Understanding and Solving the Rubik's Cube Without Algorithms
To execute the strategy, we begin by selecting a layer that contains the two corners we want to twist, which in this case are the yellow and red corners, as shown in Figure 10.5. We designate the F layer (colorful) to be preserved throughout the process.
Next, we utilize the perpendicular layer, R, to perform back and forth movements with the back layer, B. The number of movements can vary as we strive to twist the upper right corner. It is crucial to maintain the integrity of the F layer while executing these movements.
Now, you might wonder, won't these actions disrupt the rest of the cube? Indeed, they will. However, the key lies in reversing the movements, which restores the cube to its normal state.
But what is the advantage of preserving the front layer at all costs if we ultimately want to twist the corner? Here comes the secret weapon! By prioritizing the preservation of the front layer, we have an opportunity to rotate the preserved layer before executing the reverse movements. This involves placing another corner in the position of the first one. As a result, when we reverse the movements, not only does the cube realign itself, but we also twist another corner, and exactly, in the opposite direction from the first one!
Summary of the Corner Twist Strategy:
Select the layer that contains the corners to be twisted and designate it as the preserved layer. It's important to note that the corners don't have to be in the same row or column and can even be diagonally opposite.
Perform "go, swap, and back" movements using the parallel layer until the first corner is twisted.
Rotate the preserved layer in such a way that the second corner takes the place of the first corner.
Reverse and undo the exact opposite moves used to twist the first corner, ensuring they are performed in the exactly reverse order.
Undo the setup of the preserved layer, restoring it to its original position.
Figure 10.6 provides a visual representation of this process. To see a simple animation of the strategy, click the play button (▷).
In the following section, we will delve into a detailed example to illustrate the strategy further.
Curiosity (skip it if you prefer): Indeed, it's fascinating to observe the overlap of conjugates in this strategy. The first procedure involves a combination of movements to twist the corner, which we'll refer to as A. The second procedure focuses on the setup of the preserved layer, which we'll denote as B. Consequently, the conjugates utilized in this context are A () A-1 and B () B-1. It is the overlapping of these conjugates that enables us to twist only two corners in opposite directions. Therefore, we execute A B A-1 B-1 as the operation. Interestingly, this operation can be seen as a commutator as well. However, unlike commutators that swap cubelets around, this particular commutator exclusively swaps orientations. If you desire further information on commutators and conjugates click here.
When it comes to twisting a corner, the number of moves used is not necessarily important, but it is advantageous to minimize the number of moves. This is because we need to remember these moves in order to execute the reverse sequence effectively.
The minimum possible number of moves for twisting a corner consists of two "go, swap, and back" movements with a setup in between. The setup involves using the opposite layer in relation to the preserved layer, and it also allows us to determine the direction of the corner twist during the procedure.
In the following section, we will delve into the detailed movement of "go, swap and back" for the corner we wish to twist. It's important to note that while a particular explanation will be provided, it is crucial to adapt and interpret the movements in a way that feels comfortable and intuitive for you. The essence lies in following the general strategy outlined earlier, which revolves around the "go, swap, and back" sequence combined with a setup.
If you're ready, let's proceed with a detailed explanation of the "go, swap and back" movement for twisting the corner.
If you click forward (⧐) in Figure 10.7, you will initiate the "go" move (R). Pay attention to the red facelet of our corner, which will be facing upwards immediately after the "go" move. This indicates that when we perform the final swap and place the corner back in the same position before the "back" move, the facelet pointing upwards will be the one in front of the cube. Let's use this as a reference point. Therefore, the color we desire to be in front of the cube should be facing upwards just before the last "back" movement.
Continuing with the "go, swap, and back" sequence, the next move is the "swap". In this case, we have two options:
We can perform the swap by taking the corner out of the back and forth layer, or
We can perform the swap while keeping the corner within the back and forth layer.
For now, let's explore the scenario where we keep the corner in the back and forth layer, and the swap can only be done by B'. If you click forward again (⧐) in Figure 10.7, you will observe that our corner on the back layer (B) is also maintained within the back and forth layer (R).
To conclude the "go, swap, and back" movement, the only available move is R'. Click forward once more (⧐) in Figure 10.7.
It's important to observe that our corner has now rotated, with the yellow facelet facing upwards. This indicates that if we perform another "go, swap and back" movement to return the corner to its original position, the yellow facelet will be facing forward. Initially, the yellow facelet was behind the cube immediately after the "go" move. Therefore, every time after the first "go" movement, if we choose to keep our corner in the back and forth layer during the "swap", the color initially behind the cube will end up in front. It's not necessary to memorize this; simply anticipate the movements and understand what will happen.
Assuming that our goal is to place the yellow color on the front face of the cube, we need to perform another "go, swap and back" movement. However, before that, we require a setup to move our corner out of the back and forth layer.
We have performed the B move as the setup in this case. By clicking forward (⧐) in Figure 10.8, you can observe the movement. It's worth mentioning that we could have chosen to do B2 for the setup, but there's no need to complicate things when a single B move suffices.
To complete the process, we execute the last "go, swap and back" movement: R B' R'. You can click play (▷) in Figure 10.8 to see the entire sequence in action.
Let's take note of the complete sequence of movements that twisted the corner:
R B' R' B R B' R'
Later on, we will compare this sequence with another possibility.
App source: animcubejs.cubing.net/animcubejs.html