A game of basketball had brought Shen Qi and Xu Rui back together. The two of them drank ice-cold cola outside the school store, feeling refreshed.
After returning to the classroom, Xu Rui was no longer full of energy. His mysterious air of dominance had vanished, replaced by listlessness.
Shen Qi hadn't forgotten the math problems Zhang Wanbang had given him. Three remained unsolved, and they were even harder than before. They required him to combine the concepts of Frobenius matrix rank, Laplace's theorem, Cramer's rule, and the corollaries later proposed by Fourier and Poincare. They involved the theory of infinite-order matrices.
Though Advanced Algebra was abstract, it was, after all, an introductory course in the Mathematics Department. Even if he couldn't fully grasp those abstract concepts, he could just about follow the standard methods by rote.
Shen Qi struggled to follow the methods, working until 10 p.m. before finally completing the last three problems.
The next morning, Shen Qi went to Zhang Wanbang's office and handed him his answer sheets.
"Looks like you ran into some roadblocks." Zhang Wanbang took the answer sheets and quickly looked them over. He'd expected Shen Qi to hit some snags.
"Yeah, Teacher Zhang. If matrices evolved from determinants, then what on earth are quaternions good for? I spent all night thinking about it and still couldn't figure it out. To be honest, I just guessed at the last three problems."
"I studied applied mathematics first, then switched to teacher training. I didn't figure that out until my sophomore year. Shen Qi, you can head back to class for now. Come see me again at noon. Also, I have to remind you: don't put all your effort into math. Are you just going to give up on Chinese? What about English? I'm your math teacher and your homeroom teacher, so I can tell you with complete confidence that the college entrance exam won't test math alone."
"Oh, right. Got it."
Shen Qi returned to the classroom, took out Advanced Algebra, and continued studying. He didn't pay attention in Chinese class again.
I'm hopelessly addicted to math. How do I fix this?
After lunch, Shen Qi went to Zhang Wanbang's office again.
Zhang Wanbang gave him two sheets of paper. One had his corrections to the five problems from the day before; the other was a blank sheet with five new problems written on it.
After finishing lunch, Shen Qi began working on the two sets of problems.
Zhang Wanbang had marked the five problems from the previous day and left comments in the blank spaces on the answer sheet:
"On the surface, mathematics may seem to be no more than a language or a tool, but many of its vivid concepts unlock new realms of thought."
"Determinants and matrices, meanwhile, represent a complete reform of mathematical language. Shen Qi, you have to understand this deeply if you want to accomplish anything in algebra."
"Problem three: you misunderstood the Frobenius characteristic equation. Remember, the minimal polynomial is the lowest-degree polynomial satisfied by a matrix, and it is unique."
"Problem four: f(λ)=(λ-λ1)^k1(λ-λ2)^k2...(λ-λi)^ki. I didn't see you work through it, but I'm quite sure you were mechanically applying the Jordan normal form."
"But the fact that you could mechanically apply it and still get the right answer proves you have some ability. At least you know how to feel your way through a method. But that isn't the simplest method."
"The point of mathematics is to solve the most complicated problems in the simplest and most efficient way."
"Now I'll show you the simplest approach to the Jordan normal form. Let J=XXXX..."
Zhang Wanbang's comments on the answer sheet distilled a lifetime of learning into a few lines. He used limited space to explain the most essential mathematical ideas and techniques. He was giving Shen Qi guidance.
You couldn't learn this from a textbook—not at least with the same specificity, focus, and relevance as Zhang Wanbang's comments.
Shen Qi got it in an instant. The bottleneck that had held him back for over a week broke through at once. "I see! Now I get it! Everything Teacher Zhang wrote is pure gold. I've learned so much!"
The host has received guidance from an expert in a specialized field of mathematics and grasped a new professional technique. Academic Prodigy Points +100.
