Divine Problem, Divine Answer

Time shows no mercy. It crawls when you're waiting for payday, but it flies when you're taking an exam.

Shen Qi hadn't made any progress in an hour, and time was running out.

Is it over? Is everything over?

Is this my limit?

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

I've failed many courses before.

I've pulled all-nighters playing games.

I used to be a master player.

In the blink of an eye, I became a slacker.

I was once lost and disappointed, having lost all direction.

Until I found mathematics, which was the only answer.

Shen Qi thought a lot in this hour. When he was lost in thought, reminiscing about the past, it meant he was truly out of options.

What is the origin of this numerical array? Judging by its form, it looks like a group, but I've never seen such a strange group, so magical. Could it be a fake group? Or is it not a group at all?

He racked his brain, searching his mental knowledge base, but the only response he got was... the point of knowledge you've requested is currently unavailable.

There's no other way. Even the strongest brain will give up if it has a blind spot in its knowledge.

Unless I create an entirely new theory myself, that's the only way.

But Shen Qi currently doesn't have the level of a grandmaster who can establish his own school of thought.

"Perhaps I was too smooth sailing, getting perfect scores every time. How does that saying go? When heaven is about to bestow a great responsibility upon a person, it first tests their resolve, exhausts their body and mind, and starves their family."

In his predicament, Shen Qi's mind wandered. As he thought, he even laughed at himself. Damn, I'm so talented. Is this the literacy level of a first-grader?

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

"Cadre Liu, the Algebra you taught me was too flashy. I didn't use even a bit of it in this IMO. I only got this far through my own exploration."

"Teacher Zhang, I haven't seen you in a long time. You stopped giving me special tutoring when I reached Math Level 4, saying there was nothing more you could teach me."

Memories flashed through Shen Qi's mind in fragments. The experiences of the past few months felt like a dream. If he could win first place in the IMO, it would be the most beautiful dream.

"Teacher Zhang, a few months ago, I went to your office, and you asked me how far I had self-studied."

"I said I had reached the Cayley transformation matrix and the Weierstrass quadratic form. I was bragging a bit then, and I only half-understood it myself."

"You were instantly stunned."

"I can never forget that moment. From then on, I embarked on the path of a top student."

"Heh, time flies so fast. In the blink of an eye, it's my senior year."

"Heh... Holy crap!"

Shen Qi jolted. He suddenly caught a glimmer of inspiration, fleeting and elusive.

This feeling was familiar. The final question of the national Math League finals earlier this month was solved by this kind of electric shock of inspiration, which saved Shen Qi. Teacher Tian had saved Shen Qi, and he had shown off his proof that the square root of 2 is an irrational number using cuneiform script and the sexagesimal system, winning him the national championship.

"Rewind, rewind."

Shen Qi replayed his recent rambling thoughts in his mind.

"Teacher Zhang, a few months ago, I went to your office, and you asked me how far I had self-studied."

"I said I had reached the Cayley transformation matrix and the Weierstrass quadratic form."

"Although on the surface, mathematics is just a language or a tool, most of its vivid concepts can provide keys to new fields of thought."

"And determinants and matrices are a complete reform of the language of mathematics. Shen Qi, you must deeply understand this to achieve anything in Algebra."

Shen Qi smiled, incredibly happy. There's always a way out.

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

In fact, Shen Qi should thank himself the most. He never chose to give up in the face of adversity. Mathematics often requires persistence, even obsession. He and his final stubbornness saved him.

The Cayley transformation matrix and matrix theory, which he had used to show off back then, finally played their role at the most crucial moment.

No matter what kind of monster this numerical array is, whether it's a group or not, it can't escape the demon-revealing mirror in my hands, Shen Qi—the matrix.

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

According to the numerical array in the problem:

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)

Expanding it into a matrix expression:

|Ag-0|

|Ai-0|

|0-Aj|

This matrix language looks very complicated, but the meaning it expresses is very simple and direct: a matrix representation of a group G is a homomorphism mapping from the elements g of G to a set of non-singular square matrices A(g) of a fixed order.

