It is generally believed that intelligence consists of such indicators as observation, memory, imagination, judgment, deduction, and logical thinking.
That is why most IQ test questions are related to mathematics, which encompasses all of the above.
Westerners tend to place more importance on logical thinking. There's an old saying in Western academia: "Logic is invincible, because defeating logic also requires the use of another kind of logic."
The organizing committee's intention in setting this logic problem was clear: to achieve good results at the IMO, logical ability was essential, and intelligence was the entry requirement.
Every time Shen Qi leveled up his math, the system would prompt him: "Congratulations, Host. Your math has reached [level]. Your observation, memory, imagination, judgment, deduction, logical thinking, and other abilities in the field of mathematics have improved significantly compared to the previous level."
Shen Qi had raised his math to Level 5, the professional level. Even if he had only studied middle-school math, he would have had a decent chance of solving this entry-level logic problem.
But Level 5 math plus middle-school knowledge wasn't enough to solve problems involving integrals or Differential Equations. Those required knowledge of university-level algebra.
Shen Qi's understanding of the system was that it helped him continually raise his intellectual ceiling, while he had to fill his knowledge base through daily accumulation—by reading books, attending classes, and so on. The two complemented each other. Without enough intelligence, he would struggle to understand profound mathematical theories, no matter how many he read.
Back to the entry-level logic problem in Question 1. (Yesterday, I accidentally left out a few words in the problem in Chapter 46, so the conditions weren't complete. I've since updated it. If you're patient, you can go back and take a look.)
The three conditions Shen Qi deduced from the story in the problem were:
1. Tom, Jerry, and Thomas's numbers were all greater than zero;
2. The three numbers were pairwise unequal;
3. None of the numbers was twice another.
The clues supporting these three conditions were that each man could see the numbers on the other two men's foreheads, but not his own; none of the three men could answer in the first round; in the second round, Tom and Jerry still couldn't deduce their respective numbers, but Thomas, who answered last, gave the correct answer: 144.
Shen Qi imagined himself in Thomas's place. If I arrived at the answer 144 in the second round, then I must have ruled out one of the three conditions above.
If 144 was the difference between Tom's number (x) and Jerry's number (y), then the equation would be x-y=144.
In that case, x and y would both be nonzero, and x wouldn't equal y, so conditions 1 and 2 would hold.
To negate condition 3, he would need another equation: x+y=2y, which gave x=y. That condition didn't hold; otherwise, they could have found the correct answer in the first round. Therefore, Thomas's 144 wasn't the difference between the two numbers, but their sum.
That is, x+y=144.
Similarly, assuming conditions 1 and 2 held, for condition 3 not to hold, x-y=2y.
Combining the two linear equations gave a system of equations:
x+y=144
x-y=2y
Shen Qi could calculate the result in his head: x=108, y=36.
Working backward, Shen Qi replayed the scene in his mind:
Tom had 108 on his forehead, Jerry had 36, and Thomas had 144. In the first round, none of them could guess his own number. In the second round, Thomas, who answered last, gave the answer 144...
"That's it. That's the logic." Shen Qi picked up his pen and wrote 108, 36 on his exam paper.
He'd cleared the entry question and secured seven points.
Now it was time to show what he could do.
Question 2 was a problem in Plane Analytic Geometry.
The intersecting x- and y-axes were old friends to every student. Whether you knew how to use them or not, they remained steadfast in place, bearing witness to changing times and shifting fortunes.
People came and went through the coordinate plane. Mathematicians throughout history had devoted their lives to leaving their names behind in this world of one horizontal and one vertical line.
What appeared before Shen Qi's eyes were two infinity-shaped curves, one large and one small, with the larger encircling the smaller. It had a special name: the Cassini Oval.
Don't think it's useless. If you do, you definitely won't get seven points.
Shen Qi had to find the constant between the two ovals. It couldn't be too large or too small: too large, and things could go wrong; too small, and he might miss the key to solving the problem.
Analytic Geometry combined geometry and algebra. Calculating the constant required geometric methods, and vice versa.
Shen Qi launched a two-lobed curve's attack on the Cassini Oval, but he had clearly underestimated the oval's almost shamelessly stubborn defense.
The Cassini Oval could change in countless ways, displaying different properties depending on the problem setter.
