"Draw a closed curve arbitrarily, connecting end to end and not crossing itself. On this curve, prove that four points can be found so that they can be connected to form a square..."
"Creating a square within a closed curve, I think you've heard of it?"
Wang Hao thought a little doubtfully. He felt like he had seen this problem before, but he couldn't remember it.
Mainly because his initial research direction was partial differential equations, and now he is also studying analytic number theory, which does not involve too many geometric issues.
He just felt like he had seen it before and thought it was a classic question in some course.
"Is this a problem..."
"this……"
Wang Hao thought about it carefully and found that he was stumped.
There are really too many sub-disciplines of mathematics. The knowledge involved in different subjects is basically repetitive to a certain extent, but there will definitely be differences in advanced content.
Geometry has a lot to do with function theory, and it also has a certain connection with differential equations. However, for pure geometry proof questions like this, the connection is relatively small.
Wang Hao thought carefully for a few minutes, but could not think of a breakthrough in the proof. He knew that he did not have in-depth research in related fields, and there might even be some basic problems. If he wanted to solve this problem in a short time, it would be almost impossible to just rely on thinking. impossible.
In this case...
Wang Hao put down the booklet in his hand and asked, "Ding Zhiqiang, are you stuck on which step of this proof, or are there some problems that you can't solve?"
He seemed to be checking the students' problem-solving progress normally.
Ding Zhiqiang secretly exclaimed, "Sure enough," because he knew that Wang Hao would definitely ask this question. When he found that the problem could not be solved, he might suspect that the student was deliberately making things difficult for him.
That's why he worked hard all night to study the relevant content.
Now comes the use.
"Then I'll tell you."
Ding Zhiqiang said confidently, "To address this problem, I have studied the proof that closed curves have built-in rectangles and equilateral triangles."
"It's the method of using hypothetical line segments. Let me talk about the rectangle first. First, draw a rectangle..."
Ding Zhiqiang began to explain while drawing.
He spent one night carefully studying two proof papers. The proof process was not complicated, but involved some advanced knowledge.
It takes him more time to look up this knowledge and understand it.
Now he just talks about what he understands in sequence, and there is no difficulty at all. As for the formulas and theorems involved, he just says them directly and then makes transformations. There is no need to explain them in too much detail.
This little trick won't work in front of Wang Hao.
The same goes for Luo Dayong.
When he discovered that Ding Zhiqiang was explaining to Wang Hao, Luo Dayong came over curiously to see what the specific topic was, and then he discovered that it was the proof of a square within a closed curve.
Luo Dayong knows this problem and knows that no mathematician has been able to prove it so far.
Seeing Wang Hao's confident look, he did not ask directly, but listened patiently to Ding Zhiqiang's explanation.
Later, Luo Dayong was a little surprised, "This student is amazing. He knows so much advanced knowledge before he is a graduate student."
"Although I just stated the formulas and theorems, my understanding is probably not very deep. But it is still amazing."
Wang Hao also felt the same way.
He didn't know yet that no mathematician had solved the question Ding Zhiqiang asked, but when he discovered that Ding Zhiqiang said something profound, he suddenly felt that he had a really good vision.
This Ding Zhiqiang may be a genius!
Ding Zhiqiang is only a junior in the Department of Physics and a student of Xihai University. Of course, he is not looking down on Xihai University students, but the admission score of Xihai University is low. The admissions scores of other provinces are just above the key points, and the admission scores are comparable to those of Shuimu University. , the first time I went to college was only 100 points behind. On average, even in mathematics, the difference was 20 or 30 points.
And some students who are talented in mathematics, even if they just memorize a formula and do the questions at will, without studying hard or solving the questions, they will basically have no problem getting close to full marks in mathematics.
Of course.
The college entrance examination alone cannot directly evaluate a student. Some students who did not study well in high school can still stand out by working hard in college.
Ding Zhiqiang seems to be this kind of student.
At this time, Ding Zhiqiang seemed to be in a state of excitement. The more he talked, the more excited he became. What made him excited was that he realized that he was giving a lecture to Wang Hao.
Although Wang Hao was nominally inspecting his progress in solving the problem, in any case, he was also explaining the problem to Wang Hao!
Who can do this?
Who else in the world would be better than me!
Ding Zhiqiang felt a surge of domineering power in his heart, but soon felt a little nervous because many people came to listen to his explanation.
This incident spread in the corridor.
The female teacher surnamed Deng in the office next door is a more gossipy person than Zhu Ping. She saw a student giving a lecture to Wang Hao at the door of the office, and it seemed that the lecture was very exciting, so she kept standing at the door and watching. Invite others to watch together.
