Conversion probability of third-order Rubik's cube
Wang, Grade 2065438 of Chengdu Yulin Middle School1Class 2.
I. Introduction:
Rubik's cube, also known as Rubik's cube. Is this Erno from Budapest Institute of Architecture in Hungary? 6? 1 was invented by Professor rubik in 1974. The Rubik's Cube swept the world soon after it was invented, and people found this small cube really mysterious. When one side of a large cube translates and rotates, the single color of its adjacent side is destroyed to form a new pattern cube, and then it is changed to form small squares with different colors on each side. According to experts' estimation, the total number of changes in the third-order Rubik's Cube is about 4.3? 6? 1 10 19。
Restriction conditions of second-order and third-order Rubik's cube transformation
Because when rotating the Rubik's cube, one layer will be broken in one rotation, that is, 2 1 color block, so many restrictions need to be considered. That is, the Rubik's cube will never appear.
First, the Rubik's Cube cannot rotate a prism block alone.
Suppose we set our favorite direction for the six central color blocks, and we set a coordinate system. The origin of this coordinate system is the center of the Rubik's cube. Coordinates have clear positive and negative directions. We can see that each edge color block of the Rubik's Cube has an edge corresponding to the horizontal, front and back, vertical X, Y, Z, Y and Z axes respectively, and each edge has four parallel edges. We mark all four sides with an arrow pointing in the positive direction. Now, if you have a Rubik's cube, you can do this. Let's now imagine that there is such a coordinate system and 12 arrows in space. Consider the rotation of any surface, (I don't consider the rotation of three middle planes here, (because, 1, this moves the coordinate system, 2, the rotation of the middle plane can be equivalent to the rotation of both sides. ), at this time we don't consider the color of the Rubik's cube and the Rubik's cube, and regard it as transparent. We only consider arrows. Every time any surface rotates 90 degrees, we will change the direction of the two arrows (from positive to negative). We only look at the results, without considering the process of rotation, and we don't distinguish where the arrows come from. Turning your face 90 degrees is an atomic operation of the Rubik's Cube, which can only change the direction of two arrows at the same time. So in the end, we can't leave other blocks unchanged and only flip 1 arrows, which means it is impossible to flip only one prism block.
Second, you can't flip a character block alone.
First, we consider the arrangement of four numbers 1234. 1234 becomes 4 123, which is the transformation that all numbers are shifted to the right by one bit. Think about the Rubik's cube. Every time you turn a face 90 degrees, four corners and four sides are transformed.
1234 changed to 4 123. I will call it (1234) for short, which is actually easy to remember, that is, 1 to 2, 2 to 3, 3 to 4, 4 to 1. If (1432) is 1 to.
(1234) consists of several transformations of "exchanging two numbers". The answer (1234) = (1 2) (1 3) (14), (12) means1to 2,21.
Specifically, we see that the process of 1234 change is as follows:
6? 1 ( 12) 2 134
6? 1 ( 13) 3 124
6? 1 ( 14) 4 123
Just transform (1234). So we know that (1234) is obtained by odd number of exchanges.
Any transformation can be obtained by several pairwise exchanges. Because what should I do with the target arrangement of 24 13? The inner truth involves the preliminary of group theory. This may be called cyclic group, I'm not sure, because I haven't read the book. 1234 all arranged in 4! =24, there are 24 transformations of 1234. They form a group, a bunch of elements.
First of all, you need to know how the direction of the role block is defined. Because the character block will be in eight different positions, but it has only three directions, how can I define a moving coordinate and mark these three directions accurately? First, let your eyes pass through the vertex of a character block and the center of the whole Rubik's cube, and you will see a Y. With your eyes as the axis, this character block can rotate and has three positions. As follows:
0 120 240
Try to turn one side to see the orientation of the color block in the new position. If you turn the right side of a Rubik's cube 90 degrees, you will find that the orientation of the character block closest to your eyes is turned 120 degrees. Stare at this color block, turn around again, and you will turn to the end. In order to still present a y, we can turn the bottom of the Rubik's cube up at this moment, and then we find that the character block turns back to 0, and so on. The key point is that if you observe the 90-degree rotation of any side and four character blocks, the sum of their rotation angles must be an integer multiple of 360 degrees, which is exactly 120+240+240+ 120. Because rotating a surface is the smallest atomic operation, no matter how many steps we take, the sum of the angle changes of all our character blocks is 360*n, so we can't just rotate one color block 120 or 240 degrees, while the other color blocks remain the same, so we prove why we can't rotate a character block alone.