To put it more simply, groups are a set of complex ciphers that are very difficult to understand, while matrices are one of the parent codes for deciphering them.

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

If this sequence of numbers can be described using matrices, it means it is some kind of group; otherwise, it is not.

When Shen Qi expressed this sequence of numbers using regular permutations, he was very surprised: "MMP... Monster-Group... It really is a demon, a Monster-Group!"

What is a Monster-Group?

It is the largest sporadic simple group.

Compared to other groups, the Monster-Group is very young, only about forty years old.

This group is quite terrifying, which is why mathematicians named it the Monster-Group.

It is difficult for ordinary people to handle the Monster-Group; they often go crazy or break themselves while trying.

The British mathematician Bocherds made significant contributions to the theory of the Monster-Group, proving the "Monster Moonshine Conjecture," an existence that sounds magical and awesome just by its name. Bocherds received the Fields Medal for this immense achievement.

To handle the Monster-Group, the entry-level requirement is at least a Ph.D. in mathematics.

Why would such a question appear on the IMO exam paper?

Can any middle school student in the world solve it?

Of course not.

Nor is it necessary to solve it.

Shen Qi's understanding was that it was enough to provide two different mathematical explanations for this Monster-Group.

Cracking the Monster-Group and describing the Monster-Group are two different things.

No one can crack Goldbach's Conjecture, but many people can describe it: any even integer greater than 2 can be expressed as the sum of two prime numbers.

Similarly, what Shen Qi needed to do was the latter, but not in words, rather described in pure mathematical language.

He used two forms of matrix language to clearly express what

1=1

196884=196883+1

21493760=21296876+196883+1

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

was, without needing to crack it.

This question tested one's breadth of knowledge and proficiency in using matrices.

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

"Let's start with a Cayley transformation matrix," Shen Qi invoked Cayley, the founding father of matrix theory, and used a Cayley transformation matrix to express the first explanation of the Monster-Group.

"And then a Jordan normal matrix."

Soon, Shen Qi wrote out two different matrix expressions.

Seeing that he still had time, he added a third one: a Hermitian matrix.

"If three aren't enough, then three more!"

Shen Qi got into the zone and, with a click-click-click, he consecutively wrote out a Klein abstract group matrix, a Weber field matrix, and a Hensel invertible element matrix.

Six attempts!

Six attempts in over an hour!

"My body isn't hollowed out yet. If six aren't enough, then six more!"

Shen Qi had never felt so exhilarated, a sense of exhilaration from surviving a near-death experience.

Ring ring ring.

At this moment, the bell rang.

The 4.5-hour competition time was up.

"Six shots, only six shots left... what a pity." Shen Qi was out of time. He had only written six matrix languages, and he was a little annoyed.

After submitting his paper, Shen Qi saw the Russian contestant and the Indian-American contestant leaving together, chatting and laughing, seemingly full of confidence.

"There are many masters in the world; I'm not the only one who can solve this problem." Shen Qi had poured all his lifelong knowledge into this IMO. As for what results he could achieve, he would leave it to fate.

The grading work began that afternoon and continued until the early morning of the next day.

Professor Wiles, the head of the judging panel, was sixty years old but still insisted on working.

He couldn't remember how many cups of coffee he had refilled; Professor Wiles was completely devoid of sleepiness.

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

After a long hesitation, Professor Wiles finally made up his mind. He wrote a "+1" after the 42 on one of the perfect-score papers.

For the last problem, the other three perfect-score contestants had all provided mathematical explanations in two ways.

However, the contestant who got "42+1" used six methods, exactly the sum of the other three contestants' methods.

This "42+1" contestant was from China, and his name written in English was Shen-Qi.

The first contestant in IMO history to score 42+1 was born, and his name was Shen Qi.