Three Methods

Shen Qi took the sheet of white paper Cadre Liu handed him. "Sounds good. I'm new here, so I'll start with a problem to calm my nerves."

"You have one hour. I'm going out to eat." Cadre Liu stood up and checked his watch. It was past five, about time for dinner.

"I'm hungry too." The thought of food reminded Shen Qi just how hungry he was.

"You'll be allowed to eat dinner only after you solve this problem within the time limit and get the right answer." Cadre Liu's expression didn't change. He added, "That's the first rule of the Olympic Math national team."

"Cadre Liu, treating a minor like this is illegal!" Shen Qi protested vehemently. Not being allowed to eat if he couldn't solve a problem was so cruel. This was a socialist country governed by law! This was too much to calm his nerves; it was making them worse.

"Shen Qi, you're a gold medalist in math competitions, a National Champion—not an ordinary minor." Cadre Liu didn't soften at all. He pointed to the corners of the office, which was more than twenty square meters. "While solving the problem, you can consult any math reference in this room. Quite a few of them aren't for sale; the Chinese Mathematical Society got them through special channels. You should feel lucky to be immersed in a sea of knowledge."

"I'm a high school student from out of town, here in your great capital with no family and no friends. What can I do? Whatever you say goes, Cadre Liu. You're the big shot, so I can only give in." Shen Qi had no choice. He was the weaker party here.

Cadre Liu left, his back cold and unfeeling.

Shen Qi pulled over a chair and sat down. He spread the sheet of paper on the desk and carefully examined the problem.

The first thing that caught his eye was a simple diagram: a straight line running through an oval, like a flattened Chinese character for "sun."

At the twelve o'clock, three o'clock, and nine o'clock positions on the "sun," the labels y(x), x2, and x1 appeared, respectively.

Below that was the problem statement:

"Based on the diagram above, the unknown function appearing in the integrand makes the integral attain a maximum or minimum. More specifically, what is the mathematical description of the shape that a surface of revolution must have to minimize the resistance of an object moving along its axis at a constant speed?"

The problem wasn't complicated. Anyone could recognize the diagram, and anyone could read the Chinese, but it was safe to assume that more than 99 percent of high school students in China wouldn't have a clue what the hell it was asking them to do.

"Heh. Hehehe." Shen Qi laughed like an idiot.

He was laughing because he'd be able to eat dinner.

The problem wasn't hard. For Shen Qi, it boiled down to one sentence: use mathematical language to describe a simple physical motion.

The person solving it needed a solid foundation in mathematics, while a high school education in physics, including Newton's three laws, was enough.

Shen Qi picked up his pen and wrote down his mathematical description on the A4 sheet.

To put the problem simply, Cadre Liu had casually tossed a rubber ball onto a lake. The ball bounced across the surface, then eventually floated there. The task was to explain this common phenomenon in mathematical terms.

Shen Qi used Euler's brachistochrone theory: replacing the integral with a finite sum and the derivative in the integrand with a difference quotient.

This theory wasn't particularly profound, but it had a wide range of applications. It converted the integral into a function composed of a finite number of coordinates of y(x).

Shen Qi drew an arc on the sheet of paper with a few simple strokes.

This wasn't hard. Any engineering student who'd scored eighty or ninety on Linear Algebra could do it.

Shen Qi continued. By setting the variation of the integral equal to zero and using a rough limiting process to transform the resulting difference equation, he obtained the condition that the minimizing arc had to satisfy.

Shen Qi filled one side of the A4 sheet with his solution. Done.

Only fifteen minutes had passed. Cadre Liu still hadn't returned, and Shen Qi had nothing better to do, so he flipped the sheet over and kept writing.

This time, Shen Qi's method was a little more complicated. Making an integral reach a maximum or minimum didn't necessarily require using difference quotients.

Euler's method was only introductory. Lagrange's approach was more advanced.

Shen Qi sketched a crescent moon on the paper and derived the endpoint condition that a minimizing curve with variable endpoints had to satisfy. The process was rather involved.

After twenty minutes, Shen Qi had produced a second mathematical explanation.

