# 4X4X4 PARITY PDF

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 :. 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.

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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 :.

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:. 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 "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.

There are many types of parity cases which can occur during a 4x4x4 solve, but the cases which result from attempting to reduce a fully scrambled 4x4x4 into a pseudo 3x3x3 state this means an even n x n x n cube in which all of its composite edges are complete and all of its centers are complete and are in the correct center orientation, in general. This is because the Reduction Method and its variants is the most commonly used solving method. Naturally, these type of parity cases are called reduction parity.

In May , 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. 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.

It turns out that we are not limited to using well-known dedge-pairing even parity algorithms to pair dedges. Websites such as bigcubes. 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. There is actually a total of 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 cases and relatively short algorithms to solve each one directly. Algorithms for the Cage Method , as well as algorithms for theoretical purposes and general 4x4x4 exploration are present as well.

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. 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. One can split up r2 as r r or as r' r' and insert 3x3x3 moves to obtain the pure form of PLL parity.

Below is an example algorithm found in December of However, we can also just use the inner slice turns r and r' as well. 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 4x4x4 cube odd parity fix which preserves the colors of the centers essentially independently found in by Tom Rokicki and Ed Trice is f2 r E2 r E2 r f2 11,7.

Although this algorithm is not listed under a case image on this page, it would appear in the following format in an "algorithm bar" if it was. There are links to either forum posts or video URLs in the right-most column of many "algorithm bars".

That is, 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 bar above mean later. Since this algorithm 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. The shortest and well-known n x n x n cube odd parity fix which preserves the colors of the centers is r U2 4 r 13,9. The 11,7 above discolors centers on, say, the 5x5x5 cube. Finally, one of the simplest OLL parity more specifically, a double parity algorithms found in December of to remember also consists of a short repeated sequence:.

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. 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.

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". On January 24, , speedsolving. 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.

This page not only contains commonly practiced speedsolving algorithms: it also contains algorithms which illustrate the veracity of the 4x4x4 cube parity algorithm domain. Two of the most popular 15 BHTM move algorithms which flip a single dedge on the 4x4x4 are the following.

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. 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.

For example, 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.

For illustration 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. However, 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. 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.

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. Symmetrical algorithms are conjugates. A clear example of a symmetrical algorithm is Stefan Pochmann's n x n x n 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? It's worthy to note that 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.

They also happen to be supercube safe. 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.

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## Parity on the 4x4 Rubik’s Cube

Parity is something that most puzzle solvers despise. It is something that can slow speedsolvers down immensely in official solves , and is generally seen as a pain to deal with. One type of pseudo-parity is edge parity. It happens when all but two edges are solved. These two edges look the same, but inverted. In the picture, the blue-red edge on the left needs to be paired with the red-blue edge on the right, and the same for the blue-orange edges. If you know how to solve a 4x4 , you will know the flipping algorithm.

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## How to solve a 4x4 Rubik's Cube

By using our site, you acknowledge that you have read and understand our Cookie Policy , Privacy Policy , and our Terms of Service. Puzzling Stack Exchange is a question and answer site for those who create, solve, and study puzzles. It only takes a minute to sign up. YES, it works. I found it along with many other algorithms by using Cube Explorer for the 3x3x3. Unlike many 4x4x4 OLL parity algorithms which people have gotten from 3x3x3 solvers, this one was not obvious to translate to the 4x4x4. As you can see, it has some very unusual turns; and it is very unique!

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