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Understanding and Solving the Rubik's Cube Without Algorithms
9. Commutators and Conjugates (That's all you need!)
The concept of commutators is a powerful tool for solving the remaining corners of the Rubik's Cube. Actually you could solve the entire cube with commutators. However, they are particularly effective when we have limited freedom of movement or when we need to target specific transformations on the cube. Commutators allow us to manipulate certain pieces or groups of pieces while keeping the rest of the cube unaffected. They are especially useful for performing swaps, rotations, or rearrangements without disturbing the already solved parts of the cube.
At this point, there are up to 5 corners left to be solved, but it's possible that you have fewer remaining.
To solve the corners, we will follow two steps:
Place the corners in the correct positions.
"Twist" the corners so their colors are in the correct orientations.
To accomplish this, we will utilize commutators and conjugates. However, before diving into commutators, let's learn some additional "laws" of the cube.
You've already seen that it's impossible to have just one inverted edge on the cube, as in Figure 9.1. This configuration is mathematically impossible.
Another important rule to remember is that it is also mathematically impossible to have only two swapped corners like in Figure 9.2. Corners will always be swapped in trios, as well as the edges.
Key Knowledge: it is impossible to have only 2 swapped corners. Corners will always be swapped in trios, the same goes for the edges.
This means that there are specific sequences of movements capable of interchanging only 3 corners with each other. This type of movement is called a commutator.
A commutator involves intersecting procedures in such a way that the movements of one procedure partially "cancels" the movements of the other procedure, resulting in only a few specific parts being interchanged. By understanding commutators, we can interchange corners without affecting any other part of the cube. We can use the right combination of “back and forth” (chapter 4) and “go, swap and back” (chapter 5) movementes in such a way that only 3 cubelets are interchanged.
For example:
Procedure 1: go down - swap - back - go down - undo swap - back (see Figure 9.3a)
Procedure 2: go up - swap - back - go up - undo swap - back (see Figure 9.3b)
p.s.: you can choose between procedures 1 or 2 by the arrows (◁▷) in the upper right corner of Figure 9.3
Note that procedures 1 and 2, separately, are back and forth movements, everything that is done is undone.
However, there are movements in the two procedures that are equivalent. By doing the intersection below (equivalent movements highlighted in red) the final result is going to be quite interesting. Because of this intersection part of the swaps are kept while the rest of the cube comes back to its original state.
Let's take a look at an example to better understand the concept. Figure 9.4 shows the case where we want to move the pink corner to the position of the gray corner. Since corners are always interchanged in trios, this means that the gray corner will take the place of another corner (let's say the green corner), and the green corner will take the place of the pink corner.
We will use the upper (U) and down (D) layers for the swaps, and the front layer (F) for the back and forth movements. The back and forth moves will be F and F' because the desired cubelet cannot be in the back and forth layer. Let's go through the sequence step-by-step, you can click on forward (⧐) in Figure 9.4, at each step:
F ("go" of procedure 1)
D' ("swap" of procedure 1)
F' ("back" of procedure 1 and "go" of procedure 2)
U ("swap" of procedure 2)
F ("back" of procedure 2 and "go" repetition of procedure 1)
D ("undo swap" of procedure 1)
F' ("back" of procedure 1 and "go" repetition of procedure 2)
U' ("undo swap" of procedure 2)
In short, the sequence is: F D' F' U F D F' U' (go - swap - back - swap --- go - undo swap - back - undo swap).
Click on reset (∣◁) in Figure 9.4, and then click play (▷) to see the complete sequence in action.
It's important to note that the specific choice of corners does not matter. You just need to identify which parallel layers will be used for the swaps and which perpendicular layer in between them will be used for the back and forth movements. Also note that the first cubelet to go to its destination place is alone in a swap layer. The other swap layer contains the other two cubelets. This defines which perpendicular layer will be used for the back and forth movements.
The first place receiving a cubelet is the gray corner (2nd cubelet) in Figure 9.5, while the first cubelet to swap is the pink one (1st cubelet).
The parallel layers for the swaps are the upper (U) and down (D), i.e. the colored layers in Figure 9.5.
The layer used for the back and forth and which, at the same time, intersects the parallel layers of the swaps, is the front layer (F), marked with the white center in Figure 9.5.
Summary of the designing and application of a 3-cubelet commutator :
Choose any 3 cubelets that will be swapped.
Determine which position (2nd cubelet) will receive the 1st cubelet.
Ensure that the 3rd cubelet is on the same layer as the 2nd cubelet, which should be parallel to the layer of the 1st cubelet. If needed, perform a setup move to align them.
Identify the perpendicular layer for the back and forth movements, which should only contain the 2nd cubelet.
Begin the commutator with the 2nd cubelet (the first destination) "going" to the layer of the 1st cubelet.
The movements are: go – swap – back – swap --- go – undo swap – back – undo swap
Undo any setup moves performed at the beginning, before the commutator.
App source: animcubejs.cubing.net/animcubejs.html
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