There was still time, so Shen Qi took a fresh sheet of paper and started deriving a third mathematical explanation.

At six twenty, Shen Qi was so hungry he couldn't take it anymore. He put down his pen, unzipped his backpack, and looked for food. "Why hasn't Cadre Liu come back yet? I'll eat a chicken drumstick first."

Shen Qi tore open the sealed package and ate the drumstick as he strolled around the office.

"This really is a sea of knowledge. So many books. Some of these references can't be found anywhere else."

"Opera Omnia, L. Euler... What the heck is this? It's all in English... Wait, I think I know the author. Euler—this is a book by Euler... Damn, I can't read English. I'll find another one."

"Mei kan ni ke... An na lai di ke... Holy crap, this one's even more impressive—a book in French! The author is... Lagrange. Damn, I can't read French either. Next."

"Jeez... There are even references in Russian... Is Cadre Liu a foreign-language expert? There are hardly any math references in Chinese in this room."

Shen Qi was stunned. Cadre Liu's office really was a sea of mathematical knowledge, but there were hardly any books he could read. Nearly all of Cadre Liu's collection consisted of foreign-language references.

Shen Qi was very interested in the reprints of Euler's and Lagrange's mathematical works. These precious references couldn't be found in bookstores or on Taobao; he'd have to order them from overseas.

But no matter how interested he was, being unable to read the languages made it all pointless.

"Should I level up my English?" Shen Qi stood in front of a wall of books, chewing on his braised chicken drumstick as he fell into thought.

Just then, Cadre Liu returned. He saw that Shen Qi wasn't working on the problem but eating a drumstick, and he was extremely displeased. "Shen Qi! Who said you could eat that? You're breaking team rules on your very first day reporting to the national team. Your discipline is terrible!"

"I'm hungry! I'm still growing, Cadre Liu!" Shen Qi protested indignantly.

Cadre Liu spoke with forceful conviction. "Hunger is no excuse. Dying stupid is far more tragic than dying hungry! Throw away the drumstick. You don't eat until you've finished the problem!"

"Tyrant!" Shen Qi shouted. Reluctantly, he tossed the half-eaten drumstick into the trash can and muttered under his breath, "Tyrant... Dark Tyrant..."

"Wait, that's not right! I finished the problem Cadre Liu gave me. Why should I throw away my drumstick?" Shen Qi suddenly realized.

"Why didn't you say so sooner?" Cadre Liu walked over to the desk and bent down to examine the sheet of paper with the problem on it.

"I was starving and lost my mind. I shouldn't have tried to fight the Dark Tyrant at level five." Shen Qi squatted on the floor, sulking. That was a drumstick my dad bought me, full of his love. Damn, I felt bad for Dad.

"What's all this nonsense about the Dark Tyrant?" Cadre Liu frowned, then continued reviewing Shen Qi's solution. "Euler's second-order Ordinary Differential Equations—correct. Coming up with this answer in an hour is solid enough."

"Hmph..." Shen Qi lowered his head and grumbled unhappily.

"Why are there words on the other side too?" Cadre Liu flipped over the sheet. In an instant, he was astonished. "Lagrange's formula for indefinite integrals. Shen Qi... you... you used two methods to solve this problem... in one hour?"

Cadre Liu pressed his left hand to his chest, sweat beading on his forehead.

"Hmph... There's another sheet beside it." Shen Qi squatted on the floor, head bowed, and traced a character in the air with his finger: "miserable."

"You... Shen Qi... you actually know how to use Bernoulli's divergent series... You used three methods to solve this problem, in... in one hour?" Cadre Liu gasped for breath, barely able to speak.

"Who has it worse than me? I'm only allowed to solve problems, not eat." Shen Qi muttered to himself.

Thud!

A muffled sound.

Cadre Liu collapsed to the floor.

"Holy crap?" Shen Qi jerked his head up and saw Cadre Liu lying on the floor, convulsing violently.

"Cadre Liu, what's wrong? You can't die!" Shen Qi rushed to his side, stunned.

Cadre Liu's trembling hand pointed toward a corner of the office. "Quick... go get... the emergency heart pills... in the briefcase..."