U2 l f' l

(13,11)

N

Tom Rokicki

l f' l U2 S l S' U2 l f l

(13,11)

N

Tom Rokicki

l b l U2 S l S' U2 l b' l

(13,11)

N

Tom Rokicki

l b' l U2 S' l S U2 l b l

(13,11)

N

Tom Rokicki

r f r U2 S' r S U2 r f' r

(13,11)

N

Tom Rokicki

r f' r U2 S r S' U2 r f r

(13,11)

N

Tom Rokicki

r b r U2 S r S' U2 r b' r

(13,11)

N

Tom Rokicki

r b' r U2 S' r S U2 r b r

(13,11)

N

Tom Rokicki

(f2 u' r2 Uw2 S') r (S Uw2 r2 u f2)

(17,11)

N

Christopher Mowla

(f2 u r2 Uw2 S') r (S Uw2 r2 u' f2)

(17,11)

N

Christopher Mowla

r2 U2 l' E2 r E2 l U2 S2 r2 S2

(19,11)

N

Per Kristen Fredlund

r2 S2 U2 l E2 r E2 l' U2 r2 S2

(19,11)

N

Per Kristen Fredlund

r2 U2 (r S2)2 r' U2 l2 f2 M2 f2

(21,12)

N

Christopher Mowla

r2 U2 (r S2)2 r' U2 l2 b2 M2 b2

(21,12)

N

r S' L2 S (r U2)2 r S' L2 S r

(17,13)

N

Christopher Mowla

r S R2 S' (r U2)2 r S R2 S' r

(17,13)

N

Christopher Mowla

(Rw2 Uw U S Lw2 u2) r (u2 Lw2 S' U' Uw' Rw2)

(19,13)

N

Christopher Mowla

(Rw2 Uw' U' S Lw2 u2) r (u2 Lw2 S' U Uw Rw2)

(19,13)

N

Christopher Mowla

(Rw2 S) (r' U2)4 r' (S' Rw2)

(19,13)

N

Christopher Mowla

(Rw2 S') (r' U2)4 r' (S Rw2)

(19,13)

N

Christopher Mowla

r' U2 r2 U2 r U2 r' U2 r U2 r2 U2 r'

(21,13)

Alg(v1a)

Y

Rw' U2 r2 U2 Rw U2 r' U2 Rw U2 r2 U2 Rw'

(21,13)

Alg(v1b)

Y

r' U2 Rw2 U2 r U2 r' U2 r U2 Rw2 U2 r'

(21,13)

Alg(v1c)

Y

Rw' U2 r2 U2 r U2 r' U2 Rw U2 r2 U2 r'

(21,13)

Alg(v1d)

Y

l' U2 l2 U2 l U2 l' U2 l U2 l2 U2 l'

(21,13)

Alg(v2a)

Y

Lw' U2 l2 U2 Lw U2 l' U2 Lw U2 l2 U2 Lw'

(21,13)

Alg(v2b)

Y

l' U2 Lw2 U2 l U2 l' U2 l U2 Lw2 U2 l'

(21,13)

Alg(v2c)

Y

Lw' U2 l2 U2 l U2 l' U2 Lw U2 l2 U2 l'

(21,13)

Alg(v2d)

Y

Bowtie/Hourglass



• This is a 4-cycle case. Manually take the inverse of these algorithms to solve its inverse case, as it and its inverse are not the same case.
• However, this is a special case where, instead of having to take the inverse of these algorithms to solve the inverse case, you may also do a U2 turn (or y2 cube rotation) and then perform these same algorithms to solve the inverse case.

r2 U2 r B2 r B2 E2 l' B2 l B2 D2 l' y2

(21,13)

N

Oscar Roth Andersen

r U2 r U2 F2 r F2 l' U2 M r' Uw2 r2 u2

(22,14)

N

r U2 r U2 M' U2 r U2 r' U2 M r' Uw2 r2 u2

(23,15)

N

r2 U2 l' U2 l F2 U2 r2 U2 r F2 r' F2 r' F2

(25,15)

Y

(B' Rw U2 D' x) U2 r' U2 r2 (U2 r')2 (x' D U2 Rw' B)

(23,16)

N

xyzzy

r' U2 r U2 l' U2 r U2 l F2 r' F2 r F2 r2 F2

(25,16)