When he found that there were several teachers watching behind him, Ding Zhiqiang felt very nervous. Fortunately, he had reached the final stage of his lecture.
After completing the last step of explanation, Ding Zhiqiang stopped and looked at Wang Hao with some tentative eyes. He didn't know how Wang Hao would react, but what was certain was that Wang Hao definitely didn't know the proof of forming a square within a curve.
He has studied it carefully. This is a problem that mathematicians have not yet solved.
Wang Hao was silent for a moment and asked, "In other words, you have already studied the methods of proving rectangles and equilateral triangles, and you want to use this to study how to prove squares yourself, right?"
This is Wang Hao's understanding.
He didn't know that a square built into a closed curve was a problem that mathematicians had not yet solved, but by listening to Ding Zhiqiang's explanation, he knew that Ding Zhiqiang understood it based on the answer, rather than proving it himself.
So he thought about it carefully and thought that Ding Zhiqiang wanted to study the square proof method himself after seeing the two proof methods.
This is a great way to learn.
I read some proofs of the same type and then tried to complete a proof myself.
Wang Hao suddenly admired Ding Zhiqiang even more, but he still didn't know how to solve the square problem.
Square, because of the particularity of the figure, is obviously more difficult to prove than rectangle and regular triangle.
Wang Hao felt that it was okay to just say that he couldn't think of it at the moment, and it was a little embarrassing. But this was the first time for Ding Zhiqiang to ask himself a question. For such a good student, if he couldn't answer the question for the first time, it might make the student very disappointed. ?
Don't let the students down!
Wang Hao was very concerned about his image in the minds of students. He simply looked at the task system and then created a task.
【Task 2】
[Research project name: Proof of a square built into a closed curve (Difficulty: D). 】
[Inspiration value: 0. 】
"D level difficulty?" Wang Hao glanced at the difficulty level of the task and suddenly realized the problem.
He originally thought it might be an F-level difficulty question. The lowest level of difficulty in research and development is F-level, which is the kind of difficult question in textbooks. He can deduce it if he thinks about it carefully.
If it is D-level difficulty, it means that research and development issues are already involved.
In other words, no one has been able to prove this problem before. It is a pioneering question, and it can reach D level even if it is not difficult.
Many very small problems and conjectures in mathematics are at this level.
The reason why those mathematical problems and conjectures have not been proven is not because of how difficult it is, but because there are too many of them. Those who are capable do not bother to study them, and those who are not capable naturally go without saying.
The problem of a square built into a closed curve is one of them.
Top mathematicians will not spend a lot of time on this small proof, because even if it is proved, it is of little significance, and it is not easy for someone worse to prove it.
Wang Hao did not think that Ding Zhiqiang was deliberately making things difficult for him. He felt that the other party might be really researching and wanted to rely on his own ability to solve a problem that no one else had solved.
This spirit is worthy of recognition.
Wang Hao asked, "The proof of a square is much more difficult than that of a triangle and a rectangle. No one has proven it before, right?" He said and looked at Luo Dayong.
Luo Dayong nodded.
Ding Zhiqiang showed a surprised expression. In fact, he was so panicked that he couldn't say a word and didn't know how to react.
Wang Haodao, "However, just because no one has proven it, it doesn't mean that we can't prove it. The research we do is all about cutting-edge exploration."
"In this way, let's analyze it together."
"First, let's take a look at the proofs of rectangles and triangles. This is of great reference value for the proof of squares."
"Let's review it together..."
After Wang Hao finished speaking, he began to repeat Ding Zhiqiang's explanation. Because of his in-depth understanding of basic knowledge, his explanation was much more detailed. Even Luo Dayong also listened along.
When he listened to it for the second time, he found that Wang Hao's explanation could make people understand it more deeply.
Soon others came over.
Teacher Deng, who was standing at the door just now, had slowly approached, as if he wanted to know what they were talking about, but Zhang Zhiqiang simply walked over.
Zhang Zhiqiang did go to the computer lab, checked it carefully on the computer, and found that the built-in square of a closed curve is a problem that no one has proven.
So that’s it!
No one has been able to prove it before. It is normal that he cannot solve it. But why does this student ask such a question?
Zhang Zhiqiang felt strange, and he came back with renewed energy.
I am very confident when I come back this time, because no one else has proven it, and it makes sense for him not to, and he is only a professor in the computer department.
When Wang Hao started to explain, there were several people standing next to him and two teachers standing behind. It was difficult to squeeze in even if he wanted to get closer to listen.
