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Understanding the Origin of Parity Errors and Solving by Logic
3. Case 2: PLL Parity Error (2-corner permutation)
Undoubtedly, this particular parity error appears to be the most perplexing. We are well aware that swapping two corners in a 3x3x3 cube is impossible, and the same holds true for larger cubes like the 4x4x4 and beyond. So, how can we seemingly observe only two swapped corners?
Once again, this is an illusion created by the overlapping of multiple errors. These errors intertwine and partially cancel each other's effects, resulting in the misleading perception that only two corners have been swapped.
To grasp the origin of this parity error, let's revert back to a 3x3x3 cube. Even in a 3x3x3 cube, it is possible to encounter a PLL parity error. However, in this scenario, two corners are always swapped simultaneously with two edges, as depicted in Figure 3.1. You can rotate and observe the cube by dragging the mouse over the image or by clicking play (▷).
Let's examine in detail how this parity error is generated in a 3x3x3 cube. We'll begin by performing a commutation of three corners in the up layer while ensuring that the upward face (blue) remains unchanged. For those familiar with commutators, this step is straightforward. Figure 3.2 illustrates one possible commutation in detail. At the end, we rotated the cube along the y-axis to observe how the corners align with the orange and yellow faces.
Observe that, at the conclusion of the animation, the two corners in the front (layer F) are the only ones on the layer U that possess the same color on the side face (in this instance, orange). Rotate the cube using the mouse along the y-axis to verify.
Let's then try to “match” the U layer with the side faces, but without commutating the corners. In this case we start by matching the color faces, followed by a setup, and finally commutate 3 edges as ilustrated in Figure 3.3. To observe this maneuver in detail, click on the play button (▷).
Note that, in the end, we managed to hit the orange side and almost completely the white side. Rotate the cube with the mouse 180ᐤ around the y axis (y2) to observe the other side and you will see that we have reached exactly the parity error initially seen.
Therefore, this same error can be replicated in a 4x4x4 cube by assuming that the edges are "glued" together, simulating a 3x3x3 cube configuration, as depicted in Figure 3.4. However, in a 4x4x4 cube, the edges can also experience the parity error seen in case 1, resulting in the apparent permutation of two pairs of edges. To demonstrate this, let's induce the case 1 error between the yellow and red colored edges on the U layer in Figure 3.4. Click the play button (▷) to observe the animation.
And thus, we generate the seemingly absurd permutation between two corners. It's important to note that this parity error is actually a result of the overlapping of two parity errors: the parity error between two corners and two edges, which already occurs in a 3x3x3 cube, overlapped by the parity error seen in case 1.
Fundamentally, this parity error is caused by the 90ᐤ rotation of an outer layer relative to its counterpart layer on the opposite side. The final arrangement is achieved through the overlapping commutations of corners and edges, along with another parity error seen in case 1.
So, how do we resolve this parity error?
Now that you understand how it is created, the solution is to do the opposite.
The key tip in this guide is to first commute three corners of the layer where the error occurs while ensuring that one face has all the correct colors. In our previous example, this only happens for the U layer. You need to design a commutator that positions the same side color on the corners of the face where the error is observed. In our example, you should perform a commutation on the corners in such a way that both upper corners have either yellow or white colors on the red front face.
To add a slight variation to what we have already done, let's commutate the corners of the U layer in our example, but this time, we will aim to have the white color of the upper corners on the red face while maintaining the integrity of the up face (blue). Figure 3.5 provides a detailed illustration of this commutation. Click the play button (▷) to see the animation.
Now, all you need to do is a 90ᐤ turn of the U layer, aligning the corners with the corresponding color faces, and then perform a commutation on the edges as if they were "glued" together. This will effectively resolve the two edge pairs. Figure 3.6 provides a comprehensive demonstration of the entire process. Click the play button (▷) to view the animation and understand the details.
After completing the steps mentioned earlier, the only remaining error is the PLL error, which is the same as the one described in case 1. Fortunately, we are already familiar with how to solve this error using the techniques discussed previously.
Note that this specific PLL parity error, where only 2 corners are swapped, can only occur on even-sized cubes such as 4x4x4, 6x6x6, and so on. On the other hand, odd-sized cubes will always exhibit a parity error involving the swapping of 2 corners and 2 central edges, as we have seen previously with the 3x3x3 cube. To provide further examples, please refer to Figure 3.7, which showcases instances of this particular parity error on odd-sized cubes larger than the 3x3x3.
Figure 3.7
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