Understanding and Solving the Rubik's Cube Without Algorithms
7. A bit about cycles
Before proceeding with the solution for the remaining edges, it's important to understand a concept called "cycles." Due to the cube's geometrical properties, it always returns to its initial state when a specific sequence of moves is continuously repeated, regardless of the complexity of the sequence. Let's explore some examples:
Example 1:
Repeating the R move four times.
Figure 7.1 demonstrates this cycle. You can click play (▷) to observe the sequence. Note the pause (.) between repetitions.
Example 2:
Now let's try a different sequence: M2 E2 S2. If you repeat this sequence, the cube will return to its original state. Figure 7.2 showcases this cycle. You can click play (▷) to see it in action. Once again, there is a pause (.) between repetitions. Interestingly, this sequence is equivalent to Rw2 R2 Dw2 D2 Bw2 B2. You can change the sequence in Figure 7.2 by clicking the arrow in the upper right corner (▷) and then click play (▷) to observe the equivalent moves.
Example 3:
Similarly, the sequence R2 L2 U2 D2 F2 B2, repeated twice, results in the cube returning to its initial state. However, in this case, the centers remain fixed while the outer layers move. You can change the sequence in Figure 7.2 by clicking the arrow (▷) in the upper right corner and then click play (▷) to observe the cycle.
Example 4:
Repeating the sequence R2 U2 D2 four times also brings the cube back to its initial condition. Figure 7.3 demonstrates this cycle. Click play (▷) to see it in action.
The key takeaway from these examples is that any sequence, when repeated enough times, will result in the cube returning to its original state. It may seem improbable at first, but even a simple sequence like R U, repeated 105 times, will restore the cube to its initial configuration. Figure 7.4 illustrates this. You can click play (▷) to observe the sequence.
This knowledge about cycles becomes highly useful when solving the final edges of the cube, as we will explore in the following steps.
App source: animcubejs.cubing.net/animcubejs.html