Wang Hao simply stood up and walked to the small blackboard in the corner, "Since everyone wants to listen, I will talk here."
After he finished speaking, he began to explain.
Wang Hao was explaining patiently, and the news spread quickly. Several people on the floor came over. They were attracted by "Wang Hao's lectures" because Wang Hao is a very high-level mathematics professor. There are not many opportunities to give lectures, so even if you just come to join in the fun, come and have a look.
Many people have a herd mentality.
When they find a group of people standing together, they will come over and listen for a few words to understand what is going on. After they come over, they will start to follow the sound.
naturally...
Wang Hao found that knowledge and inspiration were pouring into his mind continuously. The more he talked, the more energetic he became and he completely entered the teaching state.
In fact, he just repeated what Ding Zhiqiang said just now, but he understood the content more deeply. When he encountered difficulties, he would explain them repeatedly to make everyone understand better.
[Task 2, inspiration value +3. 】
[Task 2, inspiration value +7. 】
[Task 2, inspiration value +4...]
When Wang Hao talked about the proof of a closed curve with a built-in triangle, the office was already overcrowded. If you look closely, there were about twenty people. There were also many people standing at the door listening. I don't know whether they were listening or chatting.
Because the proof process is not that complicated, it took Wang Hao half an hour to finish.
D-level difficulty research was too easy for him. In the process of teaching and watching, his inspiration value quickly soared to over one hundred points.
Wang Hao stopped talking and started to explain the next step, "Newcomers may not know that the reason why I tell you this knowledge point is because classmate Ding Zhiqiang..."
He pulled Ding Zhiqiang to his side and gave everyone a brief introduction, "Ding Zhiqiang asked me a question about a square built into a closed curve, so I wanted to analyze it with him and solve this problem."
"Just review the proofs of rectangles and triangles."
"Then the next step, I think after we have the foundation, we can continue to prove the square."
After Wang Hao finished speaking, he pushed Ding Zhiqiang back.
Ding Zhiqiang was completely confused.
During the process explained by Wang Hao just now, he found that he had a deeper understanding of the proofs of triangles and rectangles, but no mathematician had proven the square in the next step.
Now Wang Hao directly talks about the next step of proof...
this……
The development of things seems to be different from what I expected?
Ding Zhiqiang looked at Wang Hao blankly, but because of Wang Hao's introduction, he was pushed to the middle of the first row by the teachers behind him, as if he had become the leader of everyone attending the class.
Others also realized that what Wang Hao was about to talk about, and that there might be groundbreaking research, responded quickly.
Zhang Zhiqiang turned on his phone and started recording video.
Several teachers were also aware of it, and several picked up their mobile phones and started filming.
Wang Hao didn't care, it was just a small research on D-level difficulty, so he said seriously, "I thought about it carefully. In fact, the method is the same, but the specificity of the square determines the proof process will be more complicated. Some."
"We need to shape two points when we come up..."
Wang Hao made a note in the picture, "A closed curve is uncertain, but because it is a closed curve, no matter how it expands outward, there will definitely be a time when it returns."
"We can assume that when expanded outward, the farthest point is S1 and the farthest point on the right is S2."
"These two points are very important, please pay attention..."
"Wow wow wow~~"
Wang Hao began to talk very patiently. Because he had enough inspiration to support it and the proof process was not too complicated to sort out, he patiently carried out the derivation step by step, and during the derivation process, the inspiration value was still high. Keep increasing.
This is a snowball effect. The more you talk, the more inspiration and ideas you get, the deeper your understanding becomes, and the smoother you can speak.
When Wang Hao explained for ten minutes, the news spread outside the building. Even Zhou Qingyuan in the office of the School of Science knew about it. He and a doctoral student under him came over to take a look.
When he first entered the elevator, he found that the door of the office was crowded with people. He walked outside and took a look inside, and found that it was already occupied by people, and it was impossible to squeeze in anymore.
this……
"What on earth is Wang Hao talking about? I heard it has something to do with the built-in square of a closed curve?" Zhou Qingyuan asked a teacher.
"Yes, Professor Zhou, this is a question that no one has proven yet. Wang Hao proved it on the spot!"
"You can't hear it very well here, but it's good to see."
"Hurry up and shoot, hurry up and shoot, I want to send a short video."
soon.
Wang Hao completed the last step of explanation.
He was giving proofs and explanations at the same time, and the process sounded very smooth, because it did not involve too advanced set theory content, and many teachers fully understood it, including Luo Dayong and Zhang Zhiqiang.