"One hundred points! That's a ton!" Shen Qi jumped up in excitement. Teacher Zhang, you're an expert! You're teaching me and helping me rack up Academic Prodigy Points. I can sum up all my gratitude in four words: Your kindness is immeasurable.
Shen Qi absorbed the five problems from the previous day and labeled the packet "Internal Reference 1," stowing it away in his schoolbag.
Next, Shen Qi began working on "Internal Reference 2," the five problems Zhang Wanbang had assigned that day.
Unlike other natural sciences, mathematics couldn't use experimental equipment to verify theoretical hypotheses.
Mathematical experiments relied mainly on mental visualization. By thoroughly observing and understanding various theories and concepts, one could improve one's skill through derivations, proofs, and hypotheses.
For the next two weeks, Shen Qi's routine was to visit Zhang Wanbang's office once each morning and once at noon to exchange "Internal References."
To Shen Qi, Zhang Wanbang had two roles. In class, he was an ordinary math teacher who explained high school math concepts.
Outside of class, Zhang Wanbang was the "expert" who guided Shen Qi, teaching him more advanced mathematics.
The host paid close attention during a physics lesson. Academic Prodigy Points +1. The host did not pay close attention during Chinese class. Academic Prodigy Points +0. However, the host solved a math problem during Chinese class. Academic Prodigy Points +5. The host spent one hour running, playing ball, or doing other physical exercise. Academic Prodigy Points +1. The host received guidance from an expert in a specialized field of mathematics and grasped a new professional technique. Academic Prodigy Points +100. The host self-studied Mathematical Analysis for three hours. Academic Prodigy Points +10.
Shen Qi's schedule was packed, and he consistently earned 120 to 130 Academic Prodigy Points a day.
The biggest source of points was "receiving guidance from an expert in a specialized field of mathematics"—that is, Shen Qi's extracurricular sessions with Zhang Wanbang.
Zhang Wanbang's high school math lessons could no longer earn Shen Qi even a single Academic Prodigy Point.
But in more advanced mathematics, Zhang Wanbang reliably gave Shen Qi 100 Academic Prodigy Points every day.
With the "master and apprentice mode" driving him forward, Shen Qi's practical math skills improved further. His collection of "Internal References" had reached "Internal Reference 15," and his studies now covered indefinite integrals, Fourier series, the hyperboloid of one sheet, ellipsoids, and other topics across several branches, including Advanced Algebra, Mathematical Analysis, and Analytic Geometry. Shen Qi could probably sit for the first- and second-year math department exams now without failing.
April was a month of rapid progress for Shen Qi.
By April 20, Shen Qi had saved up 4,205 Academic Prodigy Points. "Finally, I can level up again."
Raising Math Level 2 to Math Level 3 required 3,000 Academic Prodigy Points.
"System, I want to level up. Raise my Math Level to 3."
"3,000 Academic Prodigy Points deducted. The host's Math Level has increased to 3."
The system classified both Level 2 and Level 3 as advanced levels, which could be understood as lower and intermediate advanced levels.
Host: Shen Qi Age: 17 1. Math Level 3 (0/5000) 2. Sports Level 1 (0/1000) 3. Chinese Level 0 (0/100) 3. English Level 0 (0/100) 3. Physics Level 0 (0/100) 3. Chemistry Level 0 (0/100) 3. Biology Level 0 (0/100) 3. History Level 0 (0/100) 3. Geography Level 0 (0/100) 3. Politics Level 0 (0/100) Remaining Academic Prodigy Points: 1205
The system ranked Shen Qi's subjects. His Math Level was far ahead at Level 3, his Sports Level ranked second at Level 1, and his other eight subjects were tied for third at Level 0.
Raising Math Level 3 to Level 5 would require 15,000 Academic Prodigy Points in total.
Shen Qi consistently earned over 100 Academic Prodigy Points a day, so he could earn 15,000 within a foreseeable period.
Doing the math, Shen Qi would have enough points in four months at the latest. He would definitely be able to boost math to Level 5 before graduating from high school.
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