After Li Xuan finished his first live broadcast, he did not pay attention to online news. It was a pure surprise that he participated in this Sudoku online competition. Of course, there are benefits, and he has a new understanding of Sudoku.

The weekend is still long, and Li Xuan suddenly became very interested in the related issues in Sudoku and was preparing to study Sudoku again.

One day on Sunday, he took out a pile of draft papers and studied them slowly in his room. First, he wrote down the Sudoku problems he encountered in the last round, trying to find other ideas.

Looking at this Sudoku problem, Li Xuan thought for a while, but still couldn't find a simpler solution. He was so tired that he thought of three mathematical problems in Sudoku.

The first question is how many types of standard Sudoku are there?

The second question is how many known numbers are required to ensure that there is a unique solution to Sudoku?

The last question is whether there is a simple way to judge the unique solution to Sudoku?

The first question is how many kinds of sudoku are there? This is a question of arrangement and combination in mathematics. Thinking of this interesting question, Li Xuan was very excited and picked up the pen and calculated it silently.

All things in Pythagoras, ancient Greece, were counted, and everything could be measured by numbers. In fact, the most obvious mathematical problems in the universe are graphs. The key is whether these problems can be extracted from the discovery and derivation of all things.

Sudoku was originally from the magic square studied by Euler. There were many mathematical problems in it. Li Xuan discovered these three problems and he thought that the most important of the Sudoku problem. Of course, his research on Sudoku was entirely due to curiosity and interest. There was no benefit or reward for solving the problem.

"How many standard Sudoku are there? It feels like it will be astronomical."

Li Xuan smiled and felt that this number would be very large. After all, there are eighty-nine spaces to fill in the number.

There is such a story that can prove Li Xuan's intuition. The ancient king fell in love with chess and decided to reward the game inventor and satisfy the wish of the game inventor.

The inventor's wish is to ask the king to reward him with rice, the first chessboard 1 grain of rice, the second 2 grains of rice, the third 4 grains of rice... and so on, the rice that fills the entire chessboard.

The king was very happy at first, but he felt very little, but he was slapped in the face, and all the rice in the world was not enough.

The reason is similar. The number of Sudoku is so large that it is unimaginable that humans can't finish it for hundreds of years.

The basis of computers is binary 0 and 1, but it forms an extremely complex virtual world. From this perspective, the complex universe is actually composed of basic atomic arrangements.

The first question seems simple, and elementary school knowledge is enough, but Li Xuan had considered it briefly before, and he couldn't figure it out even if he wanted to break it. The difficulty was beyond his imagination.

The more difficult the problem is, the more interesting it is. It is too simple, but Li Xuan has no urge to study it.

Li Xuan considered that the number of Sudoku is limited, and this problem can be calculated by programming using exhaustive methods.

Li Xuan estimated that it would take a long time to use a computer, and he didn't want to rely on computers. The point where humans are more powerful than computers lies in thinking. Mathematicians always have fantasy ideas to simply prove complex problems. This kind of logical reasoning is not available to computers.

Things in the world are all the best, and the first question is helpless. Li Xuan decided to simplify the problem first, ignore the 3x3 ninth order Sudoku, and calculate the situation of 2x2 fourth order Sudoku first.

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Fourth-order Sudoku: Fill in 1 to 4 for each space, and the numbers in each row, column, and every 2x2 palace are not repeated. How many possibilities are there?

This question is much simpler.

Through arrangement and combination, Li Xuan quickly calculated the number of fourth-order Sudoku, which was only a few hundred. According to the calculation idea of ​​fourth-order Sudoku, he began to move towards ninth-order Sudoku.

The idea is very clear, but the situation is complicated and many aspects need to be considered.

After calculating for a long time and using computer calculations, Li Xuan calculated the answer to the first question: There are 9!x72!x2^7x27704267971=6670903752021072936960 types of Sudoku.

Looking at this number, Li Xuan was stunned for a long time, thinking about how many thousand years it would take for humans to complete these sudoku...

There are only a few hundred types of sudoku in 2x2, and there are trillions of types of sudoku in 3x3... It's really annoying.

As he thought before, this is an unimaginable astronomical quantity.

Li Xuan shook his head, looked carefully at the calculation draft, and confirmed that there were no errors in the calculation process. If the results of the calculation are ignored, the results are considered to be the same Sudoku, then there are 5472730538 Sudoku.

This conclusion has no purpose, but it can definitely be published as a paper. Of course, Li Xuan felt that the mathematician must have considered this problem and did not plan to write any paper.

In fact, this conclusion was calculated and published by mathematicians abroad in 2005, but no one in China has calculated it independently, or published articles.

There are many reasons why Li Xuan doesn’t want to write a paper. The main reason is that he feels it’s troublesome. He needs to polish his writing style after publishing a paper, but there is not much royalty. It is impossible to say that such a paper can get any honor. What is not a waste of time to write a paper?

Thinking of this, I simply opened a blog, simply recorded its calculation process on the blog, and provided it to domestic Sudoku enthusiasts for free.

