Time showed no mercy. It passed slowly when you were waiting for payday, but flew by during an exam.
Shen Qi had made no effective moves for an hour. He didn't have much time left.
Was it over? Was everything over?
Was this the limit of my abilities?
No, this couldn't be the fate of a top student.
I'd failed plenty of classes before.
I'd also stayed up all night gaming.
I'd once reached the highest rank.
Then, in the blink of an eye, I'd become a lousy student.
I'd known disappointment and despair, and lost all sense of direction.
Until I found mathematics—it was the only answer.
Shen Qi had thought about all sorts of things over the past hour. When he started daydreaming and reminiscing, it meant he was truly out of options.
Where did this numerical array come from? Judging by its form, it looked like a group, but he'd never seen such a bizarre group, so full of menace. Maybe it was a fake group? Something that looked like a group but wasn't one at all?
He racked his brains and searched his mental database. The only response Shen Qi got was... The knowledge you are calling is unavailable.
There was nothing he could do. Even the strongest mind would give up if it ran into a blind spot in its knowledge, wouldn't it?
Unless he created an entirely new theory of his own. That was the only way.
But Shen Qi didn't yet have the stature of a master capable of founding a school of thought.
"Maybe it's because I've gotten full marks on every test so far, and everything's gone too smoothly. How does that saying go? When Heaven is about to entrust someone with a great task, it first makes them suffer in body and mind—and takes away their girl."
Trapped in a difficult spot, Shen Qi let his mind wander. The more he thought, the more he laughed. Damn, I'm a genius. Is this what Level 1 Chinese looks like?
"Teacher Tian, you can't save me this time. The base-60 system is useless here."
"Cadre Liu, all that algebra you taught me was so flashy. None of it's been useful in this IMO—not even a little. I made it this far by figuring things out on my own."
"Teacher Zhang, I haven't seen you in ages. When I reached Level 4 in mathematics, you stopped giving me extra lessons, saying there was nothing left to teach me."
Fragments of the past played through Shen Qi's mind. The past few months had felt like a dream. If he could take first place in the IMO, it would be the most beautiful dream of all.
"Teacher Zhang, a few months ago, I went to your office, and you asked how far I'd gotten with my self-study."
"I said I'd learned the Cayley transformation matrix and the Weierstrass quadratic form. I admit, I was showing off a little. Back then, I only half understood them."
"And then you were instantly dumbfounded."
"I could never forget that moment. From then on, I began my journey as a top student."
"Ha, time really flies. Before I knew it, I was in my final year of high school."
"Ha... damn!"
Shen Qi jolted. A spark of inspiration had suddenly come to him, fleeting and elusive.
The feeling was familiar. Earlier that month, it was this same electric spark of inspiration that had saved Shen Qi during the final problem of the National Mathematical Olympiad finals. Teacher Tian had saved him. He'd shown off a proof, using cuneiform base-60 notation, that the square root of 2 was irrational—and won the national finals.
"Go back. Go back."
Shen Qi replayed his train of thought from just now in his mind.
"Teacher Zhang, a few months ago, I went to your office, and you asked how far I'd gotten with my self-study."
"I said I'd learned the Cayley transformation matrix and the Weierstrass quadratic form."
"Although mathematics may seem to be no more than a language or a tool, most of its vivid concepts can provide keys to new fields of thought."
"And determinants and matrices are nothing less than a revolution in the language of mathematics. Shen Qi, you must understand this deeply if you want to make a mark in algebra."
Shen Qi smiled, overjoyed. There was always a way forward.
Teacher Tian had saved him last time. This time, Teacher Zhang had saved him.
But the person Shen Qi should be most grateful to was himself. Even in a desperate situation, he'd never chosen to give up. Mathematics often demanded persistence, even obsession. His own stubborn resolve had saved him.
The Cayley transformation matrix and Matrix Theory, which he'd once used to show off, finally came through at the most critical moment.
Whatever kind of monstrosity this numerical array was, whether it was a group or not, it couldn't escape the mirror in Shen Qi's hand: matrices.
The tool for understanding or translating Group Theory was matrices.
