A Genius's Flash of Inspiration

Matthew: "?"

The entire class: "?"

Mr. Harrison: "?"

The equations on the blackboard...

Weren't they a little too short?

Matthew could hardly believe it. He said, "Li Mo, didn't you just write down a few steps at random? You already knew the final answer to the problem, so you put it under a few lines of equations. Forgive me for saying so, but that seems rather improper."

Several students nodded vigorously at his words. Everyone thought the explanation made sense.

Mr. Harrison had frowned at first too, but unlike Matthew, who had questioned Li Mo on the spot, he did have some mathematical intuition.

He felt that the three lines on the blackboard had a distinctive beauty to them. They didn't look like something written at random.

"Li Mo," Mr. Harrison said, "could you explain your reasoning to everyone?"

"Of course. It's very simple. The problem asks us to find the maximum volume for a fixed surface area. Since we're looking for an extremum, we can consider establishing an inequality between the given conditions and the objective function. I split the cylinder's lateral surface area, 2πrh, into two equal parts: πrh plus πrh. Next, we introduce the axiom that the arithmetic mean is greater than or equal to the geometric mean. For any three positive real numbers a, b and c, their arithmetic mean, (a+b+c)/3, must be greater than or equal to their geometric mean, which is the cube root of their product. Applying this axiom to the three terms we've separated out-2πr², πrh and πrh-the numerator of the arithmetic mean comes to exactly the cylinder's surface area, S. Multiplying the three terms under the radical in the geometric mean gives us exactly 2π³r⁴h². Substituting the volume formula V = πr²h, the expression under the radical becomes exactly 2πV². The conclusion is obvious: equality holds if and only if the three positive real numbers are equal. In other words, for the volume V to reach its maximum, we must have 2πr² = πrh = πrh. Cancel the common factor πr, and we get 2r = h. Proof complete."

They might have been better off without an explanation. Now the students were even more bewildered!

They understood every word, so why couldn't they understand them put together?

How strange. It felt as though knowledge had flowed through their brains without leaving a trace...

Mr. Harrison stood where he was, his gaze lingering on the three lines of equations for a long time.

He quickly checked the calculations in his head.

Arithmetic mean... geometric mean... the extremum occurred when all the terms were equal...

This seemed to be a more advanced kind of algebraic thinking.

It stepped outside the framework of traditional calculus and used the symmetry of the numbers themselves to solve the problem...

Why had he never thought of that?

Mr. Harrison turned around, unable to hide his surprise. "Li Mo, where did you learn this method? Even the latest issue of the Royal Academy of Sciences' mathematics journal, published last month, doesn't seem to have included such a clever application of the inequality extremum theorem."

"Just now, sir," Li Mo said. "I simply thought the differentiation process was too cumbersome, and this approach suddenly occurred to me. I thought using an inequality would be more concise."

Mr. Harrison froze.

What did he mean, just now? What did he mean, the approach had suddenly occurred to him?

Mr. Harrison was speechless. All he could do was applaud.

A moment later, he said, "An inspiration worthy of a genius! A genius's flash of inspiration! Everyone, Li Mo's solution is completely correct, and far more elegant than our conventional method."

With the teacher saying so, the students finally accepted that Li Mo really hadn't been talking nonsense.

All sorts of emotions began to erupt.

Matthew went rigid all over, feeling miserable.

Helen, in the front row, raised her head and stole a glance at Li Mo. Then she quickly pressed her lips together and lowered her head. Once she had lowered it, she pressed her lips together even harder, afraid she might laugh out loud.

Young master Li Mo is amazing... she thought silently.

Wayne was already excited. He rose like a gentleman, pulled out a handkerchief and waved it in the air. "Magnificent! Incomparable! Masterful!!"

Outside the classroom door, the freckled girl covered her mouth in surprise. Who would have thought that the cursed Li Mo had such talent in mathematics?

Anna's reaction was different from everyone else's. She simply sat quietly in her seat.

She looked down at the equations on her scratch paper, then looked up at those on the blackboard.

For some reason, her heart ached.

In the eyes of the other students, Li Mo was an inspiring example of a fallen genius shining once again in another field.

But only Anna knew how much Li Mo had put into it.

She thought back to the days gone by. Young master's health had been so poor, yet he had still kept reading every day...

Anna bit her lower lip and silently resolved to work much harder at the occult.

Only by gaining greater power could she better protect Li Mo in the future and strive to find a way to cure the curse on his body as soon as possible.

Besides, since her talent for the occult was average, she had to work harder. From now on, she would sleep two hours less each night and spend that time meditating. She would never let young master shoulder everything alone!

The bell rang from the clock tower, as though cheering for Li Mo's equations.

Mr. Harrison gathered up his lesson plan and said, "Everyone, copy down the working on the blackboard. I need to go to the teachers' lounge. Class dismissed."

With that, Mr. Harrison strode out of the classroom, walking much faster than usual. Clearly, he couldn't wait to share the new discovery.

As soon as the teacher left, Anna and Wayne went straight to the front of the classroom.

"That was amazing, Li Mo," Anna said.

"How interesting," Wayne remarked. "Some people go to such lengths, spending a fortune on a tutor and covering the blackboard in equations, only to be beaten by three lines someone else casually wrote down. It seems mathematics really does take just a little bit of talent."

Wayne was being sarcastic purely to get under Matthew's skin.

Matthew's face instantly turned ugly.

Even so, he merely shot Wayne a cold glance and didn't reply.

He wasn't stupid. He knew that any resistance now would only make him look worse. He would put up with it for now and settle the score the next time he found an opportunity.

Anna said, "Li Mo, we have physical education next. Should we go to the lounge for a while?"

Li Mo nodded. "Mm, I won't be going to physical education. I'll go straight to Professor Bryce."

Matthew, who had been about to leave the front of the classroom, froze. He turned around and blurted out, "What are you going to see Professor Bryce for?"

"To ask him to take me on as his assistant," Li Mo said.

Matthew's mouth twitched. Then he said, "That's impossible, Li Mo. Everyone at school knows how demanding Professor Bryce is. He won't take you on as his assistant. Going to him will be a waste of effort."

There was something Matthew didn't say.

Professor Bryce was more than just a professor.

Matthew had gone to great lengths to establish that connection, yet he had still come away empty-handed.

Li Mo, who knew nothing about it, couldn't possibly succeed...

"I'll give it a try. You never know."

After tossing out those words casually, Li Mo walked out of the classroom.

Anna and Wayne exchanged a glance and immediately followed him.

Matthew watched them leave in silence for a long time, then hurried after them too.

The matter of Professor Bryce was of great importance. It was no longer merely a rivalry between students.

Considering his family's instructions, he had to keep track of it in real time...