The provincial Mathematical Olympiad competition is still held in this largest examination center in the province.

When Qin Ke got up, he felt something was wrong with his body. He secretly touched his forehead. Damn it, it was even hotter!

I only had a low fever in the morning, but after eating bread and milk and squinting for half an hour in the car, it turned into a high fever!

"Qin Ke, why don't you get out of the car?" Ning Qingyun asked from outside the car. She, Lao Zheng, and the middle-aged female teacher had already gotten out of the car.

Qin Ke took a deep breath, forced himself up, rubbed his face vigorously, and jumped out of the car with energy.

"Hey, I'm so anxious, I'll go to the examination room first and look for the restroom."

Qin Ke originally wanted to encourage Ning Qingyun, but as soon as he spoke, he found that his throat was a little hoarse and painful, so he stopped talking. He carried a document bag containing documents and stationery, put on a down jacket hat, and lazily walked

He waved his hand and strode towards the inspection office.

His steps were a bit limp, so he gritted his teeth and stepped hard. With his usually good physical condition, he didn't show anything strange.

Lao Zheng shook his head from behind and sighed: "Qin Ke, this kid, has a great attitude. He doesn't look nervous at all. I hope he can perform well this time."

Ning Qingyun looked at Qin Ke's figure, but vaguely felt that something was wrong.

"By the way, Student Ning, remember not to be reckless with the two additional questions. If you have no clue, give up quickly and focus on the main questions in front of you..." At this time, the middle-aged female teacher next to her was still talking.

Test taking skills.

When Ning Qingyun saw that Qin Ke had gone far, she quickly interrupted the middle-aged female teacher: "Teacher Liang, Teacher Zheng, I'm going to the examination room first. Goodbye!"

She hurriedly said goodbye to the two teachers and quickly chased Qin Ke: "Qin Ke, wait for me!"

Of course, Qin Ke heard Ning Qingyun's shout. It was really rare for this tender-skinned girl to shout such words on such an occasion, which proved that the relationship between the two was indeed different from before.

Qin Ke was a little pleased and a little proud, but he didn't look back or stop. He just said silently in his heart: "Come on, school committee, the hard work and sweat you have put in these days cannot be in vain."

He quickly completed the inspection and then sped into the examination building. He seemed to hear Ning Qingyun's voice behind him...

Watching Qin Ke and Ning Qingyun enter the examination room one after another, the middle-aged female teacher couldn't help but ask Lao Zheng next to her: "Mr. Zheng, I heard that this provincial competition is more difficult, how did they two review?

"

"have no idea."

"I don't know?" The middle-aged female teacher is a mathematics teacher in the third year of high school. Just because she is a woman, Vice Principal Wen considered that it would be easier to take care of Ning Qingyun, a girl, in case of any accidents, so she temporarily transferred her to

In the accompanying team.

Naturally, she didn't know much about Qin Ke and Ning Qingyun's exam preparations.

"I only know that they have worked very hard." Lao Zheng picked out a cigarette from the cigarette case, put it to his mouth and picked it up: "For us teachers, their hard work is the best answer.

It doesn’t matter what the outcome is.”

The middle-aged female teacher looked at his calm and elegant appearance, and opened her mouth. In the end, she could only say: "Teacher Zheng, you are absolutely right."

Lao Zheng was saying pretentious words, but he was more nervous than anyone else in his heart.

"Qin Ke, Qin Ke, I have staked my future on you. Whether I can get the year-end bonus depends on you. You have to bring me back a top five certificate..."

…

In this Mathematical Olympiad semi-finals, each city only has five to ten participating places, and the total number of candidates in the whole province is only about 200.

The examination rooms are much more arranged, with an average of only twenty people sitting in each examination room. Candidates from the same city are separated into different examination rooms. Qin Ke does not have to worry about being in the same examination room as Ning Qingyun. Naturally,

Chen Hanyin and Hong Xingwei, the two candidates who did not meet Chengkong.

After he entered the examination room, he lay down for a while, and the examination began soon.

