pjbooks Chapter 31: Uncle Ge's First Praise!
Chapter 31: Uncle Ge's First Praise!
"stupid!"
Looking at the words appearing on the textbook, Han Chuan twitched the corner of his mouth, not daring to refute them.
He gripped his pen and looked at the question again.
Uncle Ge's words are always sharp. Although he doesn't give you direct answers, clues are often hidden within them. He may seem to be scolding you for being stupid, but he's actually trying to enlighten you and help you break out of your conventional problem-solving mindset.
As he was thinking, a thought suddenly popped into Han Chuan's mind.
"If powers of 10 can construct numbers consisting entirely of 0s and 1s, what kind of construction can produce numbers consisting entirely of odd numbers?"
When an idea pops into his mind, it's like a bamboo shoot breaking through the soil, instantly taking root in his mind.
He picked up a pen and wrote down lines of tentative data in his notebook.
The remainders of powers of 10 modulo 20 follow a pattern: 10^1 ≡ 10 (mod 20), while 10^2 ≡ 0 (mod 20), and all remainders are 0.
This shows that powers of 10 alone are not enough.
But what if we change it to a power of 11?
After pondering for a moment, Han Chuan wrote down another line of data in his notebook: 11^1 ≡ 11 (mod 20), 11^2 ≡ 1 (mod 20).
No... no... the direction is off.
It's not 20, it should be a number modulo some number that keeps each digit odd.
As his thoughts raced through his mind, Han Chuan suddenly remembered a concept Zhou Peiyuan had mentioned to him when he was teaching him electromagnetism: the method of separation of variables.
The core idea of the separation of variables method is to break down a complex partial differential equation into several independent ordinary differential equations, solve them separately, and then combine them.
Simply put, it involves using dashed lines to cut a matrix into horizontal and vertical lines, then establishing a correct coordinate system to separate them. Each line is responsible for changes in only one direction, and they do not interfere with each other.
Coordinate system, separation, independent components...
Han Chuan stared blankly at the equations on the manuscript paper, his eyes gradually losing focus, but his mind was immersed in a state of deep thought.
Thoughts raced like high-speed gears, colliding and intertwining, tightly linking the thinking of the method of separation of variables with the number theory problem before me.
I don't know how much time had passed, but suddenly the equations and questions on the paper in front of me started to jump up.
The characters gradually formed a complete and clear physical picture in his eyes.
This time, what he saw seemed to be no longer a simple mathematical picture, but rather a picture combining mathematics and physics.
The skill of materializing knowledge resurfaced. Before his eyes, a physical image signal was decomposed into several independent frequency components, each vibrating at its own frequency without interfering with each other.
The entire signal is a linear superposition of these components.
Each of them is a decimal digit, each digit is independent, and each line shows a linear superposition...
If viewed within a physics framework, this is equivalent to requiring that every frequency component of a signal must satisfy a certain amplitude condition.
Han Chuan didn't understand what this was yet, but he saw it.
I saw how to construct a multiple of n, and I saw that the overall signal must satisfy a boundary condition.
This is essentially the same thing as the separation of variables method he used in his physics competition class.
The technique of knowledge visualization breaks down the problem into a mathematical and physical image, and then satisfies the constraints on each component individually.
Finally, put them together.
.....
The silvery-blue glow faded, and all the three-dimensional, dynamic physical images disappeared like the receding tide.
Han Chuan suddenly came to his senses, picked up his pen, and began to deduce the problem again on the draft paper.
This time, he did not follow the standard number theory route in mathematics, but instead pioneered a new direction.
He plans to use a combination of mathematics and physics to solve this problem!
Of course, strictly speaking, this number theory problem cannot be solved using physical methods.
But physical thinking can help him find a starting point in mathematics!
What he needs is not to simply copy physical formulas, but to bring the "separation of variables" mindset into his problem-solving approach.
He broke down the condition "each number is odd" that he saw in his mental visualization into "each number independently satisfies a certain modulo condition", and then combined them.
Find the entry point, use the Chinese Remainder Theorem to decompose n into the product of 2^a and the odd part m, construct numbers that satisfy the conditions modulo 2^a and modulo m respectively, and then combine them using the Chinese Remainder Theorem.
