How many states can the Rubik's Cube reach? The answer is 4325200327489856000, about 400 billion.
Algorithm: 8 cube corners are arranged in 8 positions, 12 cube corners are arranged in 12 positions, and * * * has 8! * 12 ! Kindness Each cube has two orientations, each cube has three orientations, and there are * * * 3 8 * 2 12 kinds. So the number of states of the Rubik's Cube is 8! * 12 ! * 38 * 212 = 5 19024039293878272000 species, exceeding 51902 billion.
But in these 20 squares, 18 positions have been determined, and the other two positions have also been determined. Therefore, factor 2 should be removed. Of the eight angles, seven orientations were determined, and the eighth orientation was also determined; In the 12 cube, the orientation of 1 1 is determined, and the orientation of 12 is also determined. This removes the factor of 3 * 2, which is actually112 of the above number, that is, the total is 8! * 12 ! * 3^7 * 2^ 1 1/2=43252003274489856000 .
Consider the divisor 12 above from another angle. If we determine 6 colors, each color is drawn on 9 small squares on the surface of the Rubik's Cube 1. Then we take the Rubik's Cube apart and reassemble it, so not every Rubik's Cube can be restored to its original state. Specifically, there are 5 19024039293878272000 spellings, which can be divided into 12 categories, and each category has 432520032748856000. Any two states in the same class can be transformed into each other, but not between different classes.
My classmate can turn it out in two minutes.
2. What mathematical knowledge is there in the Rubik's Cube?
The mathematical knowledge in the Rubik's Cube mainly involves combinatorial mathematics, linear algebra and group theory. The closest relationship is group theory.
If you try to play the Rubik's Cube, you will find that no matter how you turn it, it is impossible to create a single 2-cycle on the Rubik's Cube. This needs to be explained from a mathematical point of view.
Simply put, the group generally refers to the * * * entity of transactions with similar nature. Group theory was founded by Jaroy, a German mathematician, while studying and solving higher algebraic equations. Group theory is developed in practice. Essentially, it is an abstract description of symmetry, which is the same feature of many things in the universe.
Therefore, after the establishment of group theory, it has been widely used in many sciences such as physics, chemistry and biology, and has made many extraordinary achievements. After the invention of the Rubik's Cube, its structure, rotation characteristics and even the cyclic transposition of individual squares are the best interpretations of many basic concepts and theorems of group theory.
Learning group theory through the Rubik's Cube will make the theory concrete, not abstract. On the other hand, under the guidance of group theory, the six-sided dating of the Rubik's Cube becomes regular and easy to master, rather than inscrutable and elusive. Even pure Rubik's Cube players who are not interested in mathematics have a certain understanding of the mathematics in the Rubik's Cube, which will improve their skills and proficiency in playing the Rubik's Cube and contribute to a deeper understanding of the Rubik's Cube.
The direct connection between the Rubik's Cube and mathematics is the total number of changes in the Rubik's Cube: the total number of changes in the third-order Rubik's Cube is 43,252,003,274,489,856,000. Or about 4.3x1019+09. So how is this figure calculated? In fact, it is to calculate the state of corner blocks separately, and then subtract the repeated state in the symmetrical structure.
Extended data:
Different kinds of Rubik's cube
1, traditional Rubik's cube
Clockwise/counterclockwise rotation, orientation, grouping, coordinates and combination ... Whether it is basic mathematics knowledge or advanced mathematics, the idea of turning over and restoring the Rubik's Cube can help children have a more intuitive understanding of these obscure knowledge points.
2. Mirror cube
For many math teachers, the mirror cube is the best teaching aid to learn the volume and surface area of three-dimensional graphics, and there is no one! It rotates in exactly the same way as the third-order Rubik's cube. The third-order Rubik's cube is restored according to the same color, while the mirror Rubik's cube needs to determine whether they are on the same side by judging which squares have the same "height" and then restore them. This process greatly improves children's perception of volume.
3. Triangle Rubik's Cube
Triangular Rubik's Cube is the easiest to restore. Although only two steps are needed, it can play a very important role in understanding the concepts of "triangle" and "space and surface". Especially in middle school solid geometry, there is a lot of knowledge of triangular pyramid, and the triangular cube can help children understand the abstract relationship between different planes.
