Making Progress

The teachers of every subject in the second year of high school finished grading the exam papers and calculating the total scores overnight. They were an efficient team.

On Thursday morning, each of the 48 students in Class 2 of the second year had a score sheet. It was also a ranking chart.

The ranking chart had been edited in Excel and printed out. The top row consisted of seven cells: Chinese, Mathematics, English, Physics, Chemistry, Biology, and Total.

The leftmost column listed all 48 students in the class in descending order of rank.

1, Huang Zikun, Chinese 126, Mathematics 145, English 138, Physics 97, Chemistry 90, Biology 99, Total 695

2, Qian Huihui, Chinese 139, Mathematics 122, English 148, Physics 72, Chemistry 81, Biology 80, Total 642

15, Shen Qi, Chinese 88, Mathematics 150, English 95, Physics 86, Chemistry 80, Biology 82, Total 581

Chinese, Mathematics, and English each had a maximum score of 150, while Physics, Chemistry, and Biology each had a maximum of 100.

Shen Qi had scored a perfect mark in Mathematics again. He was the only student in the class to do so.

Thanks to his Mathematics score, Shen Qi had shot into the top 20 in the class rankings, climbing 32 places since the previous final exam.

"Looking at the numbers, I did about as well in Mathematics as usual. The class average on this Mathematics exam was 109.5, so of course I should score a little above average, given that my Math Level is 3."

"I did about as well as usual in Physics, Chemistry, and Biology, too. The class average was 80.7."

"An 88 in Chinese and a 95 in English, both below average. Just as expected."

The ranking chart showed every student's score in every subject. Shen Qi quietly did a little simple mathematical analysis. Fifteenth in the class—he could go home with something to show for himself.

It was worth noting that Huang Zikun had once again come first in the class overall, beating second-place Qian Huihui by more than 50 points.

"If I want to knock Huang Zikun off his throne, even perfect scores in Science and Engineering wouldn't be enough. That guy has no weak subjects."

People's ambitions grew quickly once they made progress. Shen Qi glanced at Huang Zikun, seated to the front left. To beat this all-around top student, he would have to invest in Chinese and English, and raise Physics, Chemistry, and Biology to Level 2 as well. That meant a total investment budget of 5,200 Academic Prodigy Points across five subjects.

The system suddenly announced: "Congratulations, host. You have earned a new achievement: Tied for First in Mathematics Across the School! Reward: 1,000 Academic Prodigy Points."

Shen Qi had scored a perfect 150 in Mathematics, coming first in the class and undoubtedly the entire grade. First in the grade was equivalent to first in the school. The tie meant that someone in another class had also scored 150. Nangang Second High School was, after all, a key high school, and Shen Qi wasn't the only one good at Mathematics.

What Shen Qi couldn't understand was this: "Last time, it was just a regular quiz. I scored 150 in Mathematics and came first in the class, and you gave me 2,000 Academic Prodigy Points. This time it's a major midterm exam, and I tied for first in the whole school. Why are you only giving me 1,000 points? I really need those Academic Prodigy Points!"

System: "As an advanced player fighting low-level monsters, the host must forfeit some Academic Prodigy Points. You can think of the 1,000 points as your after-tax income."

"Damn it, I can understand the experience points diminishing when an advanced player fights low-level monsters, but your taxes are way too high! What a rip-off!" As a Level 3 Mathematics player taking an ordinary high school math exam, Shen Qi had to pay a hefty tax on his Academic Prodigy Points. He hadn't paid any tax at Level 2, which meant Level 3 was a turning point. From now on, he'd have to tackle some high-level dungeons in Mathematics to earn more Academic Prodigy Points.

His deskmate, Chen Xiaoting, had been watching Shen Qi for a while. He hadn't said a word since getting his score sheet. He must have been thrilled.

"Shen Qi, you're amazing! You got a perfect score in Mathematics again! And your total ranking is higher than mine now. I'm so jealous!" Chen Xiaoting said enviously. She had scored 106 in Mathematics this time, with a total of 549, putting her twenty-fourth in the class.

"What?" Shen Qi snapped back to attention.

"I said, you're! So! A! Maz! Ing!" Chen Xiaoting raised her voice.