In the part modulo m, he constructed a periodic sequence consisting entirely of odd numbers; in the part modulo 2^a, he utilized a clever inductive construction.
The pen tip moved swiftly across the paper, leaving behind neat and dense lines of derivations.
The whole process took less than fifteen minutes, and a large portion of a single page of draft paper was filled with writing.
When he wrote the last symbol, Han Chuan himself was unaware of what he was doing.
Until Ge Jun's handwriting reappeared in the math textbook.
This time, there were no insults or harsh words, but rather a hint of silence and surprise.
"This solution...you came up with it yourself?"
Han Chuan thought for a moment and said, "I wouldn't say that. When Mr. Zhou was teaching me physics before, he mentioned the method of separation of variables, which is to break down a complex system into multiple independent degrees of freedom, solve them separately, and then combine them."
"When I was doing the problem just now, this concept suddenly flashed through my mind, and I felt... this problem can be viewed in this way too."
"Then I gave it a try, and to my surprise, I actually solved it."
Ge Jun's handwriting fell silent. The silence lasted longer than ever before.
The writing was so long that Han Chuan almost thought there was something wrong with him using physics-based thinking to solve a math problem. Just as he was about to explain, the writing reappeared.
"The standard solution to this problem is to construct a periodic sequence consisting entirely of odd numbers and then directly verify it. The process can be written more concisely using mathematical induction."
"But you've created a new route, one that I'm not familiar with."
"You transplanted a physical concept into a number theory construction problem, and found a mathematical entry point using physical thinking."
"This question..."
"You've gotten closer to the mathematical essence of the problem than the standard solution; the problem setter might never have even considered this method."
At this point, the writing paused, then slowly emerged with a heavier stroke than before:
"From the time I started teaching you until now, this is the first time you've come ahead of me in solving a problem and come up with a completely new approach."
"I really didn't expect you to use the separation of variables method from physics when solving number theory problems."
"You solved this problem very well!"
Han Chuan stared at the line of text for a long time, his smile almost reaching his lips.
Emperor Ge actually praised him?!
Holy crap! That was really tough.
This was the first time Ge Jun had praised him in over two months; it felt like...
The emotion lasted less than three seconds, and before he could even think of a suitable metaphor, Ge Jun's handwriting popped up again on the textbook, his pen regaining its familiar sharpness.
"Calm down, you're grinning like an idiot. I praised you because you actually did it, don't get carried away by those two compliments!"
"While the method of separation of variables is ingenious in solving this problem, your construction process contains a set of redundant assumptions. If n contains square factors, the construction of one set of independent solutions is redundant because you failed to check for coprimeness in the second step of merging."
"Fill in this redundancy analysis for me. If you can't, you'll be standing in my math class from now on!"
Han Chuan looked at the words, unable to suppress the smile on his face: "OK, no problem, I've got it covered!"
He replied with a message, then looked back at the calculations on the manuscript and began to think about how to fill in the gaps in his understanding according to Ge Jun's instructions.
Sensing Han Chuan's seriousness, the book spirit Ge Jun smiled as he looked at the textbook.
This kid is really talented!
The problem we just solved would have required at least two pages of standard mathematical solutions, but he finished it in just over half a page.
The entire derivation process is incredibly concise, with each step hitting the nail on the head and no unnecessary calculations.
Of course, the issue of missing coprime checks during the construction of the analytical solution is another matter. But this method, which combines mathematical physics, is indeed ingenious and a problem-solving approach he had never considered before.
Thinking about it, the book spirit Ge Jun fell into deep thought.
If he could still participate in setting college entrance exam questions, perhaps he could try incorporating this approach.
The math problems that combine physics are indeed difficult, but the final questions in the college entrance examination are meant to differentiate students and assess their logical thinking and talent.
......
Meanwhile, on the other side...
Jiangsu Province, Nanjing.
Inside the fully enclosed Education Examination Institute, a group of mathematics teachers and professors gathered from all over Jiangsu Province are currently assembled in a conference room.
The conference room was large, with a long solid wood table covered with dark green velvet, and the curtains were drawn tightly.
A dozen or so people were sitting around the table, each with a thick stack of draft exam papers spread out in front of them.