3. What mathematical knowledge is there in the Rubik's Cube?
What math knowledge is there in the Rubik's Cube?
In July 2008, many of the best Rubik's Cube players from all over the world gathered in Dubis, central Czech Republic, to participate in the important Rubik's Cube event: Czech Open. In this competition, Dutch player E. Akkersdijk set an amazing record: it only takes 7.08 seconds to restore a Rubik's cube with completely disturbed colors. Coincidentally, in August this year, people also made important progress in the research on the mathematical problems behind the Rubik's Cube. In this article, we will introduce the Rubik's Cube and the mathematical problems behind it.
1. Toys that are popular all over the world
1974 In the spring, E. Rubik, a professor of architecture at Budapest Institute of Applied Arts, had an interesting idea. He wants to design a teaching tool to help students intuitively understand the various rotations of space geometry. After thinking, he decided to make a 3*3*3 cube with small squares, and each side can rotate at will.
Although this idea is good, it faces a thorny problem in practice, that is, how to make all sides of such a cube rotate at will? Rubik tried many ways, such as connecting small squares with magnets or rubber bands, but all failed. One afternoon that summer, he was enjoying the cool air by the Danube, and his eyes inadvertently fell on the pebbles by the river. Suddenly, a new idea flashed through his mind: treating the internal structure of a cube with a round surface similar to a cobblestone surface. The new idea succeeded, and rubik quickly finished his own design. And applied for a patent with the Hungarian Patent Office. This design is the familiar Rubik's Cube (also called Rubik's Cube) [Note 1].
Six years later, rubik's Rubik's Cube, led by a Hungarian businessman and amateur mathematician, entered the western European and American markets and became a fashionable toy all over the world at an alarming rate. In the next 25 years, the sales of Rubik's Cube exceeded 300 million. Among the players of Rubik's Cube, there are both babbling children and bosses of multinational companies. Although the Rubik's Cube didn't become a teaching tool of space geometry as rubik imagined, it became the best-selling toy in history.
The biggest magic power of the best-selling Rubik's Cube lies in its amazing number of color combinations. When a Rubik's cube leaves the factory, there are six colors on each side, but after these colors are disrupted, the number of combinations that can be formed is as high as 432.5 billion [Note 2]. If we make each of these combinations into a Rubik's cube, these Rubik's cubes will be arranged together. It can be discharged from the earth to the distant starry sky 250 light years away. In other words, if we put a lamp at one end of such a row of Rubik's cubes, it will take 250 years for the light to shine on the other end. If a diligent player wants to try all the combinations, even if he doesn't eat, drink or sleep, he will find ten different combinations every second. It takes150 billion years to get it (in contrast, our universe is less than14 billion years at present). Compared with such combined figures, the adjectives of "thousands", "hundreds of millions" and "billions" commonly used by advertisers on weekdays have become rare modesty. We can be modest.
4. Knowledge about Rubik's Cube
Rubik's cube, Rubik's cube is also called Rubik's cube, also called Rubik's cube.
It was invented by Professor Ern Rubik of Budapest Institute of Architecture in 1974. The Rubik's Cube is a six-sided cube made of elastic hard plastic.
The Rubik's Cube invented by China and Huarong Road, and the Independent Diamond invented by the French, are called the three wonders of the intellectual game world. The popularity of Rubik's Cube is a miracle in the field of intelligence games.
The core of the third-order Rubik's Cube is an axis, which consists of 26 small cubes. Including six central squares, it is fixed and only one side is colored.
Eight corner squares (3 colored faces) (corner blocks) can be rotated. There is also a 12 square (colored on both sides) (edge block) that can also be rotated.
When toys are sold, the arrangement of small cubes makes each side of the big cube have the same color. 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, all possible modes are about 4.3 * 10 19. The game is to restore the disturbed cube to a single color by rotating it as soon as possible.
The total number of changes in the Rubik's Cube is 43 252 003 274 489 856 000. Or approximately equal to 4.3x1019+09.
If you can turn the Rubik's Cube three times a second, it will take 454.2 billion years to turn all the changes in the Rubik's Cube, which is about 30 times the current estimated age of the universe. Center block (6 blocks): The center block is connected with the central shaft, but can rotate freely along the direction of the shaft.