"Oh, right. It's not bad." Shen Qi was happy with fifteenth place in the class, since he'd reached his modest goal.

Chen Xiaoting cautiously asked, "Shen Qi, there's something I'm not sure I should say..."

"You're not about to call me an asshole, are you?" Shen Qi was taken aback. I beat you in the overall rankings this time, Chen Xiaoting. Are you holding a grudge? I guess if you stay a top student long enough, you find yourself with fewer and fewer friends and more and more loneliness!

But Chen Xiaoting didn't call him an asshole. Instead, she asked, "You've been going to Teacher Zhang's office every day lately..."

"Is Teacher Zhang giving you extra lessons?" Chen Xiaoting looked even more envious.

"So that's what you wanted to ask." Shen Qi laughed. He'd been overthinking it. Chen Xiaoting was a decent person after all.

"It's not exactly extra lessons. Teacher Zhang and I mostly discuss and exchange ideas." Shen Qi knew he couldn't keep his day-to-day interactions with Zhang Wanbang a secret from the other students. They weren't blind or stupid. They all shared the same classroom and spent their days together. Who could hide anything from anyone?

"So you're Teacher Zhang's prized student. Must be nice." Chen Xiaoting's impression of Shen Qi had been subtly changing. Being favored by the homeroom teacher was no small thing.

"Xiaoting, you don't have to envy me. I'm only good at Mathematics. I'm pretty average in the other subjects. And what Teacher Zhang and I discuss isn't really related to high school math. It's just a hobby of ours."

"You two sound so mysterious. I'd love to see what you talk about."

Shen Qi pulled a sheet of paper out of his schoolbag. It was "Internal Reference 10," the document he and Zhang Wanbang had put together.

"Here. This is it. There's nothing all that mysterious about it."

Chen Xiaoting took "Internal Reference 10" with almost reverent care. The paper read:

"The trigonometric formulas for imaginary angles can be replaced by hyperbolic functions. Euclidean geometry no longer applies here... Further reading: Discourse on Non-Euclidean Geometry (J. Bolyai edition), <New Principles of Geometry with a Complete Theory of Parallels> (Lobachevsky edition)."

"An infinite quadratic form—meaning the treatment of a quadratic form with infinitely many variables—is purely an algebraic problem. Advanced Algebra only touches on it in passing and does not explore it in depth... Further reading: <Principles of the General Theory of Linear Integral Equations> (Hilbert edition)."

"Speaking of integration, one has to mention Newton and Leibniz. We can start by studying Newton's lecture notes from his time at the Royal Academy of Sciences: A Treatise of Arithmetick."

Chen Xiaoting realized that she recognized all the words on the page, but couldn't understand them. "As someone who's terrible at math, all I can do is look at it... Of all those foreign names, I only know Newton, and that's from learning Newton's three laws in physics class."

"I told you, it has nothing to do with high school math."

"Shen Qi, could you tutor me in high school math?"

"Teacher Zhang explains things pretty well. He's an outstanding teacher in the city, so just listen carefully in class." Shen Qi answered sincerely. In his eyes, Zhang Wanbang really was an excellent teacher. He just had a bit of a nasty temper.

"I've been paying close attention in Teacher Zhang's math class for almost two years. He's made me cry several times, but I still haven't improved." Chen Xiaoting sighed helplessly, looking pitiful.

"Fine. If there's anything you don't understand in Mathematics, you can ask me."

"Thanks! How do you solve this one?" Chen Xiaoting seized the opportunity, pulled out a math problem, and asked Shen Qi for help.

Shen Qi took the problem and glanced at it, genuinely surprised. "The Fibonacci sequence? How could you get such a simple problem wrong? I don't understand—how did you get 25?"

Chen Xiaoting covered her face in embarrassment. "Why do you sound just like Teacher Zhang? I really don't know how to solve it..."

Shen Qi: "Sorry, Xiaoting. I came on a little strong. Here, I'll show you a trick. Once you learn it, you can solve all these types of problems the same way. Give me a sheet of scratch paper."

"Here!" Chen Xiaoting handed him her notebook.

Shen Qi opened Chen Xiaoting's notebook. Its pages were densely covered in neat handwriting.

