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Understanding the Origin of Parity Errors and Solving by Logic
2. Case 1: PLL Parity Error (permutation of edge pairs)
This parity error may initially seem absurd. In a 3x3x3 cube it would be impossible to swap two edges without affecting anything else.
Even when considering that these are four separate parts, it still appears strange that they have swapped in pairs. After all, we know that it is not possible to construct a commutator that swaps only two pieces; the minimum requirement is three. The arrows in Figure 2.1 depict the seemingly illogical permutations that have taken place.
However, this error is actually the result of the convergence of two separate errors. These errors involve overlapping commutators, each involving three edges. To understand how this error is generated, let's begin with a perfect cube and observe the process. First, we will perform a 3-edge commutation, as demonstrated in Figure 2.2. Designing commutators should be familiar to you by now, making it relatively straightforward. We will need to set up the cube properly to facilitate the execution of the commutator. One possible setup is to perform F2 Bm, which allows us to utilize the U and D layers for the swaps and the Fm layer for the back and forth movements. Figure 2.3 presents a step-by-step illustration of such a commutator. Feel free to click the play button (▷) to view the animation.
Figure 2.1
Figure 2.2
Note that two edges, in addition to changing places, have "flipped colors". Now let's perform another commutator, as depicted in Figure 2.4, to complete the parity error. Once again, this commutation is relatively straightforward. Figure 2.5 illustrates one possible sequence of moves: we start with an initial setup, rotate the cube for better visualization, apply the commutator, and then reverse the rotation and setup. To observe the process in detail, click play (▷)
Figure 2.4
The arrows depicted in Figure 2.6 represent the actual commutations of the edges.
It is important to note that the overlapping of the second commutator (indicated by the green arrows) effectively cancels out the edge orientation error caused by the first commutator (indicated by the orange arrows). As a result, it creates the illusion of only two simple permutations between pairs of edges.
To resolve this parity error, the solution is quite simple now that you understand its origin. You just need to perform two commutations: first, the one indicated by the orange arrows, and then another one indicated by the green arrows in Figure 2.6. By overlapping these commutations, you will successfully solve the parity error.
Figure 2.6
What if the swapped edges aren't in the same configuration as shown here? Well, in that case, you can design your own commutators to address the specific configuration. However, if you prefer to use the same commutators demonstrated in this guide, you can simply perform an initial setup to align the configuration with the example.
It is worth noting that this parity error can also occur in cubes larger than 4x4x4, but only for edges that have a pair equidistant from the center of a side border. In other words, this error does not occur for edges located in the border centers of odd-sized cubes, such as 3x3x3, 5x5x5, 7x7x7, and so on. Figure 2.7 provides some examples of this parity error in cubes larger than 4x4x4.
Figure 2.7
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