Understanding and Solving the Rubik's Cube Without Algorithms
In the first "swap" movement, we explored the alternative of getting our corner out of the back and forth layer. If we choose this option, there are two possibilities for the setup: B or B2. Let's stick with the simpler one, B. By clicking forward (⧐) in Figure 10.9, you can observe the movement.
Notice that our corner is now positioned outside the back and forth layer. To complete the first "go, swap and back" movement, we simply perform R'. You can click forward (⧐) again, in Figure 10.9, to see the next step.
We are left with the final "go, swap and back" movement, but this time we want our corner to twist during the "go" move. To achieve this, we need to set it up in such a way that it remains in the swap layer (B) after the "go" movement. The only possibility for the setup is to do B'. Click forward (⧐) once more, in Figure 10.9, to see the setup movement.
Take note that the setup will force the corner to twist during the subsequent "go" move. Finally, we perform the last "go, swap" sequence: R B. If you double-click forward (⧐) in Figure 10.9, you can see the final result.
Observe that after the last "swap" movement, the facelet pointing upwards is the blue one. This means that this face will be in front of the cube immediately after the final "back" movement: R'. Click forward (⧐) one last time, in Figure 10.9, and observe the result.
We obtained a twist in the opposite direction compared to the previous one! This means that, in this case, after the first "go" move, the color that was initially on the right side of the cube ended up facing upwards before the last "back" move, and as a result, it is now on the front face. Again, there's no need to memorize this! Just anticipate what will happen to the corner after the movements.
One way to approach this example is as follows:
Question: Which color do I want to end up on the front face of the cube?
Answer: If, just after the first "go" movement, the color is behind the cube, then I'll set up the corner downwards. On the other hand, if the color is on the right side of the cube, then I'll set up the corner to the left.
Once again, there's no need to memorize this! Simply anticipate the moves, and you'll know what to do. Let's write down the complete sequence of movements that twisted the corner in the opposite direction:
R B R' B' R B R'
Now, let's compare the two sequences of moves:
1st case: R B' R' B R B' R'
2nd case: R B R' B' R B R'
Note that one case is the exact reverse of the other! To check this, take one of them and reverse the movements from right to left. For example, R becomes R' and R' becomes R. If we reverse the first case, we will have: R B' R' B R B' R' >> reversing >> R B R' B' R B R', which is precisely the second case!
Try doing the same with the second case, and you will see that reversing the movements gives you back the first case. This is exactly what we wanted - for the procedure of twisting one corner in a given direction to be the reverse of the procedure for twisting another corner in the opposite direction, according to the proposed strategy!
Remember, this is the chosen solution for this guide. You could use, for example, B2 for the setup in the back layer, which would result in different procedures, but ultimately they should cancel each other out. Similarly, it is not necessary to use the R layer for the back and forth movement; you can choose the layer that suits you best. The layer to preserve doesn't have to be the front layer either - you have the flexibility to choose!
The tip is to always keep in mind the "laws" of the cube and the general strategy presented in the summary of the corner twisting strategy. The rest is up to you!
If the corners are not on the same layer, you already know what to do, setup, twist the corners normally and then undo the initial setup i.e. use a conjugate.
Just for comparison, here are the algorithms that you should memorize for a 2-corner twist, considering only the case presented here:
R B' R' B R B' R' F R B R' B' R B R' F' to twist corners in one direction
R B R' B' R B R' F R B' R' B R B' R' F' to twist corners in the opposite direction
And all of this even without the slightest idea of what is happening or why each step of these sequences is being performed.
To conclude this chapter, here's a challenge: Using the general strategy of twisting corners, find a way to solve the cube below by flipping the edges that are not in the correct position. Tip: It is possible to flip one edge by applying two "go, swap, and back" movements, with a setup in between.
Feel free to interact with the image by rotating the cube and its layers. If you want to start over, simply click the reset button (∣◁).
To view a possible solution, click the downward arrow (∨) to reveal the hidden tab.
(x' z') (M' B2 M B' M' B M) F' (M' B' M B M' B2 M) F (z x)
(x' z') (M' B2 M B' M' B M) F' (M' B' M B M' B2 M) F (z x)
The parentheses serve only to highlight the conjugates. Note within the second and third parentheses the 2 "go, swap and back" movements with a setup in between. Also notice that these two parentheses have reverse movements regarding each other.
Note: The first and last parentheses are just rotations of the whole cube in space.
Click reset (∣◁) then play (▷) to see this sequence.
Remember, this is just one possible solution. If you have found another one, congratulations!
Congratulations on completing this chapter! By solving the Rubik's Cube without relying on memorized algorithms and using logical reasoning, you have demonstrated a great achievement.
Even if you haven't solved the cube in a long time and may have forgotten some specific move sequences, remember that understanding the fundamental principles and "laws" of the cube, along with general strategies, is key. By combining these principles with simple movements like the "go, swap and back," you can navigate through the solving process.
Keep practicing and applying your logical reasoning skills, and you'll continue to improve your ability to solve the Rubik's Cube efficiently. Well done!
App source: animcubejs.cubing.net/animcubejs.html