This is the closed-off area for setting the 2009 Jiangsu Province College Entrance Examination Mathematics paper. It has been closed for nearly two months since the question-setting process began.
As is customary, mid-May is the key window for finalizing the draft.
The initial draft, trial tests, collective refinement, and multi-level review of the college entrance examination papers, including the final determination of Paper A (official paper) and Paper B (backup paper), must all be completed by May 25th at the latest.
Today is May 20th.
In four or five days, the college entrance examination papers for Jiangsu Province will begin to be printed and stored in secrecy, and then transported to various examination centers under armed escort before the exam.
At the other end of the long table in the office, Ge Jun, who was invited to participate in this year's college entrance examination question setting, had a stack of printed drafts spread out in front of him.
He had been staring at the first draft for almost an hour, the pen between his fingers, motionless.
Other members of the question-setting team occasionally exchanged a few whispers, flipped through the papers, scribbled on the edges of the drafts, and then handed them to the team leader for signature.
No one noticed Ge Jun's unusual behavior—he was not talkative to begin with, and when reviewing manuscripts, he was as silent as a stone statue.
But only Ge Jun himself knew that he wasn't silent, but rather confused.
Starting this morning, it felt like something had been pried open in his mind.
At first it was just a vague idea, like an inspiration flashing in the distance that he tried to grasp but it slipped away.
But as I continued to examine the question, that feeling became clearer and more specific.
Now, what appears in his mind is no longer fragments of inspiration, but a complete and rigorous conception of the topic.
Yes, what came to his mind today was the thought process behind a math problem.
He picked up his pen and quickly wrote down the prototype of the problem on the manuscript paper. The derivation process was so smooth that it seemed as if he had already calculated it countless times.
After finishing writing, he put down his pen and stared at the equations on the manuscript paper.
This is a comprehensive problem about the intersection of sequence limits and probability theory, using concepts such as motion, vibration, and energy from physics as the background for mathematical modeling.
It also creatively combines multiple objects, multiple processes, the composition and decomposition of motion, extreme value problems, etc., with mathematical tools to form an extremely ingenious problem.
This problem is extremely difficult to solve using traditional mathematical methods, which combine eigenfunctions and eigenvalue distributions.
However, another solution inexplicably appeared in Ge Jun's mind, which was to solve the problem by combining it with simple harmonic motion from high school physics.
After thinking for a moment, Ge Jun picked up his pen again, found a blank A4 sheet of paper, and wrote down two completely different problem-solving processes.
The first approach is the traditional mathematical problem-solving method, which uses a combination of eigenfunctions and eigenvalue distributions to find the solution.
The solution process took up most of the draft paper.
Another approach is to calculate the extreme values of the displacements after the superposition of simple harmonic motions.
The idea is ingenious, and the solution is elegant. It can be said that as long as you get to the right point, the whole problem-solving process will be as easy and effortless as a knife cutting into butter.
But Ge Jun stared at the problem and the solution steps in front of him for a long time.
It went on for so long that he himself began to doubt himself.
This question is completely different from any of his previous question styles; it is more unconventional and bolder.
At the same time, in a certain key step, he even subtly bypassed his usual rigorous approach and directly grasped the core of the problem from another angle.
However, its mathematical core is rigorous and impeccable, and while it is extremely difficult, it can also challenge the limits of students.
To be honest, he never imagined that math problems could be presented like this.
But it did happen.
This surprised Ge Jun greatly.
It was as if someone had directly implanted a problem-solving approach that he had never tried before into his brain.
"This question could perhaps be included in the exam."
Looking at the test questions in front of him, Ge Jun suddenly had an idea: he wanted to put this question into this year's college entrance examination paper!
Just then, another teacher from the question-setting team, carrying a thermos, walked over, glanced at the draft paper in front of Ge Jun, and asked curiously after seeing the math problems on it.
"Teacher Ge, haven't this year's questions all been confirmed?"
"Why are you still setting questions?"
Upon hearing this, several members of the question-setting team gathered around with interest.
One of them glanced at the title on the manuscript paper and immediately raised his eyebrows.
"Teacher Ge, this question... isn't it a bit too difficult?"