Y

(B' R Uw' u' r' f2 Rw2 E) f' (E' Rw2 f2 r u Uw R' B)

(21,17)

N

Christopher Mowla

(B' R Uw' u' r f2 Rw2 E) f' (E' Rw2 f2 r' u Uw R' B)

(21,17)

N

Christopher Mowla

r U2 M' U2 r2 U2 r' U2 r U2 r' U2 r' U2 l U2 r'

(26,17)

N

In adjacent dedges

Checkerboard



• This is a 4-cycle case. Manually take the inverse of these algorithms to solve its inverse case, as it and its inverse are not the same case.

F2 Uw' F2 Rw2 Uw2 R2 z' S u F' B' Uw2 Rw2 Uw' F2

(22,14)

seq3002

N

Tom Rokicki

(F U' F' Rw2 U2) r' E2 r E2 r' (U2 Rw2 F U F')

(21,15)

N

Tom Rokicki & Ed Trice

(R2 Uw' u' r' f2 Rw2 E) f (E' Rw2 f2 r u Uw R2)

(21,15)

N

Christopher Mowla

(R2 Uw' u' r f2 Rw2 E) f (E' Rw2 f2 r' u Uw R2)

(21,15)

N

Christopher Mowla

(Rw R U2 D' x)(r U2)2 r2 U2 r U2 (x' D U2 Rw' R')

(23,16)

N

xyzzy

(Rw' F2 M2 F) U2 l U2 l2 U2 l U2 l (F' M2 F2 Rw)

(25,16)

N

Christopher Mowla

(Lw' U2 M2 U) M U2 l U2 l2 U2 l U2 r (U' M2 U2 Lw)

(26,17)

N

Christopher Mowla

(R B) r U2 M' U2 r2 U2 r' U2 r U2 r' U2 r' U2 l U2 r' (B' R')

(30,21)

N

Bowtie/Hourglass



• This is a 4-cycle case. Manually take the inverse of these algorithms to solve its inverse case, as it and its inverse are not the same case.

(R B) Rw2 U2 r' E2 r E2 r' U2 Rw2 (B' R')

(19,13)

N

Tom Rokicki & Ed Trice

(R Fw f r u2 Rw2 S) u' (S' Rw2 u2 r' Fw' f' R')

(19,15)

N

Christopher Mowla

(R Fw f r' u2 Rw2 S) u' (S' Rw2 u2 r Fw' f' R')

(19,15)

N

Christopher Mowla

F' Lw U D Lw2 Bw2 Rw' R' U' D B2 Lw2 Fw2 D2 Lw' F

(21,15)

Y

Tom Rokicki

(Lw' U2 M2 U' x) r' U2 r' U2 r2 U2 r' U2 (x' U M2 U2 Lw)

(25,16)

N

Christopher Mowla

R B r U2 r2 U2 r' U2 r U2 r' U2 r2 U2 r B' R'

(25,17)

N

(r U2 M2 U') M' U2 l' U2 l2 U2 l' U2 r' (U M2 U2 r')

(26,17)

N

Christopher Mowla

y2 Fw' r2 Uw2 L' Uw' U' R U F' R' F r' F' R F U' R' Uw U L Uw2 r' Fw Uw Fw' r' Fw Uw' y2

(31,28)

N

Christopher Mowla




Move count comparison of all last layer 2-cycles and 2 dedge 4-cycles

• Since there is disagreement among cubers about whether lowest slice quarter turn moves (BQTM) or lowest slice half turn moves (BHTM) defines the "optimal algorithm", the table below categorizes all 2-cycle and 4-cycle cases (in two dedges) on the 4x4x4 by the average of the two.
• Each case's "Optimal Move Average" was found by taking the average of BQTM and BHTM from all algorithms on this page which solve it and choosing the smallest result.
• The table shows that the "worst case" is the One Dedge Flip, and that the "easiest case" is the Checkerboard case in two opposite dedges.

• These results will change in the future if shorter algorithms are found for any of these cases.

Arranged by Rank (Worst last layer cases are first.)