The surface of the central block is square, and the structure is slightly cuboid, but the interior of the cuboid is not flat, and there is a cylinder connected with the central axis in the center. Seen from the side, there will be an arc-shaped groove on the inner side of the central block, and the groove on the central block and the side block can form a circle after combination.
When rotating, the edge block and corner block will slide along the groove. Prism (12): The surface of the prism is two squares, and the structure is like a cuboid protruding from one side of the cube, so that the prism can be embedded between two central blocks.
The radian of the cuboid surface is the same as that of the central block, and it can slide along it. There are missing corners inside the cube. After combination, the grooves on the center block and the edge block can form a circle.
When rotating, the edge block and corner block will slide along the groove. In addition, this missing corner is also used to fix the corner block.
Corner block (8 blocks): The surface of the corner block is three squares, and the structure is like a small cube protruding from one side of the cube. This structure allows corner blocks to be embedded in three edge blocks. Like the prism, the surface of the small cube has the same curvature, which makes the corner block rotate along the groove.
5. What mathematical knowledge is there in the Rubik's Cube?
What math knowledge is there in the Rubik's Cube? In July 2008, many of the best Rubik's Cube players from all over the world gathered in Dubis, central Czech Republic, to participate in an important activity in the Rubik's Cube world: Czech Open. In this competition, Dutch player E. Akkersdijk set an amazing record: it only took 7.08 seconds to restore a Rubik's cube with completely disturbed colors. Coincidentally, in August this year, people also made important progress in the research on the mathematical problems behind the Rubik's Cube. In this article, we will introduce the Rubik's Cube and the mathematical problems behind it. 1. Toys popular all over the world 1974 1974 In the spring of 1974, rubik, a professor of architecture at Budapest Institute of Applied Arts, had an interesting idea. He wants to design a teaching tool to help students intuitively understand the various rotations of space geometry. After thinking, he decided to make a 3*3*3 cube, which is composed of small squares, and its faces can be rotated at will. Such a cube can easily demonstrate various spatial rotations. Although this idea is good, it faces a thorny problem in practice, that is, how to make all the faces of such a cube rotate at will? Rubik tried many ways, such as connecting small squares with magnets or rubber bands, but all failed. One afternoon that summer, he was enjoying the cool air by the Danube, and his eyes inadvertently fell on the pebbles by the river. Suddenly, a new idea flashed through his mind: treating the internal structure of a cube with a round surface similar to a cobblestone surface. The new idea succeeded, and rubik quickly finished his own design. And applied for a patent with the Hungarian Patent Office. This design is the Rubik's Cube, also called Rubik's Cube. Six years later, rubik's Rubik's Cube, led by a Hungarian businessman and amateur mathematician, entered the western European and American markets, and became a toy popular all over the world at an alarming rate. In the following 25 years, the sales volume of Rubik's Cube has exceeded 300 million. Among the Rubik's Cube players, there are both babbling children and bosses of multinational companies. Although the Rubik's Cube didn't become a teaching tool of space geometry as rubik imagined, it became the best-selling toy in history. The biggest magic power of Rubik's Cube's best-selling lies in its amazing number of color combinations. When a Rubik's cube leaves the factory, each side has colors, and there are always six colors. However, after these colors are disrupted, the number of combinations that can be formed is as high as 432.5 billion [Note 2]. If we make each of these combinations into a Rubik's cube, then these Rubik's cubes can be arranged together from the earth to the distant starry sky 250 light years away. In other words, if we put a lamp at one end of such a row of Rubik's cubes, it will take 250 years for that beam of light to reach the other end. If any diligent player wants to try all the combinations, even if he doesn't eat, drink or sleep, it will take 65.438+05 billion years to get them (in contrast, our universe is currently less than 65.438+04 billion years old). Compared with the number of this combination, the adjectives commonly used by advertisers, such as "thousands", "hundreds of millions" and "billions", which bluff customers on weekdays, have become rare modesty. We can safely say that even if a person starts playing the Rubik's Cube from BIGBANG, there is almost no hope to restore a Rubik's Cube whose color is disturbed.
6. Mathematical formulas and skills for playing Rubik's Cube.
The third-order Rubik's Cube has 26 pieces, which are divided into three parts. Six central blocks, this is not moving. Eight corners and twelve sides.
There are generally three commonly used methods: layered method, angle priority method and edge priority method. However, I think the edge first method is relatively simple and practical.