Shen Qi held back his attack. The weapon he'd brought out—the lemniscate, his nunchucks—couldn't take down the monster before him, the Cassini Oval.
It wasn't just that it wouldn't die. The oval wasn't even taking damage.
The Cassini Oval, with its seventy-two transformations, had to have a true form. He needed to find the monster's real body and kill it before he could journey west to obtain the scriptures.
If one move didn't work, he'd try another.
Shen Qi brought out his ultimate combination artifact: the catenary and the cycloid, the strongest duo.
At Shen Qi's current level, the catenary and cycloid were the finest magical artifacts he could forge. He wouldn't use such a devastating move—one that could smite the heavens, the earth, and even the air—unless absolutely necessary, because it consumed too much mana. Use it too often, and his brain couldn't take it.
But there was no helping it. This was the IMO, and Shen Qi had no choice.
Empowered by Shen Qi's magic, the catenary-and-cycloid combination artifact packed a powerful physical attack and an unblockable magical one. Under this combined assault, the Cassini Oval finally revealed a weakness. Its true form emerged: it was nothing more than a mechanical curve.
"You maddening little vixen. Think putting on a bull's hide will turn you into the mighty Bull Demon King? Ha! How naive. Monster, take a blow from Old Shen!"
After plotting the final section of the curve, Shen Qi found the constant b^2 for the Cassini Oval's fixed points and distance.
"Whew, that really made my head hurt. I'm exhausted."
Two and a half hours had passed. After solving two questions, Shen Qi's lips were parched, and his mouth was desperately dry.
"Time for a break. Just a little one."
Shen Qi took a small sip of mineral water to wet his lips. He didn't dare drink too much in case he needed to pee.
Twenty contestants had been assigned to take the exam in this classroom. Shen Qi was seated in the back row. He looked around at the others; most were staring into space, looking utterly hopeless.
A small national flag had been placed on each contestant's desk—the flag of their respective country.
Shen Qi noticed that only a handful weren't staring into space. They were the contestants from the United States, the Soviet Union, and Kazakhstan.
"Are these guys really Americans?" Shen Qi noticed that the American contestant seated diagonally ahead of him on the left had darker skin and curly black hair, unmistakably South Asian features. He was most likely of Indian descent.
"The deputy team leader was right. America digs up talent from everywhere. They just take what they can get." Shen Qi knew that the American math team was strong, a formidable rival to the Chinese team. Indians were pretty good at math, too, and deserved to be taken seriously.
He looked at the two handsome contestants from Russia and Kazakhstan. Both were white, and the Russian guy stood out in particular. He was probably left-handed, writing solutions at high speed with his left hand.
Left-handed people were generally smarter, so he deserved to be taken seriously. If Russia and Kazakhstan hadn't split up, their Soviet Union or CIS math team might have been the best in the world. The Chinese team would have been challenging them, not defending its title.
Shen Qi felt the pressure. They were all formidable—every one of them!
He wanted to win the team championship, but he wanted the individual IMO title even more.
The Chinese math team was strong overall, but in one-on-one competition, Shen Qi wasn't sure he could beat the Russian guy, or the Indian or other foreign-born contestants on the American team. It seemed there were also Chinese Americans on this year's U.S. team.
Shen Qi didn't dare let his guard down. After a short break, he immediately turned to Question 3.
Author's Note:
Some readers have said they don't quite understand the mathematical theories in this book.
I'm writing a novel, and the mathematical concepts I bring up in the main text are mostly the most concise parts of each theory. If I explained them in detail in the story, it would inevitably interrupt the flow.
My original intention was to describe some basic academic subjects in an interesting, engaging way. I definitely don't want to turn this book into an academic paper. I could write about how to calculate a circle's diameter, or which side sin is equal to divided by which other side, and so on—but I'm sure you wouldn't be too keen on reading that.
My knowledge is limited, and mistakes are bound to crop up as I write. Some of my explanations of certain theories may be biased, so please feel free to criticize and correct me. I welcome your valuable suggestions.
When I refer to a theory, I'll do my best to list its source at the end of the chapter. Readers who are interested can look it up themselves.
The references for the problems in this chapter are:
IQ Test Question Bank
<High School Mathematics Textbook, Compulsory Course>
University textbook Analytic Geometry
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