Last Layer Case	Minimum BQTM	Minimum BHTM	Optimal Move Average	Rank
One Dedge Flip 2-cycle	18	15	18	1
Adjacent Checkerboard 4-cycle	21	14	18	1
Double Parity 4-cycle	21	14	17.5	2
Bowtie/Hourglass 4-cycle	21	13	17	3
Close Unoriented 2-cycle	17	15	17	3
Far Unoriented 2-cycle	17	15	17	3
Adjacent Double Parity 4-cycle	17	16	16.5	4
Adjacent Bowtie/Hourglass 4-cycle	19	13	16	5
Oriented 2-cycle	16	15	16	5
Diagonal 2-cycle	16	14	15.5	6
Opposite 2-cycle	16	12	15.5	6
Checkerboard 4-cycle	13	9	12	7




Algorithms which do not preserve the centers

• These algorithms are for fixing the wing edges, but they do not preserve the centers. Many of these algorithms can be used in solving methods such as the Cage Method.

One dedge flip



Rw2 U' r' U B2 U' r U' r' U2 B2 Rw2

(17,12)

N

Michael Gottlieb

Rw2 U r' U' B2 U r U r' U2 B2 Rw2

(17,12)

N

Christopher Mowla

Rw2 3Dw l U' R2 U l' U y r U2 F2 Rw2

(17,12)

N

Christopher Mowla

Rw2 F r F' Uw2 B l' B l B2 Uw2 Rw2

(17,12)

N

Christopher Mowla

Rw2 U2 r U2 l' U2 l U2 r' F2 r F2 Rw2

(21,13)

N

Rw2 U2 r U2 l' U2 l U2 r' B2 l B2 Rw2

(21,13)

N

l U2 l' U2 Lw2 F2 r' F2 Rw2 U2 l' U2 r x2

(21,13)

N

r2 Uw Bw' Uw' r Uw Rw Bw r' Bw' Rw' Bw Uw' r

(15,14)

N

Christopher Mowla

r2 U2 r' U2 r U2 l' U2 r U2 r' U2 l r

(21,14)

N

Lucas Garron

r2 U2 r U2 l' U2 l U2 l' U2 r U2 l r

(21,14)

N

Nicholas Ho

Rw' l' Uw Rw' Uw' l Uw y Lw Dw r' Dw' Lw' y' Rw Uw' Rw

(15,15)

N

Christopher Mowla

x r R' U' Rw U' r U Rw' r' U R U' r U r' x'

(15,15)

N

Opposite double parity



• This is a 4-cycle case. Manually take the inverse of these algorithms to solve its inverse case, as it and its inverse are not the same case.
• Most of these algorithms are exactly the same as the one dedge flip ones except that the extra quarter turn is inverted.

Rw' F' u F' D2 F u' F' l D2 F2 Rw

(15,12)

N

Christopher Mowla

Rw2 U r' U' B2 U r U r U2 B2 Rw2

(17,12)

N

Christopher Mowla

Rw2 U' r' U B2 U' r U' r U2 B2 Rw2

(17,12)

N

Christopher Mowla

Rw2 3Dw l U' R2 U l' U y r' U2 F2 Rw2

(17,12)

N

Christopher Mowla

x' Rw2 F2 r F2 r U2 l' U2 l U2 r' U2 Rw2 x

(21,13)

N

l' y' Uw2 Rw2 Uw' l' Uw Rw2 Uw' l Uw' y U2 l U2 l'

(18,14)

N

Christopher Mowla

Rw2 b2 D' f' D B2 D' f U' f U D Bw2 Rw2

(19,14)

N

Christopher Mowla

l r U2 l U2 r' U2 r U2 r' U2 l' U2 l2

(21,14)

N

Nicholas Ho

Adjacent double parity



• This is a 4-cycle case. Manually take the inverse of these algorithms to solve its inverse case, as it and its inverse are not the same case.