It was clear that Chen Xiaoting worked hard at Mathematics. She had carefully written down every point Zhang Wanbang had taught.

But diligence alone wasn't enough for Mathematics. Without much natural talent or a system, even someone as hardworking as Chen Xiaoting couldn't raise her math scores.

"To use the Fibonacci sequence properly, Xiaoting, you only need to remember this: as n approaches infinity, the decimal part of the ratio of one term to the previous term gets infinitely close to 0.618. So you can just divide by 0.618 and follow the method." Shen Qi wrote a formula in Chen Xiaoting's notebook and explained it.

The Fibonacci sequence, a constant mathematical pattern derived from animal reproduction, was covered in small print in the "Reading and Thinking" section of the high school math textbook.

When Shen Qi had studied Advanced Algebra and Mathematical Analysis on his own, he'd looked into sequences in particular. He thought the Fibonacci sequence was an interesting theory with plenty of real-world applications. "Xiaoting, have you seen Wong Kar-wai's In the Mood for Love, Zhang Yimou's Hero, or Andrew Lau's Infernal Affairs?"

"I have. They're all classics. I've seen In the Mood for Love nine times. I absolutely adore Tony Leung." Chen Xiaoting, who liked art-house films, was puzzled. Wasn't he supposed to be teaching her how to solve a math problem? How had they gotten sidetracked into movies?

"They're all famous directors, and all three films won a lot of awards. Take a closer look: all three films—In the Mood for Love, Hero, and Infernal Affairs—won the same award: Best Cinematography Award."

Chen Xiaoting: "So?"

Shen Qi smiled. "So we have a cinematographer named Christopher Doyle to thank. He's a foreigner, and he was the cinematographer for all three films: In the Mood for Love, Hero, and Infernal Affairs."

"When Christopher Doyle filmed In the Mood for Love, he divided by 0.618 in his composition. When he filmed Hero, he divided by 0.618 in his composition. When he filmed Infernal Affairs, he divided by 0.618 in his composition."

"We can complain about the plot of Hero, but we have to admit that it looks absolutely perfect. It gives you a beautiful, harmonious visual experience, because Christopher Doyle understands math. His framing and staging reflect the Fibonacci sequence's golden spiral, based on 0.618, everywhere."

"So people who don't understand math can't make blockbuster films." Shen Qi concluded, then asked, "Xiaoting, can you solve that sequence problem now?"

"I can! I get it!" Chen Xiaoting suddenly understood. She felt the joy of learning. "You explain things in a more interesting, vivid way than Teacher Zhang. I had no idea that the great cinematographer Christopher Doyle was a math whiz! Come on, I have a whole stack of questions I want to ask you."

Shen Qi quickly waved her off. "No, no. Make sure you've fully absorbed what we covered today before we talk about more. Xiaoting, don't rush or panic. Great things take time. Let's make a deal: one question a day, no more. Learning never ends; little by little, it all adds up, and quantitative change leads to qualitative change." I'm busy. How could I possibly have time to be your full-time math tutor?

"How about two questions a day?" Chen Xiaoting was desperate to learn.

"No. One question a day." Shen Qi shook his head firmly.

"Since you like movies, I'll give you a movie ticket. How about two questions a day?" Chen Xiaoting secretly took out her phone to send him a QR code.

Shen Qi was displeased. "Chen Xiaoting, can scientific knowledge be measured in money? Can you trade math skills for movie tickets?"

"I... I'm sorry." Chen Xiaoting lowered her head in embarrassment. Knowledge was priceless, and science was sacred. She really had been shallow.

"Which cinema is it for?" Shen Qi asked.

"Feiyang Cinema. 3D-MAX."

"It's no fun going alone. Come with me. You don't have to treat me, either—we'll split the cost and each buy our own ticket."

"But I don't have much time to go to the movies with you. You know we only get one day off a week in the second year, and I still have to practice violin..." Chen Xiaoting trailed off.

Shen Qi handed Chen Xiaoting's notebook back to her. "Xiaoting, you really can't rush when you're trying to improve your math. Take it slow, or trying to force growth will only backfire."

"Then I'll see if I can ask my violin teacher for half a day off... I only have half a day at most."

"Just this once, okay? Two questions a day. So, which math problem don't you understand?"