Third, you can't change a pair of color blocks casually.
1. closure: A and B are elements in a group, so is A * B.
2. There is also the element E (actually 1 in analogy multiplication). a*e=e*a=a
3. every element a has a unique inverse A- 1, a * a-1= a-1* a = e.
4. Constraint law (a*b)*c=a*(b*c)
6? 1 First of all, 1234 is an arrangement, which corresponds to a transformation, that is, unchanged. I use (1) to indicate that the element E meets the second definition.
6? 1 is closed, which is obvious, because there are only 24 permutations and corresponding transformations, and it is impossible to run out.
6? There are 1 inverses, that is, each step is reversed and then reversed, and it must be among these 24 transformations.
6? 1 The associative law seems to be quite troublesome, but in fact, it is obvious, because (a*b)*c and a*(b*c) both represent A first, then B, and then C. So they form a group.
Why? Actually, I can't tell you what it's like to form a group now. I'm just saying that I can take advantage of some properties of groups. Know some characteristics of this structure. You can also use some perspectives and ideas of the analysis group to analyze this system. First, let's look at these 24 changes.
6? 1 (1), even number
6? 1 (12), (13), (14), (23), (24), (34), odd numbers.
6? 1 ( 123), ( 132), ( 124), ( 142), ( 134), ( 143), (234).
This is 15, leaving 9. If you don't understand what it means, just look ahead. When I say a (243), I mean 2 to 4, 4 to 3 and 3 to 2. He shifted 1 of 1234 to the left, and shifted the three digits of 234 to the left by one place to become 1342. Obviously,
6? 1 (1234), (1432), odd number
6? 1 (14) (23), (13) (24), (12) (34) even numbers.
There are four left. they are
6? 1 ( 13) ( 12) (24), ( 12) ( 14) ( 13), ( 14) (23) (65438)
We call the transformation formed by odd pairwise exchange odd transformation, and vice versa, in fact, it is to mark the parity of group elements. We see that two odd transformation operations get even transformation, while two even transformation operations never get odd transformation.
Such even-numbered transformations actually constitute a subgroup. In other words, their business is closed. they are
6? 1 (1), even number
6? 1 ( 123), ( 132), ( 124), ( 142), ( 134), ( 143), (234).
6? 1 (14) (23), (13) (24), (12) (34) even numbers.
These 12 elements constitute a subgroup. I seem to think something is wrong, hehe. But everything written in front is correct. I may use it later, and go back to why we can't just change a pair of color blocks.
Why? Because an atomic operation rotates a surface by 90 degrees to make four angles (1234) or (1432) are three interchangeable odd transformations, and four sides are also three interchangeable odd transformations, the overall effect of his transformation on all color blocks is even transformation. Therefore, for the arrangement of all color blocks, we can achieve even transformation, while switching only one pair of color blocks is odd transformation. Impossible to achieve. Therefore, we prove why we can't just switch a pair of color blocks.
(At this point, we have finally completed the complete proof of the total number of changes in the Rubik's Cube, which is necessary and sufficient:)
1. Calculate how many changes there are in the Rubik's Cube.
2. Based on the proof of the above limitations, the formula for calculating the total number of changes of the third-order Rubik's cube is obtained:
Fourth, summary.
The reason for the total number of changes in the third-order Rubik's cube is this: after the six central blocks are oriented, we can't turn the Rubik's cube, they just form a coordinate system. In this coordinate system, all 8 character blocks are arranged in 8! And each character block has three orientations, so it is 8! * 38, 12 prism blocks are all arranged, each with two orientations, that is, 12! *2 12, so multiplication is a numerator, and 3*2*2 on the denominator means to keep other color blocks still, and it is not allowed to change the direction of a character block, change the direction of a prism color block, and exchange the positions of a pair of prism color blocks or a pair of character blocks separately.