Given 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 down a matrix homomorphism:
A(gi*gj)=A(gi)* A(gj)
He expanded it into a matrix expression:
|Ag-0|
|Ai-0|
|0-Aj|
This matrix notation looked complicated, but what it meant was very simple and direct: a matrix representation of a group G was a homomorphism from the elements g of G to a set of nonsingular square matrices A(g) of a fixed order.
Put more simply, a group was a complicated set of passwords that was extremely hard to understand, while matrices were one of the templates for deciphering them.
The only requirement was that you had to master as many decoding methods as possible.
If the numerical array could be described using matrices, then it was some kind of group. If not, it wasn't.
After expressing the numerical array as a regular permutation, Shen Qi was astonished. "MMP... Monster-Group... It really is a monster after all—a Monster-Group!"
What was a Monster-Group?
It was the largest sporadic simple group.
Compared with other groups, the Monster-Group was very young, only about forty years old.
It was so fearsome that mathematicians had named it the Monster-Group.
Ordinary people couldn't handle the Monster-Group. They'd go mad or ruin themselves trying to work with it.
The British mathematician Bocherds had made major contributions to the theory of the Monster-Group. He proved the "Moonshine Conjecture," something whose very name sounded fantastical and impressive. Bocherds received the Fields Medal for this tremendous achievement.
To work with the Monster-Group, you needed at least a PhD in mathematics.
Why would a problem like this appear on the IMO exam?
Were there any middle schoolers in the world who could solve it?
Of course not.
And they didn't need to.
Shen Qi understood that for this Monster-Group, it was enough to give two different mathematical explanations.
Deciphering the Monster-Group and describing it were two different things.
Nobody could prove the Goldbach Conjecture, but plenty of people could describe it: every even integer greater than 2 can be written as the sum of two primes.
Shen Qi needed to do the latter, except not in words. He had to describe it using pure mathematical language.
He only needed to clearly express what the numerical array
1=1
196884=196883+1
21493760=21296876+196883+1
864299970=842609326+21296876+2*196883+2*1
represented, using two kinds of matrix notation. He didn't need to decipher it.
The problem tested breadth of knowledge, as well as mastery of matrices.
We all knew that a group could have many different matrix representations, because the order of the matrices could vary.
"First, the Cayley transformation matrix." Shen Qi invoked Cayley, the founding master of Matrix Theory, and used the Cayley transformation matrix to give the first explanation of the Monster-Group.
"Next, the Jordan normal matrix."
Before long, Shen Qi had written down two different matrix expressions.
Seeing that he still had time, he went for another. His third was the Hermitian matrix.
"If three aren't enough, I'll do three more!"
Shen Qi was caught up in the thrill of it. Rapidly, he wrote down the Klein abstract group matrix, the Weber field matrix, and the Hensel invertible element matrix.
Six!
Six in a little over an hour!
"My tank isn't empty yet. If six aren't enough, I'll do six more!"
Shen Qi had never felt so good, exhilarated to have survived a close call.
Ding-a-ling.
Just then, the bell rang.
The four-and-a-half-hour exam was over.
"Six... I could only do six... What a shame." Shen Qi was out of time. He'd only written down six kinds of matrix notation, and he was a little frustrated.
After handing in his paper, Shen Qi saw the Russian contestant and the Indian-American contestant chatting merrily as they left together. They seemed full of confidence.
"There are plenty of strong contestants in the world. I'm not the only one who can solve this problem." Shen Qi had poured everything he'd learned in his life into this year's IMO. He could only leave the result to fate.
Grading began that afternoon and continued until the early hours of the next morning.
Professor Wiles, the head of the judging panel, was sixty years old, but still stayed at his post.
He'd lost count of how many cups of coffee he'd had. Professor Wiles was wide awake.
There were four exam papers in front of him, all with perfect scores of 42, from four different countries: Russia, the United States, Korea, and China.
After a long hesitation, Professor Wiles finally made up his mind. He wrote "+1" after the 42 on one of the perfect-score papers.
On the final problem, the other three contestants with perfect scores had each provided mathematical explanations in two ways.
But the contestant with "42+1" had used six—exactly as many as the other three contestants combined.
This contestant with "42+1" was from China. His name was written in English as Shen-Qi.
The first contestant in IMO history to score 42+1 had emerged. His name was Shen Qi.
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