4×4 Parity

While solving the last layer of the 4×4, you will sometimes encounter parity cases that don’t exist in the 3×3 puzzle. There are two types of parity.

OLL Parity

When orienting the last layer of the 3×3 puzzle, you will always have an even number of correctly oriented last-layer edges (zero, two, or four). But in the 4×4 puzzle, it’s possible that an odd number of edge pairs are correctly oriented, such as one edge pair or three edge pairs. Before you can complete OLL, flip one of the edge pairs in place so that an even number of edge pairs are correctly oriented.

First, decide which edge pair to flip. As a beginner, which edge pair you flip is less important than learning to identify and resolve parity. As your skill increases, you’ll learn how to select an edge pair that produces a more favorable OLL case.

Next, turn the U side so that the edge pair to be flipped is in the top front edge position, then perform this algorithm.

Rw′ U2 Rw U2 Rw U2 Rw2 F2 Rw′ U2 Rw′ U2 F2 Rw2 F2

However, I find this algorithm easier to execute with its left-handed equivalent.

Lw U2 Lw′ U2 Lw′ U2 Lw2 F2 Lw U2 Lw U2 F2 Lw2 F2

Whichever form of the algorithm you use, the important aspect is that it transforms an odd number of oriented edge pairs into an even number. With this OLL parity case resolved, you can solve OLL as a 3×3 puzzle.

PLL Parity

A second parity case is possible during PLL.

With the last layer corners of a 3×3 puzzle solved, zero, three, or four edges can require permutation. But in the 4×4 puzzle, it’s possible that two edge pairs will need to be permuted. You’ll need to swap two opposite edge pairs, or two adjacent edge pairs.

You can recognize PLL parity by touch when the edge positions don’t conform to any PLL algorithms you’ve learned. If all four last-layer corners are correctly positioned but exactly two edge pairs need to exchange positions, you’ve encountered PLL parity.

To swap two opposite edge pairs, turn the U side so that the pairs to be swapped are in the top front and top back edge locations. Then perform this algorithm.

r2 U2 r2 Uw2 r2 Uw2

Read that algorithm carefully. The r2 is a lowercase R, indicating a 180-degree turn of the right inner layer.

Note that a final U2 is required to restore last layer corners to their original positions.

To swap two adjacent edge pairs, turn the U side so that the edge pairs to swap are in the top front and top right locations. Then perform these three algorithms.

R U R′ U′

r2 U2 r2 Uw2 r2 Uw2

U′ R U′ R′

The first algorithm is the sexy move. This temporarily relocates several pieces, but it’s necessary to transform the adjacent edge case to an opposite edge case. The second algorithm swaps the two opposite edge pairs, top front with top back. The third algorithm is the inverse sexy move (modified to account for the U2 displacement of the parity algorithm), which reverses the effects of the initial sexy move.

Final Thoughts on Parity

When solving the 4×4, you might encounter OLL parity, PLL parity, both, or neither. To prepare for competition, learn to identify both parity cases and solve them quickly.

With practice, you can learn to solve centers experimentally. Now that you know how to pair edges and solve parity, you have everything you need to solve the 4×4 using the Reduction Method.