Understanding and Solving the Rubik's Cube Without Algorithms
Understanding conjugates is quite simple. Remember the setup move we sometimes use before a sequence of movements? Well, a conjugate is essentially a setup move that is reversed after performing any sequence.
A conjugate can involve a single movement or a sequence of movements that are then reversed (performed in the opposite order). Let's take a look at a couple of examples:
Consider the sequence U A U', where A represents any sequence of moves. In this case, the moves U and U' act as conjugates for A.
Another example is B D' A D B'. Here, the moves B D' and D B' are conjugated with A.
Conjugates prove to be handy tools in various algorithms, such as the 3-edge swap example shown below.
In this scenario, we aim to rearrange the edges on the cube. Specifically, we want to place the cyan edge where the green edge is, the green edge where the pink edge is, and the pink edge where the cyan edge is. To achieve this, we can utilize a commutator sequence.
Based on the summary of commutators, we observe that the S layer could be utilized to swap the U and D layers back and forth. However, in this case, we encounter an obstacle because the first piece we need to swap is located precisely in the S layer. To overcome this, we must remove the first piece from the S layer while keeping it in the D layer.
To proceed, we begin by performing a setup move of D' to adjust the initial piece. Then, we apply the commutator sequence (S D S' U' S D' S' U) to perform the desired edge swaps. Finally, to complete the process, we undo the initial setup move by performing D.
To visualize this sequence, you can click the play button (▷) on Figure 9.4.
To enhance your understanding of conjugates, let's engage in the following commutations for practice:
Exercise (a)
Place the pink edge in place of cyan, the cyan in place of green, and the green in place of pink.
You can interact with the image by rotating the cube and layers. If you need to start over just click reset (∣◁).
To see a possible solution expose the hidden tab by clicking ∨.
S (M D' M' U2 M D M' U2) S'
Click reset (∣◁) then play (▷) to see this sequence.
Please note that this is just one of the possible solutions. If you have discovered an alternative solution, congratulations on your achievement!
Exercise (b)
Place the pink edge in place of green, green in place of cyan and cyan in place of pink.
You can interact with the image by rotating the cube and layers. If you need to start over just click reset (∣◁).
To see a possible solution expose the hidden tab by clicking ∨.
B' (F E' F' U F E F' U') B
Click reset (∣◁) then play (▷) to see this sequence.
Please note that this is just one of the possible solutions. If you have discovered an alternative solution, congratulations on your achievement!
To represent the inverse of a sequence of moves called A, we use the notation A-1. That's why a conjugate is always indicated by A Z A-1, where A-1 is the reverse sequence of movements of A and Z is any other sequence.
App source: animcubejs.cubing.net/animcubejs.html