Chapter 16

Gu Lu noticed that Zhou Haihong had been really depressed in recent days.

I don’t even have the heart to practice my favorite Baduanjin.

Gu Lu also had the good sense not to touch Zhou Haihong's bad luck, and just let him wait for a while.

It’s Wednesday again.

Gu Lu was on duty studying at night as usual.

Gu Lu was the only one left in the math team's office under the darkness.

Gu Lu had just turned on his computer, ready to watch a movie and relax, when he saw Bi Qi knocking on the door and poking his head in.

"Classmate Bi Qi, what's wrong?" Gu Lu looked at Bi Qi with a smile.

Bi Qi walked up to Gu Lu with long strides, handed a test paper to Gu Lu, and said, "Teacher, I have already completed the question you gave me!"

Gu Lu picked up the test paper full of formulas and glanced at it briefly, "Well, not bad, the answer is correct."

Bi Qi looked happy.

In order to solve this problem, he put a lot of effort into it.

"Well, teacher, I have something else to trouble you with." Bi Qi rubbed his hands.

Gu Lu raised his eyelids and said, "Just tell me, I won't bother you.":

Bi Qi took out a piece of paper and put it in front of Gu Lu, "I want to ask the teacher for the solution to this question."

On the piece of paper, a question was written with a carbon pen.

Gu Lu looked at:

[Question: Dye each small square in 33X33 graph paper with one of three colors so that the number of small squares in each color is equal. If the colors of two adjacent small squares are different, then they are called

The common edge is the "separating edge". Try to find the minimum number of separating edges?]

The topic is very short.

But the difficulty is unusually high.

Gu Lu raised his head and glanced at Bi Qi, "Where did you get this question from?"

"The competition tutor gave us homework after class." Bi Qi answered honestly.

"Homework?" Gu Lu chuckled, "Your tutor really trusts you. There are no such difficult questions in the math league."

"Then...even Teacher, you can't solve it?" Bi Qi looked hesitant.

He has already asked many people for advice on this topic.

But all the replies given to him were helpless.

He had no choice but to find Gu Lu.

If even Teacher Gu can't solve it...

"No, no, no." Gu Lu smiled, "Biqi, I'm afraid you misunderstood what I meant. Although this question is considered to be at the top of the pyramid of difficulty in high schools, for me, I can only say that it is just that.

That’s all.”

That's all?

Bi Qi was stunned when he saw Gu Lu taking out a carbon pen from the pen holder on his desk.

Gu Lu found a piece of draft paper on the table and casually turned a pen around, "I'll write it down and talk about it again, but you must pay attention and listen."

Bi Qi nodded subconsciously, and then reacted, "Teacher, don't you need to do the calculation first?"

Gu Lu shook his head, "That's not necessary."

"Concentrate, I'm going to start."

This question is a typical graph theory problem in combinatorial arrangement, which can be regarded as a branch of topology.

Geometric topology happened to be one of Gu Lu's several areas of interest.

Therefore, this kind of question does not require Gu Lu to think carefully at all.

Gu Lu slowly spoke, "Find the minimum number of dividing edges. The easiest thing to think of is that the grids of three colors are gathered in three areas of the entire graph paper. In this way, the following will appear.

Two situations."

Gu Lu drew two pictures on the draft paper.

Figure 1

Figure 2

"It can be clearly seen that the three-color vertical partition arrangement in Figure 1 has 66 dividing lines, while the three-color T-shaped arrangement in Figure 2 has 56 dividing lines."

Bi Qi said, "Teacher, I have also calculated the number 56, but the key point is that I don't know how to prove that this is the 'minimum number of dividing lines'."

Indeed, the number 56 is only obtained through conjecture, and there is no rigorous proof process.

Gu Lu waved his hand, "Don't worry, just listen to me and explain slowly."

He wrote down rows of formulas on the scratch paper and slowly said, "Suppose the number of dividing lines is L, and the following is to prove that L ≥ 56. Record the rows of the graph paper from top to bottom as A1, A2, A3...

, the columns are in order from left to right

Denote it as B1, B2, B3... The number of colors appearing in the square in row Ai is denoted as n(Ai), and the number of colors appearing in the square in column Bi is denoted as n(Bi). The three colors are denoted as c1, respectively.

c2,c3…”

"...define δ(Bi,cj), then ∑(n(Ai)n(Bi))=∑∑(δ(Ai,cj)δ(Bi,cj))=∑∑(δ(Ai,cj)

δ(Bi,cj))=∑n(cj), due to..."

Gu Lu explained every step in detail.

Bi Qi's eyes were bright and he listened very seriously.

"...To sum up, the minimum number of separating edges is equal to 56!"

Gu Lu finished writing the last stroke and rubbed his sore wrist.

"Do you understand?"

"I understand." Bi Qi nodded repeatedly, looking at Gu Lu with eyes full of admiration.

"Teacher, you are really amazing. I thought about a question that I couldn't come up with all day long, and you came up with it in just a few minutes!"

Bi Qi obviously wasn't good at flattering him yet, but it already made Gu Lu feel comfortable and narrowed his eyes.

He glanced sideways at Bi Qi and said, "By the way, Bi Qi, let me ask you, what do you think of your competition tutor?"

"You mean Teacher Gao?" Bi Qi raised his head and thought carefully, "Teacher Gao's teaching is not bad, you can say that he is conscientious of his duties."

Bi Qi gave a relatively fair evaluation.

"That's it." Gu Lu raised his eyebrows and said with a smile, "What about you, student Bi Qi? After this period of training, do you think it is possible for you to achieve that 'small goal'?"

"Well..." Bi Qi scratched his head in embarrassment, "It should be no problem to join the provincial team, but the national team is still more difficult."

"Teacher, don't talk about me, let's talk about you. Teacher, what were the 'small goals' you set back then?" Bi Qi changed his words.

He knew that the age gap between Gu Lu and himself was very small, and he must have participated in the National High School Mathematics League back then.

"My small goal is relatively ordinary. I just want to get the IMO gold medal."

IMO? Gold medal?!

The prerequisite for participating in the IMO (International Mathematical Olympiad) is to be selected for the national team. Then you will compete with mathematical geniuses from all over the world, and the best among them will win the gold medal.

Bi Qi swallowed hard, "Teacher, have you achieved your small goal back then?"

Gu Lu didn't say anything, just opened the drawer and threw three golden plaques on the table.

Bi Qi recognized it at a glance. It was a special IMO gold medal.

And it’s still three dollars!

Bi Qi almost knelt down on the spot.

Three IMO gold medals, doesn’t that mean that Teacher Gu has been...

What kind of magical power is this!

Bi Qi suddenly hugged Gu Lu's thigh.

Gu Lu was startled and shook it hard, but did not shake it away.

"What are you doing?" Gu Lu held his forehead, his face full of helplessness.

"Teacher, let me follow you!" Bi Qi shouted loudly.
Chapter completed!