Later, I stopped playing for more than ten years, and the Rubik's Cube seems to have disappeared from every household. 1997 Summer, I went to my elder sister-in-law's house to be a guest, and I was surprised to find that there was a Rubik's cube in her house. Anyway, it's okay to live there, just twist the Rubik's cube every day. When you get older, your mind will be different and your understanding of the problem will be deeper than before. When you can see that four corners and four shoulders are different, you need to consider the four faces adjacent to this face. After doing this right, the surrounding areas are almost the same.
The natural way of thinking is to get it right layer by layer, so that the second layer is only four pieces short. Because I haven't learned any formulas, I still have a lot of brains. I created two formulas myself (each formula is twisted 8 times, and these two formulas are symmetrical and mirror images of each other), so I can put these four pieces well without affecting the first layer.
On the third floor, you will be in big trouble. Moving each block will destroy the upper two layers. After thinking for a long time, I suddenly got in touch with the group theory I had learned. Because I am a math major, I studied advanced algebra in my freshman year, and I used the textbook of Peking University. The last chapter introduces group theory. Engineering students study linear algebra instead of group theory. There are permutation groups in group theory. I think every twist of the Rubik's cube is actually a transformation of permutation groups.
For example, take a face as an example. The number is
1 2 3
4 5 6
7 8 9
Then twist it counterclockwise
3 6 9
2 5 8
1 4 7
And the position where 1 becomes 7 is recorded as1->; 7, in the same way 7->9, 9->; 3,3->1,these four blocks form a cycle.
The permutation group is (1 7 9 3), the other four blocks are (2 4 8 6), and the position of 5 is fixed and can be ignored.
Together, it is recorded as (1 7 9 3)(2 4 8 6).
This representation is concise and easy to operate. If you twist it counterclockwise, you can multiply it by this permutation group.
( 1 7 9 3)(2 4 8 6)x( 1 7 9 3)(2 4 8 6)=( 1 9)(7 3)(2 8)(4 6)
Give it another twist
(19) (73) (28) (46) x (1793) (2486) (you can try it yourself)
One more twist and you'll get it.
(1)(2)(3)(4)(6)(7)(8)(9) means to return to the original state.
In this way, we can twist all Rubik's cube numbers from 1 to 26 into corresponding permutation groups. Only the position is considered here, not the rotation (too complicated).
So I created the third formula, wrote out the permutation groups of the three formulas respectively, and then combined and multiplied them separately. I found that my formula is the most concise according to the combination of 32 1. As a result, only three shoulders are round, that is to say, as long as they are twisted in the order of 32 1, only these three blocks will rotate, and the positions of other blocks will not move (the angles may rotate). Therefore, Equation 4 is defined as the sequential combination of Equation 32 1.
So I used this method to look at three blocks to be rotated at a time, and twisted them many times to put all the blocks in their own positions, so that the third floor was just at the wrong angle.
Flipping the angle is also very troublesome. I know nothing but my own recipe. So I thought, since Formula 4 is the rotation of three squares, I twisted Formula 4 three times to make each square return to its original position and observe the change of their angles. Sure enough, several blocks were rotated, so I wrote down the result of angular rotation and used this result to adjust the block I wanted to rotate. After countless times, I finally got a six-sided Rubik's cube.
I used it for three days before and after, and because I had few formulas, it was particularly troublesome. It took me an hour or two to screw it up after I became proficient.
Now look at the Rubik's Cube. You can number each side of each cube from 1 ... 54, and each rotation is also a permutation group (including position and angle), so you can use a computer to calculate at will and find a concise formula.