Understanding and Solving the Rubik's Cube Without Algorithms
10. Solving the 5 last corners
With a grasp of commutators and conjugates, the remaining task is to solve the last 5 corners. This can be achieved through two steps:
Placing the corners in their correct positions using commutators, disregarding the color orientations.
Adjusting the corners' orientations to ensure the colors are correctly aligned.
The first step becomes straightforward once you understand commutators and conjugates.Â
For the second step, conjugates come into play, but due to the limited range of possible moves, it can be challenging to find the right combination without knowing two additional "laws" of the cube and lacking a strategy. However, there's no need to memorize algorithms as long as you understand the cube's "laws" and have a strategy.
Now, let's delve into the second step in more detail. As we explored earlier, there are certain impossible configurations for the cube, such as a single edge flipped over. Here are two more of these "laws":
Key Knowledge: It is impossible to have only one twisted corner. The corners will always be twisted in pairs, with opposite directions of twisting.
See the examples below:
Figure 10.1: Impossible configuration! Only one corner twisted.
Figure 10.2: Impossible configuration! Two twisted corners, both twisted in the same direction.
Figure 10.3: Allowed configuration! Two twisted corners, twisted in opposite directions.
Figure 10.4: Allowed configuration! Three twisted corners, one twisted twice in one direction and the other two twisted once each in the opposite direction to the first.
Indeed, it is possible to find a combination of movements that twists only two corners in opposite directions. This can be achieved through the proper combination of conjugates.
Now, let's take a look at the strategy using an example to better understand the process.
App source: animcubejs.cubing.net/animcubejs.html