(r2 U' Rw U' R' U') r' (U R U Rw' U r2)

(15,13)

N

Christopher Mowla

(r2 U F2 U' r' F2) r (F2 r U F2 U' r2)

(19,13)

N

Christopher Mowla

Rw' U Rw U F2 U' r' U F2 r' U' R' U' Rw

(16,14)

N

Christopher Mowla

(x Uw F U r' U R' U) r (U' R U' r U' F' Uw' x')

(15,15)

N

Christopher Mowla

x' r2 U' Rw U' l' U2 l' U2 l' U2 l2 U' Rw' U r2 x

(21,15)

N

Christopher Mowla

Three flips

Single parity



M r' U B2 l2 B2 3Dw' Rw2 U2 r' U2 Rw2 y' M' r

(21,14)

Pattern 1

N

Per Kristen Fredlund

(r2 F2 U) r U2 r' U2 r U2 r' U2 r (U' F2 r2)

(23,15)

N

Christopher Mowla

(r2 U2 F' M' U) r U2 r' U2 r U2 r' U2 r (U' M F U2 r2)

(27,19)

N

Christopher Mowla

Double parity



y M r' U B2 l2 B2 3Dw' Rw2 U2 r U2 Rw2 y' M' r y'

(21,14)

Pattern 1

N

Per Kristen Fredlund

(R2 D')(r' U' F' R' F R2 U') r' (U R2 F' R F U r)(r U2 r' U2)(D R2)

(29,22)

N

Christopher Mowla

2-cycles in two adjacent dedges (in the M ring)

Opposite



l U' l' U' F2 U l U' F2 U2

(13,10)

N

l U l' U F2 U' l U F2 U2

(13,10)

N

z2 y' U2 R2 U l U' R2 U l' U y r z2

(13,10)

N

l F2 r' U2 l U2 l' U2 r U2 F2

(17,11)

N

l' U Lw' U L U' l' U L' U' Lw l U'

(13,13)

N

Diagonal



• Some of these are nearly the same as the one dedge flip algorithms.

Lw' U' l U B2 U' l' U' l U2 B2 Lw

(15,12)

N

Christopher Mowla

Lw' U l U' B2 U l' U l U2 B2 Lw

(15,12)

N

Christopher Mowla

Lw' 3Dw' r' U L2 U' r U' y' l' U2 F2 Lw

(17,12)

N

Christopher Mowla

x' U Rw' U r' U' Rw r U' R' U r' U' R x

(13,13)

N

Marc Waterman

x' R U Rw' U r' U' Rwr U' R' U r' U' x

(13,13)

N

Marc Waterman

x' r' Uw Bw' Uw' r Uw Rw Bw r' Bw' Rw' Bw Uw' x

(13,13)

N

Christopher Mowla

Lw' 3Dw' r' U L2 U' r D' r' U D L2 y' Lw

(15,13)

N

Christopher Mowla

Lw' U l U' B2 U l' U' B2 D2 r' D2 Lw

(17,13)

N

Christopher Mowla

Lw' 3Dw' r' U L2 U' r U D2 y' r' D2 F2 Lw

(17,13)

N

Christopher Mowla

(x') r' U2 r U2 l' U2 l U2 l' U2 r U2 l (x)

(19,13)

N

Lw' U2 l' U2 r U2 r' U2 l F2 l' F2 Lw

(19,13)

N

Lw' U2 l' U2 r U2 r' U2 l B2 r' B2 Lw

(21,13)

N

M' l' U2 l' U2 l U2 r' U2 l U2 l' U2 l

(20,14)

N

Nicholas Ho

2-cycles in two opposite dedges (in the M ring)

Directly opposite



y Lw' Uw' r Fw r Fw' r2 Uw Lw Uw' r Dw

(13,12)

N

Christopher Mowla

r' U' r' U r' U r U2 r U r' U r' U'

(15,14)

N

Ben Whitmore

U' Rw' U r' U' Rw r U' R' U r' U' R U2

(15,14)

N

François Courtès

Opposite/Diagonal



• These are the same as the one dedge flips but without first and last moves.

U r' U' B2 U r U r' U2 B2

(13,10)

N

Christopher Mowla

U' r' U B2 U' r U' r' U2 B2

(13,10)

N

Christopher Mowla

3Dw l U' R2 U l' D l U' D' R2 y

(13,11)

N

Christopher Mowla

U' r' U B2 U' r U B2 D2 l D2

(15,11)

N

Christopher Mowla

3Dw l U' R2 U l' U' D2 y l D2 F2

(15,11)

N

Christopher Mowla

x' U2 l U2 r' U2 r U2 l' F2 r F2 x

(17,11)

N

U2 r U2 l' U2 l U2 r' F2 r F2

(17,11)

N

U2 r U2 l' U2 l U2 r' B2 l B2

(17,11)

N

4-cycles in opposite dedges

Checkerboard



• This is a 4-cycle case. Manually take the inverse of these algorithms to solve its inverse case, as it and its inverse are not the same case.

