Rubik's Cube

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:

  1. Place the corners in the correct positions.
  2. "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 groups of three, as well as the edges.

Key Knowledge: it is impossible to have only two swapped corners. Corners will always be swapped in groups of three, 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 commutator.

A commutator involves intersecting "back-and-forth" procedures in such a way that everything that is done is subsequently undone, except for a few specific parts that end up 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 three cubelets are interchanged.

For example:

Procedure 1: go - swap - back - go - undo swap - back (see Figure 9.3a)

Procedure 2: setup - undo setup (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.

Let's carefully check the first procedure. Click Reset (∣◁) in Figure 9.3a and play it again, step by step, pressing forward (⧐) each time. We already knew (from chapter 5) that the go-swap-and-back does not affect the upper layer, except for the corner that was intentionally swapped. Unfortunately, the two bottom layers become scrambled. Of course, we can fix that by reversing the go-swap-and-back procedure.

So what's the point, since everything returns to its original position?

Now comes the trick! Before reversing the go-swap-and-back procedure, we change the upper corner by applying a setup move. But not just any setup move—a specific one that preserves the two bottom layers. In other words, a setup move using a rotation of the upper layer, as shown in Figure 9.3b. With this setup, we place another upper corner into the position of the first one and then reverse the go-swap-and-back procedure. By doing this, everything in the two bottom layers returns to its original position except for the first corner. Notice that the reversed go-swap-and-back procedure does not affect the upper layer. Finally, we reverse the setup move, and the upper layer returns exactly to its original state except for the two corners affected by the setup.

Abracadabra! We have found a way to interchange only three corners and nothing else.

Let's take a look at a detailed example to better understand the concept. Figure 9.4 shows the case in which we want to move the pink corner into the position of the gray corner. Since corners are always interchanged in groups of three, 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) layer for the setup and the down (D) layer for the swap. The front layer (F) will provide the "go" and "back" moves between the upper and the down layer. Let's go through the sequence step by step. You can click forward (⧐) in Figure 9.4, at each step:

  • F ("go" of procedure 1)
  • D' ("swap" of procedure 1)
  • F' ("back" of procedure 1)
  • U ("setup" of procedure 2)
  • F ("go" repetition of procedure 1, since we want to reverse it)
  • D ("undo swap" of procedure 1 to reverse it)
  • F' ("back" repetition of procedure 1)
  • U' ("undo setup" of procedure 2 to reverse it)

Click reset (∣◁) in Figure 9.4, and then click play (▷) to see the complete sequence in action.

If you look carefully, you will notice that the upper layer (U) and the down layer (D) are actually used to perform and undo swaps. The front layer (F) is used to make back-and-forth movements between these parallel U and D layers.

In short, the sequence is: F D' F' U F D F' U' (go - swap - back - swap --- go - undo swap - back - undo swap).

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 (that one used for the setup) 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.

Click on play (▷) in Figure 9.5 to see, once again, the complete and detailed sequence in action.

Summary of the designing and application of a 3-cubelet commutator :

  1. Choose any 3 cubelets that will be swapped.
  2. Determine which position (2nd cubelet) will receive the 1st cubelet.
  3. 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.
  4. Identify the perpendicular layer for the back and forth movements, which should only contain the 2nd cubelet.
  5. Begin the commutator with the 2nd cubelet (the first destination) "going" to the layer of the 1st cubelet.
  6. The movements are: go - swap - back - swap --- go - undo swap - back - undo swap
  7. Undo any setup moves performed at the beginning, before the commutator.