Subsequently, the Rubik's Cube became popular all over the world, and the greatest magic power was its amazing number of color combinations. When a Rubik's cube leaves the factory, each side has colors, with a total of six colors (generally: yellow, white, green, blue, red and orange). However, after these colors are disrupted, the number of combinations that can be formed is as high as 432.5 billion (note that it is really 2 billion words). 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 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, it will take him 65.438+050 billion years to get them (in contrast, our universe is currently less than 65.438+040 billion years old). 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 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. There are many players in the Rubik's Cube, so it is natural to compete with each other. From 198 1, Rubik's cube lovers began to hold worldwide Rubik's cube competitions, thus creating their own world records. This record is constantly being refreshed. As of the writing of this article, the fastest record of restoring the Rubik's Cube-as we mentioned at the beginning of this article-has reached an astonishing 6.77 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 of repeated recycling. At present, the world record for this average score is 6.54 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, how many turns does it take to ensure that no matter what color combination can be restored? 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", of course, we must first study the reduction method of the Rubik's cube. In the process of playing the Rubik's Cube, people have long 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, does not have 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 not 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 speaking, if someone can find such a minimum number of laps for each color combination, then the maximum number of laps 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 adopted, 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 combination of the Rubik's cube is ever-changing, it is actually produced by a series of basic operations (namely rotation), and those operations also have several very simple characteristics. For example, any operation has an opposite operation (for example, the operation opposite to clockwise rotation is counterclockwise rotation). For such operations, mathematicians have a very effective tool in their arsenal to deal with them. This tool is called group theory, which appeared more than 40 years ago when the Rubik's Cube appeared/kloc-0. It is said that German mathematician D Hilbert once said that the key to learning group theory is to choose a good example. Since the advent of the Rubik's Cube, mathematicians have written several books on group theory through it. Therefore, although the Rubik's Cube has not become a teaching tool of space geometry, it can be used as a "good example" for learning group theory to some extent.
For the study of Rubik's cube, group theory has a very important advantage, that is, it can make full use of the symmetry of Rubik's cube. When we mention the huge figure of 432.5 billion, there is actually an omission, that is, the symmetry of the Rubik's Cube as a cube is not considered. Therefore, many of the 432.5 billion color combinations are actually exactly the same, only from different angles (such as looking up with different faces or looking through a mirror). Therefore, the daunting figure of 432.5 billion is actually "pork with water". So, what percentage of this "pork" is "water"? Say it to scare everyone: it accounts for nearly 99%! In other words, mathematicians can reduce the color combination of the Rubik's cube by two orders of magnitude only through symmetry.
But reducing two orders of magnitude is not enough to find the "magic number", because it only reduces the aforementioned 10 million years to10 million years. For solving a mathematical problem, 1 00000 years is obviously too long, and we don't expect anyone to use1100 million computers to count the number of gods. Mathematicians are clever, but they are not necessarily rich in other ways. Maybe what they can really use is the machine on their desk. Therefore, in order to find the "number of gods", people need to find more ingenious ideas. Fortunately, the power of group theory tools is far from being used to analyze something as obvious as the symmetry of a cube. With its help, new ideas soon appeared. At present, only the number of gods in the third-order Rubik's cube has been found, but there are still many classic alien gods of order 4 and above. Seeing the third-order solution process, it is conceivable that finding out all the magic numbers in the Rubik's Cube is a great test of human wisdom.