I Just Want to Be a Quiet Top Student

Time has no regard for human beings, it passes slowly while waiting to receive a salary, and it passes very quickly during exams.

Shen Qi didn't make any effective actions within an hour, and the time left for him was running out.

Is it over, is it all over?

Is this my limit?

No, this shouldn't be the fate of a top student.

I have failed many subjects.

Also played games all night.

I used to have the rank of king.

In the blink of an eye, he became a scumbag.

I've been down and down, lost all direction.

Until you see mathematics, that is the only answer.

Shen Qi thought a lot in this hour, when he was thinking wildly and recalling the past, it showed that he was really helpless.

What is the source of this array of numbers? It looks like a group in form, but I have never seen such a weird group, full of magic. It may be a fake group, right? Is it plausible that it is not a group at all?

I racked my brains to search in the knowledge base of my brain, and the only response to Shen Qi was...the knowledge point you called is not in the service area.

There is no way to do this, no matter how strong the brain is, if there is a blind spot in knowledge, it will give up treatment.

Unless you create a whole new theory yourself, this is the only way.

But Shen Qi currently does not have this kind of self-contained master level.

"Perhaps I got full marks every time in the previous exams, and my journey was too smooth. How do you say that, the sky will send a great mission to the people, and you must first work hard to win their sisters."

Shen Qi in the predicament was thinking wildly, and he laughed when he thought about it, I am so talented, is this a level 1 Chinese level?

"Teacher Tian, ​​you can't save me this time, the sexagesimal system is useless here."

"Officer Liu, the algebraic knowledge you talked about is too exaggerated. This time IMO, I didn't use it at all. I can get to this point all by myself."

"Mr. Zhang, I haven't seen you for a long time. When I was in the fourth grade of mathematics, you stopped giving me little trouble and said that there was nothing to teach me."

Scenes of the past played in Shen Qi's mind in a fragmented manner. The experience of these few months is like a dream. If he can get the number one in the IMO world, it will be the most beautiful dream.

"Mr. Zhang, I went to your office a few months ago, and you asked me what I learned by myself."

"I said, I learned about the Cayley transition matrix and the Weierstrass quadratic form. It was indeed a bit pretentious at the time, and I didn't know much about it at that time."

"Then you're instantly stunned."

"I can't forget this scene. From then on, I started the road of being a master."

"Hehe, time flies so fast, it's time for senior year in a blink of an eye."

"Hehe... what the hell!"

Shen Qi was shocked, he suddenly caught a sliver of inspiration, which was fleeting and inseparable.

This feeling seems to be familiar. The finale of the national math competition at the beginning of this month, it was this kind of electric inspiration that saved Shen Qi. Mr. Tian saved Shen Qi. The irrational number showed up to the national championship.

"Back back, back up."

Shen Qi played back the random thoughts just now in his mind.

"Mr. Zhang, I went to your office a few months ago, and you asked me what I learned by myself."

"I said, I learned about Cayley transition matrices and Weierstrass quadratic forms."

...

"While mathematics appears outwardly to be no more than a language or a tool, its most vivid concepts can provide keys to new fields of thought."

"The determinants and matrices are completely reforms in the language of mathematics. Shen Qi, you must have a deep understanding of this before you can make a difference in algebra."

...

Shen Qi smiled, very happy, there is no unparalleled road.

Teacher Tian saved him last time, and Teacher Zhang saved him this time.

In fact, Shen Qi should be most grateful to himself. He never chose to give up in adversity. Mathematics often requires persistence and even madness. He and his last stubbornness saved him.

The Cayley transition matrix and matrix theory that were used in pretense back then finally came into play at the most critical moment.

No matter what kind of demons and ghosts this digital array is, or whether it is a group, it cannot escape the demon mirror in Shen Qi's hand—the matrix.

The tool that can comprehend or translate the group theory is the matrix.

According to the number array of the title:

1=1

196884=196883+1

21493760=21296876+196883+1

864299970=842609326+21296876+2*196883+2*1

...

Shen Qi wrote a matrix homomorphism:

A(gi*gj)=A(gi)*A(gj)

Expand it into a matrix representation:

|Ag-0|

|Ai-0|

|0-Aj|

...