Because of English, there are communities abroad for science enthusiasts to communicate with. China lacks such scientific communities. Ordinary people who want to communicate with others and learn science cannot find a suitable place.

Li Xuan published these articles on his blog, and he didn't expect many people to read them. There were not many people on the blog, and there were not many people playing them lately.

While the enthusiasm was still there, I began to briefly write the calculation process on my blog, but when Li Xuan was editing the text, he thought of a new question.

There are 6670903752021072936960 possibilities for nx order sudoku, so how many possibilities are there for 4x4 order sudoku... How many types are there for nxn order sudoku? Is there any general formula?

Don't think about it and know that the explosion is difficult.

After thinking a little in his mind, Li Xuan's scalp became numb, "Oh my god, I feel like I'm crazy."

There are only a few hundred types of 2x2 order Sudoku, and there are only a few trillion of 3x3 order Sudoku. According to this explosion-up trend, please imagine how many megazines of the number of 4x4 order Sudoku?

How many types of Sudoku's quantity formulas do you think are there in the nxn order?

Come on, try to figure it out.

It's hard to describe...

Li Xuan took the written test and calculated the 4x4-order Sudoku. He felt that his IQ was not enough, and he once suspected that there was something wrong with his head.

"Damn, I might be a fool! Purebred!" Li Xuan looked very painful.

He couldn't think of this problem for a while, and the scientific intuition given to him by the system also judged that it was difficult for him to calculate. It was not that he did not understand the calculation method, but that the calculation was too large. He simply wrote the problem under the blog and threw this mathematical problem to the public to see if anyone had calculated it.

How many kinds of nxn order sudoku?

If mathematicians had enthusiasm for studying Riemann's conjecture world problems, this problem should have been found by mathematicians... Now, no mathematician would calculate such a painful problem.

The first question of the number of Sudoku is a headache for Li Xuan, let alone the only solution to the second Sudoku problem.

The first problem was mostly solved, but it was not completely solved. Li Xuan decided not to think about it first and look at the second problem first. How many known numbers should there be at least in order of 3x3 Sudoku to ensure that there is a unique solution to Sudoku?

Li Xuan was a little confused when he thought about this question, so he finally considered a simpler 2x2 order Sudoku first. 16 spaces look simpler than 81 spaces.

I don’t want to know, but it’s a crash when I think about it, because the spaces appearing in Sudoku are random, and it’s difficult to find rules that can be used for logical reasoning.

He used all his brain power to fight with mathematics and finally successfully killed him... He could only confirm that he needed at least 17 known numbers to ensure the unique solution of the sudoku, and he could not think of the proof process anyway.

"Damn, I'm paramecium!"

Li Xuan covered his head and looked at the second question, making sure there was something wrong with his head. He felt that there was no hope of proof of the second question.

Apart from using the exhaustive method, he didn't expect other proof methods. If he used the exhaustive method to calculate this problem, his home computer would probably crash. This proof was too ugly, and Li Xuan didn't want to prove it like this psychologically.

Li Xuan is not very familiar with the current new theories of mathematics at present, and I don’t know if there are any theories to solve this problem. Without these theories, would he try to refine the Sudoku law and create new mathematics?

After all, mathematics is created from a small problem, such as solving linear system of equations from the source of determinants.

Li Xuan thought for a moment.

"I'm so stupid, really." Li Xuan suddenly giggled and raised his eyes without any spirit. "I can only get full marks in a simple test paper like high school mathematics. I can answer easy questions like CMO. I can calculate the six-digit multiplication and division in one second. I don't know the method of proof of the only solution of Sudoku, nor can I find the general formula for the number of Sudoku, nor do I know how the Riemann conjecture proves..."

Then Li Xuan grabbed his head, and there was no smile on the corpse's face and he couldn't say anything.

Looking at such a simple question, he thought about it for a long time without any idea and was crushed into scum by mathematics. On this day, he remembered the fear of being dominated by mathematics. It is no exaggeration to say that it is paramecium on the road of science.

The real difficulty of mathematics is not to do the questions repeatedly, but to do the questions using methods that have never appeared before.

System: [This system also thinks that humans in the 21st century are quite stupid.]

Li Xuan was a little helpless: "System..."

System: [This problem is too complicated for you. Xuanmei, your brain capacity is not enough. If you want to calculate this problem, please borrow the national supercomputer Tianhe 1. The number of calculations per second is 1,000 megabytes. It is no problem to calculate it.]

Li Xuan lost his temper: "Can I use supercomputers? If you ask the supercomputers in the country to calculate the Sudoku problem, you may not be unintelligent."

Unwilling to give up after not proving it, Li Xuan scratched his ears and looked at the Sudoku on the paper, thinking about what method to prove it, waiting for the moment of enlightenment.

But there is no idea. Li Xuan has a strong sense of numbers and has intuition to know what the conclusion is, but he doesn’t know the proof process. Mathematics is rigorous, and it is not possible to have only conclusions without proof.

In fact, many problems cannot be solved. It is an ideal state to solve any problems. When a person is the stupidest, he thinks he is smart. The smarter he is, the more stupid he feels.
Chapter completed!