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. 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. 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.

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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. 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.

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. 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 "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 nxnxn 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. 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.

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. 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 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.

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.

Below is an example algorithm found in December of 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.

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## Parity on the 4x4 Rubik’s Cube

Step 1: Centers Step 2: Edges Step 3: Fix parity Theory Just a little theory here, but one thing I think that is important to realize is that the power of centers first solving is that you reduce to a 3x3x3 cube, where you can use all of your normal speedsolving tricks. The major drawback of centers-first solving though are the two parity errors that come up. However, I think that the ability to really use your specialized 3x3 tricks makes up for that Taking care of parity cases So this last step is a page on how I handle the parity cases. Also check out bigcubes.

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## 4x4x4 parity algorithms

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