Trimming is to put a cross on each face. When you spell the cross, you don't rely on the surface, but on the layer.
Go back to the first floor, that is, put a cross on the first side. It's simple, but you must spell the cross correctly.
That is, the colors of the four sides of the cross must be consistent with the colors of the front, back, left and right center blocks.
That's right. I forgot to tell you the direction. Call up, the right on the right hand side is right, the left on the left hand side is left, and so on, which one?
Can be used in future formulas.
After the first side is ready. Now it's easy to restore the second floor. The formula is also front+bottom+front-front+bottom-front-
One is very simple. After restoring this, there will be four inverted T's on all sides.
Now it's time to turn the Rubik's cube upside down, that is, turn the lower level into the upper level. At this time, if you are lucky, the bottom floor is ready.
If not. Now we really need to use formulas.
Cross formula
Formula 1 right-up-front-up+front+right+
Formula 2 Right-Front-Front+Up+Right+
When these two formulas are used. Spell two opposite sides with 1 point, and then you need 2. Think of the upper layer of the Rubik's cube as a clock.
Think of its two upturned edges as an hour hand and a minute hand, which should be placed on the children's table at six o'clock sharp. Thus, Equation 2 can be used.
When using 2, two adjacent edges will be spelled out. When using the formula 1 again, the Rubik's Cube will be placed at nine o'clock sharp.
The cross spelled out at this time is not necessarily in the right position. It's possible for one person, and it's possible for two people. Maybe none of them are right, because the upper level can
Free rotation. At this time, the formula will be changed. When using the formula, only one edge is crossed. That is, when the other three are wrong.
Cross formula
Formula 1 right-up-right+up-right-up 2
Formula 2 Left+Up+Left-Up+Left+Up 2
Using the formula 1 will move these three wrong edges clockwise by one position. Equation 2 is the opposite.
When it's done. The hexagonal cross has been put together, and now it is necessary to restore the angle.
Rotation angle formula
Formula 1 Up+Right+Up-Left-Up+Right-Up-Left+
Formula 2 Up-Left-Up+Right+Up-Left+Up+Right-
Usage: The formula 1 is used to move the left front, left rear and right rear one position counterclockwise, but the left rear angle mainly turns to the left front.
Equation 2 is to move these three angles clockwise. But mainly turn right back to the right front.
When using 1, the right rear corner will move. If this angle has been restored. Just turn the one on the right. Using 2 will upset the left back corner.
The processing method is the same as that of 1.
When the Pentagon recovers. At this time, the remaining three corners can be turned around at once, but it is easier said than done. For beginners, it's still
Restore another corner. There will be several situations at this time. First, two adjacent angles are not in the right position. Put those two confused corners in the left front corner and the left back corner.
These two positions, then you will find that the two corners will have two colors on the same side. You should face the same color up, and you will find these colors.
Consistent with the color on the left. That is, it can be flipped directly to the left.
Use the formula 1 first. After+. Then turn the whole Rubik's Cube 90 degrees clockwise. It is a whole. Not one side. Then use equation 2.
If you complete the above steps. Congratulations. It's over.
The second situation. There are two opposite corners left. At this time, as long as the two corners are turned to adjacent positions, it will become the first situation.
Of course, there will be another situation. It's two opposite corners of the Rubik's cube, not one side, but the whole Rubik's cube. The treatment method is the same as above.
