Ultra Instinct Hao

After the teacher nodded to him, Matthew stood up and said:

"Mr. Harrison, may I do this problem by myself? I happen to have two solutions, and I'd like to take this opportunity to demonstrate both in full on the blackboard."

Li Mo was stunned when he heard that:

What in the Ultra Instinct Hao!

Then he glanced at a girl sitting toward the front left of the classroom.

Mr. Green's daughter, Helen.

Helen had lowered her head and looked terribly uncomfortable.

Matthew's motive was simple.

Only one month remained before graduation.

He wanted to make the most of the time he had left, draw more of Helen's attention and show her how outstanding he was.

Unfortunately, the boy had gone about it entirely the wrong way.

Helen was an introverted girl with social anxiety, and she clearly disliked making a big spectacle.

Matthew was already bold about showing his interest in her, and now his showy behavior at the front of the classroom would only make her squirm...

After a moment's thought, Mr. Harrison nodded.

"Come on up, Matthew. Show us your approach."

Matthew went to the front, elegantly rolled up his cuffs, and picked through the chalk until he found a slender piece.

Watching him pick out the chalk made Li Mo put a hand to his forehead again.

Hao got me again... I suggest Brother Matthew take that chalk out into the rain and open Heaven's Gate.

Matthew finally began writing. "Class, I'll be going fairly fast, so do your best to keep up. For the first solution, I'll follow the standard procedure Mr. Harrison just taught us."

First, he wrote down the given surface area formula, followed by the formula for the cylinder's volume.

After a series of calculations, he arrived at h = 2r.

Li Mo sighed quietly. He calls that fast? What's fast about it? Sure enough, most men always expect to last longer than they actually do...

Mr. Harrison, however, said approvingly, "Very good, entirely by the book. Now, what's your second solution?"

Matthew said casually, "For the second solution, I've used an algebraic technique I learned outside class. I introduced an auxiliary variable, setting a new intermediate ratio k = h/r.

"Then I converted both the surface area and volume formulas into functions of the radius r and the variable k.

"By separating out the constant terms, I expressed r² as S /(2π(1+k)), then substituted it into the volume formula: V =πr³k.

"After a series of complicated exponent manipulations, the volume can be expressed as an irrational fractional function dependent only on k."

Li Mo was getting a little sleepy watching him.

It was still a calculus framework.

Matthew had substituted one geometric variable partway through the calculation.

It was the same old thing in a different guise.

The introduction of fractions and square roots even made the differentiation more prone to error than in the first solution.

It was like a teacher back on Earth saying, "The answer is 0.5!"

Then someone raising a hand and asking, "Teacher, does it count if I wrote one-half?"

That was almost exactly what it felt like.

Matthew finished simplifying the derivative on the last line, arriving at k = 2 and thus h = 2r.

He put down the chalk and turned around gracefully, signaling that everyone could start applauding.

Mr. Harrison smiled and said approvingly:

"Introducing a geometric ratio variable to reconstruct the function-you must have spent a great deal of time studying after class, Matthew. An excellent performance."

The students broke into applause.

Wayne, who sat on the right side of the classroom near the corridor, rolled his eyes in exasperation and muttered, "Didn't he just pay for a tutor and get an extra two weeks of math lessons? Now he's come up to the blackboard to parrot the method his tutor taught him, putting on a show..."

Wayne thought, Keep showing off. In a moment, I'll show you what the story of 1729 is and scare you to death.

At the front of the classroom, Matthew's gaze discreetly swept over Helen.

Unfortunately, not only was Helen not applauding, she hadn't even raised her head.

Matthew frowned slightly.

Then he turned his gaze toward Li Mo, who sat in the back row.

At that moment-

An inexplicable desire to compete rose in his heart.

Matthew suddenly said, "Li Mo, you've missed so many classes and only just come back today. Can you still keep up? If you think these derivations skip too many steps and there are parts you don't understand, that's all right. Would you like me to explain them to you again in private?"

Li Mo: "?"

How can there be someone like this?

I was going to treat you as an ordinary person... but you just had to offer up your face for a slapping?

Li Mo briefly weighed his options.

He needed to establish himself as a genius, and he couldn't miss an opportunity to show his abilities when it had come right to him.

So he slowly stood up. "Thank you for the offer, Matthew. I understood it... But Mr. Harrison, I also have a new idea for this optimization problem. May I come up and work through it?"

Mr. Harrison was momentarily taken aback.

He hadn't expected Li Mo to say that.

As he remembered it, Li Mo was the town's occult genius, but as far as math went, he was only average.

But considering that Li Mo had only just recovered from a serious illness, Mr. Harrison decided to go along with him for now. "Since you have a new approach, come on up."

Seeing this, the students in the classroom grew excited one by one.

Fight! Fight! Fight!

Everyone knew about the conflict between Matthew and Li Mo.

This was practically a gladiatorial arena!

The contest was a math problem!

Whoever had the more elegant solution would win Helen's heart!

Anna was a little worried.

She hurriedly took out a fresh sheet of scratch paper and began thinking hard, desperately trying to recall any algebraic transformations that might prove useful.

The girl's plan was to raise her hand and rush to the front to help Li Mo out if he ran into trouble.

Alongside his concern, Wayne felt a little anticipation.

He hoped Li Mo would once again display his amazing ability, as he had when he saw through the house number 1729, and give Matthew a resounding face slap.

Helen quietly raised her head in the front row and cast a quick glance at Li Mo as he came up the aisle, then swiftly lowered her eyes again as if startled.

Outside the classroom's front door.

The freckled girl had just composed herself and reached the doorway when she heard what Li Mo said.

She stopped, pressed herself against the doorframe, and peeked into the room through the crack.

Li Mo walked to the front.

First, he erased Matthew's equations.

Matthew frowned, wanting to speak but saying nothing.

Immediately afterward,

Li Mo picked up a piece of chalk and began to write.

On the first line, he wrote: S = 2πr²+πrh +πrh.

The students below were all baffled.

Why split the complete 2πrh into two terms? What was the point?

Li Mo continued writing.

He introduced the basic principle of the arithmetic mean-geometric mean inequality: the arithmetic mean of three positive numbers was greater than or equal to their geometric mean.

On the second line, he wrote: S≥ 3 *∛(2πr²*πrh *πrh).

Li Mo combined like terms as he went, making the expression under the radical 2πV².

The third line read: S≥ 3 *∛(2πV²).

At this point, the problem was entirely clear.

Equality held if and only if all the terms were equal, meaning 2πr²=πrh.

Canceling the common terms gave the conclusion: 2r = h.

Li Mo put down the chalk and said softly, "I'm done, Matthew. Did you understand it?"