Rw2 U2 r U2 Rw2

(9,5)

N

Marc Waterman

Bowtie/Hourglass



• This is a 4-cycle case. Manually take the inverse of these algorithms to solve its inverse case, as it and its inverse are not the same case.

(y L' U B' Rw2 U2) r' (U2 Rw2 B U' L y')

(15,11)

N

(y L F' U Rw2 F2) r (F2 Rw2 U' F L' y')

(15,11)

N

r U2 r' U2 x' U2 r U2 r2 U2 r' U2 r' x

(19,12)

N

Nicholas Ho

r2 U2 r U2 x' D2 r' U2 r D2 r' U2 r' x

(19,12)

N

Kenneth Gustavsson




Algorithms which do not preserve F3L OR the centers

• These algorithms are least practical when it comes to use of parity algorithms, and therefore they are mentioned here for theoretical purposes only. They have fewer moves than any other OLL parity algorithm forms.

Single parity

Rw2 U r' U' B2 U r U r' B2 Rw2

(15,11)

N

Christopher Mowla

Rw2 U' r' U B2 U' r U' r' B2 Rw2

(15,11)

N

Christopher Mowla

Rw2 U f U' B2 U f' U l B2 Rw2

(15,11)

N

Christopher Mowla

r' U' R U' r U2 r U2 r U' R' U' r'

(15,13)

N

Christopher Mowla

r' U' R U r' U2 r' U2 r' U R' U' r'

(15,13)

N

Christopher Mowla

r' U' R U r' F2 U2 r U2 F2 r' U R' U' r'

(19,15)

N

Christopher Mowla

Double parity

r 3Dw r' 3Dw F2 3Dw' r 3Dw l' F2 l

(13,11)

N

Christopher Mowla

Rw U R B R' l Fw2 r' Fw2 U' Rw2 x

(14,11)

N

Bruce Norskog

Rw U R B l Fw2 r' Fw2 R' U' Rw2 x

(14,11)

N

Bruce Norskog

Rw2 U r' U' B2 U r U r B2 Rw2

(15,11)

N

Christopher Mowla

Rw2 U' r' U B2 U' r U' r B2 Rw2

(15,11)

N

Christopher Mowla

Rw2 U f U' B2 U f' U l' B2 Rw2

(15,11)

N

Christopher Mowla

Rw U2 Rw U2 Rw U2 Rw2 U' R U Rw2 U R U' Rw2

(21,15)

Y

Christopher Mowla

Rw U2 Rw2 U' Rw' U2 Rw U2 Rw' U' Rw2 U' R2 U' Rw'

(21,15)

Y

Kåre Krig & Christopher Mowla

Either single parity or double parity

• The following list of algorithms can be considered either single parity or double parity. Just exchange the last move Rw' with Lw', Rw with Lw, and so forth.

Rw' U' R' U' Rw F' R' F' Rw'

(9,9)

Y

Christopher Mowla

Rw F R F Lw' B R B Lw

(9,9)

Y

David Gilday

Rw' U' R U' Rw F' R F' Rw'

(9,9)

Y

Christopher Mowla

Rw U' R U' r' D' R D' Rw

(9,9)

N

Christopher Mowla

Rw' U R' U Rw U F2 U Rw'

(10,9)

Y

Christopher Mowla

Rw U' R U' Rw' D' R2 D' Lw

(10,9)

Y

Christopher Mowla

Rw' U' F2 U' r B' R B' Lw'

(10,9)

N

Christopher Mowla

Rw U' L U' F2 Lw U' R U' Rw'

(11,10)

Y

Christopher Mowla

Rw U' L' U' F2 l U L U Rw'

(11,10)

N

Christopher Mowla

Rw' U' L' U' Rw F2 U L U Rw

(11,10)

Y

Christopher Mowla

Rw' U' L' U' B2 Rw' B R B Rw

(11,10)

Y

Christopher Mowla

Rw' U R2 U Rw F2 U L' U Rw

(12,10)

Y

Christopher Mowla

Rw' U F' U l U2 F' U2 F Rw'

