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.

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Edge parity 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. If you do this to a solved cube, you can see how the algorithm affects the rest of the puzzle, but this is not noticeable during edge pairing.

Now solving this parity should be simple. All this algorithm does is slice along so that the other red-blue piece is above the orange-blue piece, flips the edge, then slices back, solving both edges. It is important to note that these parities can ONLY occur on even layered cubes 4x4, 6x6 etc. This will be explained later. This image shows OLL parity in its purest form, but any state where there is an odd number of yellow edges facing upwards is the same.

This is the indication that you have parity. It is called so because it is first noticed during the OLL stage of a solve.

It is caused when solving the edges. This means when you have reduced the cube to a 3x3 with the reduction method, although it may just look like a scrambled 3x3, picture it as a scrambled 3x3 after one edge has been removed and put back in the wrong way around. An even number of flipped edges means there is no OLL parity. If you want to see for yourself, take out two edges from a 3x3 and put them back in their places but flipped.

An odd number of flipped edges means that there is OLL parity. Do the same thing: take a 3x3 cube and this time flip three edges in their places. No matter what you do, there will always be one flipped edge remaining when you try to solve it. This is combined with the fact that you have to solve the centres, meaning they can all be in different places.

When solving the first two centres, you can solve them on any side, as long as they are opposite one another. Therefore there is absolutely no way of telling whether an edge is solved correctly or not, meaning OLL parity is completely random. There are many algorithms that are all unique apart from one thing — They are all very long and difficult to memorise. Even with the best memory and the best algorithm, most speedcubers will struggle to solve OLL parity in under seconds, meaning that it can be the cause of a ruined average.

OLL parity is also known for the large risk of pieces popping from the puzzle during execution. Because it involves almost exclusively slice turns turns that would be impossible on a 3x3 and is executed as fast as possible, a pop during this stage can be much larger than a normal pop, and combined with the strong reliance on muscle memory that most speedcubers have, this can prevent the solver from completing the rest of the algorithm without error when the cube is put back together.

These images show PLL parity in their pure forms. One of the best examples for this is the pseudo-T permutation. After finishing OLL, you will notice the two bars that indicate a T permutation. But instead of having headlights on the side that the bars point towards, you will have a solved bar. The main algorithm people learn for PLL parity swaps the two edges shown in the first picture.

This is called the Half-H perm case. The other case is the Half-Z perm case, and the same algorithm is used but with a setup beforehand. It occurs for a similar reason to OLL parity.

During edge pairing it is impossible to tell whether edges are swapped, therefore having two swapped edges results in some form of PLL parity. These parities can happen on any even NxN layered cube. For a 4x4 there are two edges that need pairing, but with a 6x6 there are 4. Parities occur, but they are exactly the same as 4x4 parity, they just look bigger. If you want to find a list of these, check out this page for a full list of all 4x4 Parity algorithms.

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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. The stuff below is how I would do each case, but there are lots of different option for how to handle them. OLL parity This is the parity that is caused by solving the centers such that the edge permutation is odd.

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## 4x4x4 parity algorithms

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