Ding Zhiqiang understood part of it, at least the whole proof process, but it involved some content that he had never been exposed to, so he didn't understand it very well.
After Wang Hao finished explaining, he asked Luo Dayong again, "No one has ever proven this, right?"
Before Luo Dayong could speak, Zhang Zhiqiang rushed to say, "Yes, I checked carefully and no one has proved it."
Wang Hao said, "Then I will post the proof process on my blog."
After he finished speaking, he looked at Ding Zhiqiang and said, "I would like to thank classmate Ding Zhiqiang for this small proof. You should understand it, right? If you don't ask me again."
"correct!"
Wang Hao suddenly thought of something. He went to find two teaching textbooks on his desk, one was "Basics of Discrete Mathematics" and the other was "Ordinary Differential Equations". He handed it to Ding Zhiqiang and said, "Classmate Ding Zhiqiang, you There is great potential in mathematics, and most importantly, you are talented.”
"If you don't know how to learn, I recommend you to fully understand these two books, just this semester."
"It is best to fully understand these two books within three months. I will check them out at that time, and if possible, I will recommend other books to you. Your knowledge base is still lacking..."
Ding Zhiqiang took the two books in a daze, looked down at the titles of the books, and then realized what was happening.
Teacher Wang Hao...
He actually completed the proof of a square inside a closed curve on the spot. In addition, he also left himself homework. It was not just an ordinary homework, but to understand two difficult mathematics books in three months?
Ding Zhiqiang suddenly felt the urge to cry. He realized that he had lost everything by stealing the chicken.
I came to the complex and took away two books.
Three months?
In such a short period of time, if you want to thoroughly understand the contents of these two books, you probably won’t even have any leisure time, right?
He thought of his miserable life in advance.
The other teachers in the office had a completely different understanding. They looked at Ding Zhiqiang with surprise.
This little fat guy is definitely not simple!
Wang Hao has completed the demonstration of Artyn's constant and the achievements of Mersenne prime numbers, and has a certain reputation in the world. Can the students he is optimistic about do worse?
Ding Zhiqiang left the office with envious eyes.
He didn't know whether he should be happy or disappointed. When he left, several teachers asked him his name, asked about his major, and said words of encouragement to him.
All the teachers seemed to be very optimistic about his future prospects. This was the first time that he had received so much attention, and it was attention in a positive sense.
However, his purpose is not like this!
And he doesn't want to spend three months understanding irrelevant and complex mathematical content...
…
The explanation is completed.
The people gathered in the office slowly dispersed, but things had just begun. Wang Hao's performance was really amazing.
He made a proof that a closed curve has a square inside it. It's not a big deal, and it's nothing compared to other results, but the thing itself is very interesting.
Because of a student's question, he proved it on the spot.
This is really amazing.
Some teachers even jokingly said, "How about I ask Professor Wang Hao how to prove the Riemann Hypothesis? Maybe he can prove it on the spot."
Of course, this is just a joke.
Others were also shocked by Wang Hao's genius.
There were some unproven minor mathematical problems mentioned on the Internet before. To Wang Hao, they were really minor problems. When people saw such remarks, they just laughed it off.
It seems true now.
Wang Hao demonstrated this level of ability on the spot and made a proof that no one else had proven before. It seemed that there was no difficulty for him.
The matter was quickly posted on the Internet, and there was even a video of the whole process to prove it. In the video, Wang Hao's face was blurred, but it was no different from not doing it, because everyone knew that he was Wang Hao.
There was a heated discussion on the Internet.
During the heated discussion, Wang Hao published the third blog content--
"A small proof that a closed curve has a square inside it".
The content of this blog is a little different. The first sentence is to introduce the source of the proof, "This is a question from a student named Ding Zhiqiang. He is studying this problem, and his research has given me a lot of inspiration."
Next is the content of the proof.
Putting aside the discussion on the Internet, Wang Hao was very excited because he discovered that at some point, the inspiration value of the 'Research on the Probability of Prime Number Distribution' had exceeded one hundred points.
At the same time, I also have a clear idea in my mind and know the specific content of the mathematical method developed, but it is just that it is too complicated and it seems difficult to summarize.
This will be a bit of a difficult job.
But, it doesn't matter.
It is indeed difficult to summarize mathematical methods, but using mathematical methods to solve the Kakutani conjecture has already given us a clear idea.
"The summary of mathematical methods is put aside for now. You can take your time. Let's prove the Kakutani conjecture first."
"This guess is very valuable!"
Wang Hao happily started his research work.
(Ask for monthly ticket)