After the three invigilators read out the rules of the examination room, they began to distribute the test papers. Qin Ke glanced at him. None of the three invigilators knew him. He didn't know if the three invigilators saw his name and deliberately avoided it.

go.

But Qin Ke didn't have the time to think about this. His brain was buzzing and he felt like he was rusty. His thinking ability was less than 70% of the usual level. Moreover, his body felt more and more afraid of the cold, and his hands became colder and colder.

Qin Ke tried hard to keep his brain awake, but he knew that his cold was getting worse and his current state would not last long. It would probably get worse over time, so he had to hurry up and answer the questions.

He flipped through the main paper and the additional paper. As Lao Zheng said, the main paper has ten big questions, each worth 20 points, and the additional paper has two big questions, each worth 50 points.

Qin Ke had already decided on his exam strategy when he was lying down at the table before the exam started, which was to take advantage of his current condition to solve the two most difficult additional questions of the national competition level first, and then do the provincial exam main paper.

Even if the situation becomes worse by then, I should still be able to cope with the problem.

He shook his head and focused on the first additional question.

"Additional Question 1: The graph formed by n points and several edges on the plane is not a Hamiltonian graph, but if one point and the edges connected to it are removed arbitrarily, the remaining graph is a Hamiltonian graph. Find the minimum value of n

.”

Qin Ke gasped. It was indeed a difficult national competition, and it was a Hamilton picture.

It is estimated that 99% of high school students across the country have never noticed the name Hamilton.

Hamilton was a famous British mathematician in the 18th century. At that time, he proposed a game called "Around the World", using the twenty vertices of a regular dodecahedron to represent twenty big cities, and asked him to move along the edges from

Starting from a city, passing through each city only once, and then returning to the starting point, this is the famous "Hamiltonian problem".

Later, the mathematical community called "a cycle that passes through each vertex on the graph once and only once" a "Hamiltonian cycle". If a graph contains a Hamiltonian cycle, then the graph can be called a "Hamiltonian graph".

On the surface, this Hamiltonian problem seems to be similar to Euler's Seven Bridges of Königsberg problem (the Seven Bridges of Königsberg problem means that there are two islands in the river, and there are seven bridges on the river connecting the two islands and both sides of the river.

, is it possible to cross each bridge once and only once. It is also known as the "one-stroke" problem) is very similar, but there is an essential difference between the two.

The Seven Bridges of Königsberg problem has been solved by Euler himself, thus creating a new branch of mathematics - "graph theory".

The Hamiltonian problem has not been solved so far. For more than a hundred years, countless first-rate mathematicians have tried hard but have not found the necessary and sufficient conditions to judge it. They have only proposed some proven necessary and sufficient conditions and applied them to it.

different occasions.

The difficulty of this question is that it not only requires the solver to understand the characteristics of the Hamiltonian graph and those proven necessary and sufficient conditions, but also to be able to use them flexibly.

As soon as Qin Ke saw this question, he knew that Ning Qingyun couldn't answer it - because of limited time, he only explained two examples to Ning Qingyun about the Hamiltonian diagram, which was not in-depth. Ning Qingyun's answer to Hamilton

It is impossible to answer the question without understanding the diagram.

Not only Ning Qingyun, but probably no one else in the entire examination room besides him could answer the question.

Qin Ke rubbed his slightly swollen temples and pondered for more than three minutes before starting to write:

"Solution: First of all, the degree of each point is at least 3. Otherwise, there is a point a that only connects to at most two sides. Then after one of them is removed, the remaining point a must not be on a certain circle. This is inconsistent with the conditions, so it can

It turns out that n≥3..."

"When n=4..."

"..."

"The condition is only true when n=10, so the answer to this question is 10. The specific diagram is as follows:"

Qin Ke drew a regular pentagon with a "one-stroke" pentagram in the middle. Each vertex of the pentagram is connected to the vertices of the pentagon surrounding it.

This is the graph that best fits the meaning of the question when n=10. Remove any point and the edges connected to it, and the remaining graph is a Hamiltonian graph.
Chapter completed!