7. What are the mathematical laws in the Rubik's Cube?
In July 2008, many of the best Rubik's Cube players from all over the world gathered in Dubis, central Czech Republic, to participate in the important Rubik's Cube event: Czech Open. In this competition, Dutch player E. Akkersdijk set an amazing record: it only took 7.08 seconds to restore a Rubik's cube with completely disturbed colors. Coincidentally, in August this year, people also made important progress in the research on the mathematical problems behind the Rubik's Cube. In this article, we will introduce the Rubik's Cube and the mathematical problems behind it. 1. Toys popular all over the world 1974 1974 In the spring of 1974, rubik, a professor of architecture at Budapest Institute of Applied Arts, had an interesting idea. He wants to design a teaching tool to help students intuitively understand the various rotations of space geometry. After thinking, he decided to make a 3*3*3 cube, which is composed of small squares, and its faces can be rotated at will. Such a cube can easily demonstrate various spatial rotations. Although this idea is good, it faces a thorny problem in practice, that is, how to make all the faces of such a cube rotate at will? Rubik tried many ways, such as connecting small squares with magnets or rubber bands, but all failed. One afternoon that summer, he was enjoying the cool air by the Danube, and his eyes inadvertently fell on the pebbles by the river. Suddenly, a new idea flashed through his mind: treating the internal structure of a cube with a round surface similar to a cobblestone surface. The new idea succeeded, and rubik quickly finished his own design. And applied for a patent with the Hungarian Patent Office. This design is the Rubik's Cube, also called Rubik's Cube. Six years later, rubik's Rubik's Cube, led by a Hungarian businessman and amateur mathematician, entered the western European and American markets, and became a toy popular all over the world at an alarming rate. In the following 25 years, the sales volume of Rubik's Cube has exceeded 300 million. Among the Rubik's Cube players, there are both babbling children and bosses of multinational companies. Although the Rubik's Cube didn't become a teaching tool of space geometry as rubik imagined, it became the best-selling toy in history. The biggest magic power of Rubik's Cube's best-selling lies in its amazing number of color combinations. When a Rubik's cube leaves the factory, each side has colors, and there are always six colors. However, after these colors are disrupted, the number of combinations that can be formed is as high as 432.5 billion [Note 2]. If we make each of these combinations into a Rubik's cube, then these Rubik's cubes can be arranged together from the earth to the distant starry sky 250 light years away. In other words, if we put a lamp at one end of such a row of Rubik's cubes, it will take 250 years for that beam of light to reach the other end. If any diligent player wants to try all the combinations, even if he doesn't eat, drink or sleep, it will take 65.438+05 billion years to get them (in contrast, our universe is currently less than 65.438+04 billion years old). Compared with the number of this combination, the adjectives commonly used by advertisers, such as "tens of thousands", "hundreds of millions" and "billions", which bluff customers on weekdays, have become rare modesty. We can safely say that if we don't master the tricks, even if we start playing the Rubik's Cube from BIGBANG, there is almost no hope to restore a Rubik's Cube with its colors disrupted. 2. There are many players in Rubik's Cube and Shenshu Rubik's Cube, so it is inevitable to compete with each other. From 198 1, Rubik's cube lovers began to hold a worldwide Rubik's cube competition, thus starting to create their own world records. This record is constantly being refreshed. Until the time of writing this article, the fastest record of restoring the Rubik's Cube-as we mentioned at the beginning of this article-has reached an astonishing 7.08 seconds. Of course, a single recovery record is accidental. In order to reduce this contingency, since 2003, the champion of Rubik's Cube competition has been decided by the average score restored many times [Note 3]. At present, the world record for this average score is 1 1.28 seconds. The appearance of these records shows that although the Rubik's Cube has astronomical color combinations, as long as you master the tricks, there are not many rotations required to restore any combination. So, at least how many rotations are needed to ensure that any color combination can be restored [Note 4]? This problem has aroused the interest of many people, especially mathematicians. The minimum number of rotations needed to restore any combination is dubbed by mathematicians as the "magic number", and the Rubik's cube, the darling of the toy industry, has invaded the academic world at one fell swoop because of this "magic number". To study the "magic number", we must first study the reduction method of the Rubik's cube. In the process of playing the Rubik's Cube, it has long been known that it is easy to restore any given color combination, which has been proved by the excellent records of countless players. The reduction method used by Rubik's cube players, although easy to master by human brain, is not the method with the least number of rotations, so it is not helpful to find the "magic number". Finding the method with the least number of rotations is a difficult mathematical problem. Of course, this problem is very difficult for mathematicians. As early as the mid-1990s, people had a practical algorithm, and the minimum number of rotations to restore a given color combination could be found in about fifteen minutes on average. Theoretically, if one can find such a minimum number of rotations for each color combination, then the largest one of these rotations is undoubtedly the "magic number". But unfortunately, the huge number of 432.5 billion has become a stumbling block for people to peek at the number of gods. If the algorithm mentioned above is used, even if 1 100 million machines are used at the same time, it will take1100 million years to complete. It seems that brute force doesn't work, so mathematicians turn to their old job: mathematics. From a mathematical point of view, although the color combinations of the Rubik's cube are ever-changing, they are actually composed of a series of basic operations (that is, rotation)