(12,10)

N

Christopher Mowla

Rw U' R' U' Rw' D' L D L' D' Lw

(11,11)

Y

Christopher Mowla

Rw B U F' R b' M R E' R Bw

(11,11)

N

Bruce Norskog & Christopher Mowla

Rw U R' U R r' U2 B' L' B' Rw'

(12,11)

N

Christopher Mowla

l' B' u B' D2 B u' B' l D2 l'

(13,11)

N

Christopher Mowla

Rw F U' R U Rw' F U R F' U Rw

(12,12)

Y

Christopher Mowla

Rw' U F' L F Rw F R B' E y B Rw'

(12,12)

Y

Christopher Mowla

Rw S' F L F' U b' U E M U Bw

(12,12)

N

Bruce Norskog & Christopher Mowla

Rw' U' F R' F' Rw F' U' F2 R U Rw'

(13,12)

Y

Christopher Mowla

2-cycles

• Although the following list of algorithms is not "OLL parity", where dedge preservation is important, they are the briefest 2-cycle algorithms yet to be found. They do not preserve F3L or the centers.
• These algorithms are nothing more than 2-cycle algorithms which don't preserve centers with beginning and ending outer layer turns omitted.

r' U' B2 U r U r'

(8,7)

N

Christopher Mowla

r' U B2 U' r U' r'

(8,7)

N

Christopher Mowla

r' U L2 U' r D' r'

(8,7)

N

Christopher Mowla

r' U' R2 U r D r'

(8,7)

N

Christopher Mowla

External Links

PLL parity algorithms

• 10 Ways to Swap Two Corners on a 4x4x4
• Speedcubedb PLL parity algs
• Re: 4*4 PLL Alg 5 moves cancelled (Yahoo Groups)
• Interesting 4*4 PLL Alg (Yahoo Groups)
• http://frederickbadie.free.fr/444PLLparity.html
• https://www.speedsolving.com/forum/showthread.php?26357-PURE-Corner-swap-parity-algorithms
• https://www.speedsolving.com/forum/showthread.php?23182-full-4x4-pll
• https://www.speedcubingtips.eu/4x4x4-pll-et-parites/

OLL parity algorithms

• Speedcubedb OLL Parity algs
• Both 4x4x4 parities in one (Yahoo Groups)
• Some tricks for 4x4x4 (Yahoo Groups)
• double parity 4x4 (Yahoo Groups)
• http://www.speedsolving.com/forum/showthread.php?11311-4x4x4-OP-DP-algorithms-%28more-finger-friendly%29
• http://www.speedsolving.com/forum/showthread.php?15614-Odd-parity-Algorithms-%28specifically-single-edge-quot-flip-quot-%29
• http://www.speedsolving.com/forum/showthread.php?2727-4x4-Orientation-parity
• http://www.speedsolving.com/forum/showthread.php?30127-New-4x4-parity-algs-using-R-Rw-U
• https://www.speedsolving.com/forum/showthread.php?10037-4x4-Parity-Algorithms
• https://www.speedsolving.com/forum/showthread.php?27783-4x4-pure-double-parity
• http://www.speedsolving.com/forum/showthread.php?11409-4x4-Optimal-Solver-v2
• http://www.twistypuzzles.com/forum/viewtopic.php?f=8&t=8829
• http://www.stefan-pochmann.info/spocc/other_stuff/4x4_5x5_algs/?section=FixOrientationParity
• http://www.stefan-pochmann.info/spocc/other_stuff/4x4_5x5_algs/?section=FixBothParities
• http://www.mementoslangues.fr/CubeDesign/4xCubes/DoubleMidgeFlip.pdf
• http://apelgam.se/Rubik/4x4parity/
• https://www.speedcubingtips.eu/4x4x4-oll-et-parites/

Supercube parity algorithms

• 5x5x5 help needed (Yahoo Groups)
• http://www.speedsolving.com/forum/showthread.php?4659-4x4-algorithm
• http://www.speedsolving.com/forum/showthread.php?37308-Is-a-SuperCube-Safe-Single-Dedge-Flip-Algorithm-Possible-in-lt-U-Rw-gt
• http://www.speedsolving.com/forum/showthread.php?26622-God-s-algorithm-discovered-for-OLL-parity-edge-flip-%28theory%29

