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Understanding the Origin of Parity Errors and Solving by Logic
4. Case 4: OLL Parity Error (misorientation of two edges)
What about case 3?
Don't worry, let me clarify the situation. Case 3 is actually mixed up with case 4, although it may not appear that way at first. We'll focus on case 4, but we'll address case 3 along the way.
Undoubtedly, case 4 is the most intriguing parity error, and it's also the one that is most deceptively concealed in its origins.
For those familiar with commutators and conjugates, solving case 4 might seem straightforward, as it gives the impression that the two edges are simply flipped. However, that's not the case. Despite being called OLL (Orientation of the Last Layer), this error is not about the orientation but rather the permutation of the edges. Yes, the edges have been swapped, and not only that, some center pieces have also been exchanged, but this fact eludes us because they all appear identical.
Surprisingly, the fundamental origin of this error is the same as the one presented in case 2.
In case 2, we started with the commutation of three corners within the same outer layer. Similarly, for the OLL error, we'll begin by commuting three edges within the same inner layer.
Let's focus on the second vertical layer on the left side of the 4x4x4 cube (referred to as the Lm layer) as shown in Figure 4.1. The orange arrows indicate the desired exchange of edges. To facilitate this, we need to perform an initial setup, such as B2 Rm' D, allowing us to utilize the U and D layers for the swaps, while the Lm layer will be used for the back and forth movements. For a more detailed illustration of the setup and commutation, simply click play (▷) on Figure 4.2.
Figure 4.1
At the end of the animation in Figure 4.2, we successfully accomplished the planned exchange of three edges on the Lm layer. Similar to case 2, our next step is to "solve" this layer, but without commuting the edges. It's important to note that, within the Lm layer, only two edges share the same color on one face. Specifically, these are the edges with the red color located on the top of the cube. To confirm this arrangement doesn't occur on the other faces, rotate the cube around the x-axis. Now, let's proceed by rotating only the Lm layer, aligning the red colors with the correct face. To observe this action, please click play (▷) on Figure 4.3.
By performing the last move, we can observe that only two edges within the Lm layer are swapped, along with some center pieces. Take a moment to rotate Figure 4.3 using your mouse and observe the result at the end of the animation.
Now, if we focus on the centers that have been interchanged, as illustrated in Figure 4.4 (displaying only the left half of the cube), we can perform a permutation to restore them to their proper positions. By doing so, the cube will be considered "correct," except for the two remaining misaligned edges.
Figure 4.4
You have the flexibility to interchange the centers in any desired manner using commutators and conjugates. By now, you may have gained some experience with center commutations in the 4x4x4 cube.
To keep our example simple, we will focus on commutations involving only two colors at a time, allowing us to "solve" one face at a time. While this approach may be slower, it simplifies the process.
Let's start by swapping the blue centers with the red ones. Before proceeding, we'll rotate the cube 90ᐤ around the x-axis for better visibility, as shown in Figure 4.5 (using the x rotation). The orange arrows indicate the intended commutation of the centers. To facilitate this, we'll set up the cube and utilize the top three layers for the "swaps," while the Fm layer (located just behind F) will be used for the back and forth movements. For a detailed demonstration of this commutation, please click play (▷) on Figure 4.6.
Figure 4.5
To continue, we rotate the cube once again by 90ᐤ along the x-axis (applying x) and repeat the previous procedure to swap the green centers with the blue ones. Following that, we rotate the cube once more by 90ᐤ along the x-axis (applying x), and there remains one final swapping to be executed, this time between the blue and orange centers. Figure 4.7 provides a comprehensive demonstration of the entire procedure. Simply click play (▷) to observe the animation.
And now, we have arrived at the PLL parity error involving the permutation of two edges, which we refer to as Case 3! This case has been waiting in the background until now.
To transform this error into the OLL parity error, wich we refer to as Case 4, we need to perform one final edge commutation, as depicted in Figure 4.8. There are multiple ways to design this commutator, and an example is provided in Figure 4.9. Feel free to click play (▷) to witness the animation and better understand the process.
Figure 4.8
And now, we arrive at the intriguing OLL parity error in the 4x4x4 cube!
Similar to case 2, this error is primarily caused by a 90ᐤ rotation of an inner layer in relation to its equidistant "pair" layer from the center of the cube. The resulting configuration is a consequence of edge and center commutations that effectively "cancel out" the majority of the errors. Just like in the previous case, the apparent misorientation of only two edges is merely an illusion.
Curious about resolving this parity error through logical deduction? Please turn to the next page for additional tips and an illustrative example.
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