This matrix language looks complicated, but the expression is very simple and direct, that is, the matrix representation of a group G is a homomorphic mapping from the element g of G to a set of non-singular square matrices A(g) of fixed order.

To put it more simply, a group is a complex set of ciphers that are very difficult to understand, and a matrix is ​​one of the mother copies of deciphering ciphers.

The only requirement is that you must be proficient in various decoding methods, the more the better.

If the array of numbers can be described by a matrix, it means that it is some kind of group, otherwise it is not.

When Shen Qi expressed this array of numbers in a regular permutation method, he was very surprised: "MMP...Monster-Group...is really a monster, a monster group!"

What are demons?

That is, the largest scattered simple group.

Compared with other groups, the demon group is very young, about forty years.

This group is quite scary, so it was named Monster-Group by mathematicians.

It is difficult for ordinary people to play with the demon group, and they will drive themselves crazy when they are playing, and they will break the game.

The British mathematician Bocheltz has made a major contribution to the theory of monsters. He proved the "moonlight conjecture of monsters", a very magical and powerful existence just by looking at its name. Bochelz won the Fields Medal for this great achievement.

Demon group, if you want to play with it, you need at least a doctorate in mathematics at the entry level.

Why does this kind of question appear on the IMO exam paper?

Is there a middle school student in the world who can handle it?

of course not.

No need to fix it either.

Shen Qi's understanding is that for this monster group, it is OK to give two different mathematical explanations.

Deciphering demons and describing them are two different things.

No one can crack Goldbach's conjecture, but many people can describe Goldbach's conjecture: any even number greater than 2 can be written as the sum of two prime numbers.

Similar to it, what Shen Qi wants to do is the latter, but it cannot be described in words, but in purely mathematical language.

He uses two matrix languages ​​to convert

1=1

196884=196883+1

21493760=21296876+196883+1

864299970=842609326+21296876+2*196883+2*1

...

It is enough to express clearly what it is, and there is no need to crack it.

This question examines the knowledge and the proficiency in the use of matrices.

We all know that a group has many matrix representations, because the order of the matrix can be changed.

"Let's start with the Cayley turning matrix." Shen Qi sacrificed the founder of matrix theory, Cayley, and used the Cayley turning matrix to express the first demon group explanation.

"Let's send the Jordan standard matrix again."

Soon, Shen Qi wrote two different matrix expressions.

Seeing that there is still time, he made another shot, and the third shot was the Hermitian matrix.

"If three shots are not enough, then three more shots!"

Shen Qi killed the sex, Ka Ka Ka, he successively wrote the Klein abstract group matrix, the Weber prime field matrix, and the Hensel reversible element matrix.

Six rounds!

Six shots were produced in more than an hour!

"My body has not been hollowed out. If six shots are not enough, then six more shots!"

Shen Qi has never been so happy before, a kind of refreshing feeling after a catastrophe.

Jingle Bell.

Then the bell rang.

The 4.5-hour competition time has come.

"Six shots, only six shots... what a pity." Shen Qi ran out of time, he only wrote six matrix languages, a little annoyed.

After handing in the test paper, Shen Qi saw the Russian contestant and the Indian-American contestant leaving together, chatting and laughing, and seemed to be full of confidence.

"There are many masters in the world, and I am not the only one who can solve this problem." Shen Qi devoted all his life's learning in this IMO, and it is up to fate what kind of results he can achieve.

Marking work began in the afternoon of the same day and continued until the early morning of the next day.

Professor Wiles, the leader of the judging panel, is sixty years old and still insists on working.

I can't remember how many cups of coffee I refilled, and Professor Wiles lost all sleepiness.

There are four exam papers in front of him, all with a full score of 42, from four different countries, Russia, the United States, South Korea, and China.

After hesitating for a long time, Professor Wiles finally made up his mind. He gave a "+1" after 42 in one of the full-score test papers.

For the last question, the other three full-score contestants all used two methods to give mathematical explanations.

And the "42+1" player used six kinds, which is exactly equal to the sum of the other three players.

This "42+1" player is from China, and his name is written in English as Shen-Qi.

The first player with 42+1 points in IMO history was born. His name is Shen Qi.