OLL parity algorithms which do not preserve F3L

• http://www.speedsolving.com/forum/showthread.php?37962-Fixing-4x4x4-orientation-parity-earlier-than-usual
• http://www.speedsolving.com/forum/showthread.php?26564-4x4x4-edge-parity-is-there-a-shorter-alg-that-doesn-t-preserve-corners
• http://www.speedsolving.com/forum/showthread.php?17839-WANTED-New-Dedge-Flip-Algorithm!
• http://twistypuzzles.com/forum/viewtopic.php?f=8&t=9502

K4 method parity algorithms

• http://tomas.rokicki.com/ell4x4.html
• https://www.speedsolving.com/forum/showthread.php?708-A-Collection-of-Algorithms&p=811284&viewfull=1#post811284

Cage method parity algorithms

• http://twistypuzzles.com/forum/viewtopic.php?f=8&t=6552
• http://twistypuzzles.com/forum/viewtopic.php?f=8&t=7140

Comprehending and making your own parity algorithms

• 5x5x5 help! (Yahoo Groups)
• http://twistypuzzles.com/forum/viewtopic.php?f=1&t=9720
• http://twistypuzzles.com/forum/viewtopic.php?f=1&t=995
• https://www.speedsolving.com/forum/showthread.php?15643-Can-we-really-solve-big-cubes&p=239619
• http://www.speedsolving.com/forum/showthread.php?22969-Methods-for-Forming-2-Cycle-Odd-Parity-Algorithms-for-Big-Cubes
• http://www.speedsolving.com/forum/showthread.php?21725-New-Two-Corner-Swap-Algorithm-Technique-for-Big-Even-Cubes-%28PLL-Parity%29
• 10 Ways to Swap Two Corners on a 4x4x4
• http://www.speedsolving.com/forum/showthread.php?43745-4x4-fixing-one-edge-parity-without-algorithms
• https://www.speedsolving.com/forum/threads/help-in-solving-4%C3%974%C3%974-parities-intuitively.13572/
• http://www.speedsolving.com/forum/showthread.php?26477-Contructing-Algorithms-%28not-a-how-to-sorry%29
• http://twistypuzzles.com/forum/viewtopic.php?f=8&t=31410
• http://twistypuzzles.com/forum/viewtopic.php?f=8&t=32717

General parity theory

• http://twistypuzzles.com/forum/viewtopic.php?f=8&t=31472
• http://hlavolam.maweb.eu/parity-problem
• https://www.speedsolving.com/forum/showthread.php?41972-Parity-Errors/
• https://www.speedsolving.com/forum/showthread.php?9996-understand-parity
• http://twistypuzzles.com/forum/viewtopic.php?f=8&t=26759
• http://twistypuzzles.com/forum/viewtopic.php?f=8&t=22201
• http://twistypuzzles.com/forum/viewtopic.php?f=8&t=12841
• https://www.youtube.com/watch?v=wmOyuROOANI
• https://www.speedsolving.com/forum/showthread.php?21045-Difference-between-OLL-and-PLL-parity-on-4x4
• https://www.speedsolving.com/forum/showthread.php?52852-Can-all-parity-be-explained

Preventing/Avoiding parity

• Challenge for those who enjoy the 4x4x4 (Yahoo Groups)
• http://twistypuzzles.com/forum/viewtopic.php?f=8&t=7366
• https://www.speedsolving.com/forum/showthread.php?892-avoiding-4x4-pairity-problems
• https://www.speedsolving.com/forum/showthread.php?15880-Can-odd-parity-be-detected-in-a-big-cube-prior-to-being-solved
• https://www.speedsolving.com/forum/showthread.php?19965-Avoiding-parity-errors-in-4x4x4
• https://www.speedsolving.com/forum/threads/avoiding-parity-problems-in-big-cubes.4894/
• https://www.speedsolving.com/forum/threads/parity-error-prediction-and-new-simulator.306/

Miscellaneous

• (No title) (Yahoo Groups)
• The corner cycle alg is no longer "magical" (Yahoo Groups)
• http://twistypuzzles.com/forum/viewtopic.php?f=1&t=9133
• http://www.mementoslangues.fr/CubeDesign/