Understanding and Solving the Rubik's Cube Without Algorithms
8. Solving the edges of the last layer
In this step, we will solve the remaining edges of the Rubik's Cube using a similar principle to what we learned in Chapter 5. However, there is one key difference: we need to preserve the equator layer (E), which means it won't undergo any rotation. Instead, we'll focus on the back and forth movement of the auxiliary column (R and R' or F' and F) to swap its edge with the upper layer. The goal is to solve all the edges of the top layer, and by doing so, the missing edge in the equator layer (E) will also be solved. It's important to note that having all the edges solved except one with flipped colors is mathematically impossible.
There are a couple of precautions to keep in mind during this stage:
Whenever you perform the "go" movement of the vertical column, it is mandatory to follow it with the "back" movement.
Since the "back" movement is mandatory, you should avoid swapping with an edge that has already been solved in the upper layer.
Let's walk through a practical example in Figure 8.1:
Since none of the upper layer edges are solved, we can start with any one of them. In this case, we have one of the upper edges in the vertical column used for the back and forth movement. The only option we have is to apply R ("go"), then perform the "swap" and "back."
Example: R U' R'
Click play (▷) on Figure 8.1 to see the sequence.
After the swap, rotate the upper layer so that the solved edge matches its side facelet with the corresponding cube face having the same color. This positioning makes it easier to decide on the next swap. In the example, we apply U2 to rotate the upper layer to the correct position.
Click play (▷) on Figure 8.2 to observe the sequence.
With the upper layer in the correct position, we can now determine the next swap. In the example, since the edge in the back and forth column has an orange face (opposite to the solved edge's red face), we do U2 before the next "go, swap, and back" movement.
Click forward (⧐) in Figure 8.3 to proceed.
Repeat the "go, swap and back" process, selecting another edge with a yellow facelet to swap. In the example, we choose the yellow/blue edge. Therefore, we perform F' ("go"), U' ("swap") and F ("back").
Click forward (⧐) 3 times on Figure 8.3 to see the sequence.
Rotate the upper layer to the correct position again, determining the next swap. In the example, we apply U'.
Click forward (⧐) again, in Figure 8.3 to proceed.
Repeat the process until there is only one yellow-faced edge left to solve. Always remember not to use already solved edges in swaps.
At the end of this step, you will have three possibilities:
a. You were lucky, and the last swap solved everything (a rare case).
b. You end up with two solved yellow edges, one of them facing up but in the wrong position, along with the last unsolved edge in the back and forth column.
c. You end up with three yellow edges in the correct places and orientations, along with the last unsolved edge in the back and forth column.
For case b, proceed as before: set up the upper layer and perform the "go, swap, and back" movement. The only available option for swapping will be the edge without the yellow facelet since the others have already been solved. In this case, we will do:
U' ("setup")
R ("go")
U ("swap")
R' ("back")
Click play (▷) on Figure 8.4 to see the sequence.
Once the "go, swap, and back" movement is completed, all the yellow edges will be correct, along with the edge in the back and forth column. It may be necessary to rotate the upper layer to the correct position.
For case c, there might be some uncertainty. As soon as you perform the "go" movement, all the yellow edges will be correct, but you are forced to perform the swap and come back. So what should you do?
Let's apply the knowledge from Chapter 7 about cycles. We know that by repeatedly making a specific movement, everything will return to its original condition. In this case, as we have 4 sides, the cycle must involve the upper four edges. We can initiate the swap by applying U or U' and then make an additional quarter turn of the upper layer in the same direction used in the swap. After that, keep repeating the same procedure of "go, swap and back" with an additional quarter turn at the end. Before everything returns to the original condition, there will be a moment when all the yellow edges are solved. Stop at that point and rotate the top layer to the correct position if necessary.
For our example in case c, apply R U' R' U' three times or R U R' U three times. You'll notice that in the end, when you do the third repetition, it won't be necessary to complete it as the movement R U' R' or R U R' will solve everything.
Click play (▷) on Figure 8.5 to see the sequence.
At this point, you have solved 80% of the Rubik's Cube! The most fascinating aspect is that you've achieved this by understanding the cube's "laws" and using logical reasoning with simple moves, without the need to memorize complex sequences.
Now, only the last 5 corners remain. Solving them requires knowledge of two important concepts: commutators and conjugates.
App source: animcubejs.cubing.net/animcubejs.html