A variation of the Seven Bridges of Königsberg?

On the morning of July 3rd, at the second teaching building of Shuimu University.

The second teaching building is an old structure, built before 1949.

Rumor has it that strange events often occur in the second teaching building; it's commonly known to be haunted.

The second teaching building is never open at night, which the school claims is to save on electricity and water.

Shuimu University, as a top research and academic institution, naturally reveres science. Ghosts and monsters are destined to be crushed by the mighty power of science; they are not allowed to take form after the founding of the People's Republic.

At this time of year, over half of the university students have completed their final exams, leaving the second teaching building empty. This place is the venue for this year's High School Mathematics League national preliminary and final rounds.

At precisely 9 AM today, the national preliminary round will commence.

With half an hour left until the start, a total of 192 contestants from 32 provinces and cities are waiting in rooms 201 and 202. Ten teams, totaling 60 people, will qualify for the national finals.

"Honestly, I'm already very satisfied just to be able to participate in the national competition, really." said Qi Jianhong, the youngest member of the Nanyue Province Math Olympiad team.

Shen Qi retorted, "Qi Jianhong, you know nothing. If we don't make it to the national finals, it's like we came to Shuimu University for nothing. Do you think you're a member of the 'Five Hegemons' – E, Xiang, Zhe, Jing, Hu – that just getting into the provincial team guarantees you admission to Shuimu University or Yanjing University?"

Qi Jianhong retorted, "What if I don't rely on recommendations? I can get into Shuimu or Yanda through the college entrance examination."

Shen Qi asked, "Are your grades in other subjects very good?"

Qi Jianhong said proudly, "Of course. In our high school, my total score has consistently been first in the entire grade. If I'm not ahead by more than 50 points, I consider it a loss. Math is just my hobby; my dominance over the entire grade comes from my invincible comprehensive strength."

"You little cutie are quite impressive, but I don't care how amazing you are in your high school. For this national preliminary, you must perform at your absolute best, or I'll beat you," Shen Qi said resolutely.

"You, you... how barbaric!" Qi Jianhong, barely over 1.5 meters tall and lacking any strength, certainly couldn't beat the 1.7-meter-tall Shen Qi.

"What good would beating you do? Since the training camp started, I've repeatedly emphasized that we are a team, from the same place, with a shared dream. I can't beat all five of you alone, but I have a champion's heart. As for you all, you're trash." After speaking, Shen Qi grabbed his stationery set and left classroom 202 without looking back.

"Shen Qi, you... you actually called us trash!" Qi Jianhong felt both ashamed and angry. He turned to his four teammates and said, "Shen Qi called us trash, how hateful!"

"Hmph, Shen Qi is such a show-off."

"We're not trash, we're strong!"

"Shen Qi, you jerk, I'm going to defeat you!"

"In the national preliminary, I'll definitely score higher than you!"

"+1!"

An unprecedented atmosphere of unity and shared hatred against a common enemy emerged among Qi Jianhong and the other four members of the team.

Shen Qi, Qi Jianhong, and the other six were assigned to different classrooms. At 5 minutes to 9, the invigilators began reading the competition rules: "The rules are simple. You have 3 hours to complete the national preliminary exam paper. You are not allowed to look around during the exam. Raise your hand if you need anything. The competition rules have been announced. Now, we will begin distributing the exam papers and scratch paper."

Shen Qi was taking the exam in classroom 101. He received the national preliminary paper and quickly scanned the three problems. Each problem was worth seven points, for a total of 21 points on the paper.

From the provincial preliminary to the provincial semi-final, and then to the national preliminary, the point value of the paper decreased, while the difficulty increased.

The first problem featured a diagram on the paper.

A river flowed with two small islands, connected by bridges. Bridges also connected the islands to the riverbanks.

In total, there was one river, two islands, and eight bridges.

Question: How can a pedestrian cross all eight bridges exactly once, without repetition or omission, and return to the starting point?

"Heh, who designed this question? Did Euler allow you to do this?"

Shen Qi saw through it immediately. This was a variation of the "Euler's Seven Bridges" problem, Shuimu's Eight Bridges?

Euler, a legendary master in mathematical history, was energetic in his youth. He loved mathematics and women.

In his twenties, Euler fell in love with a woman, a beautiful and gentle art teacher. He pursued her passionately, and they eventually married and had 13 children... This shows that Euler was not only academically top-tier but also possessed remarkable vitality.

On a bright spring day in 1736, Euler was waiting for his art teacher girlfriend in a park in Königsberg.

Being late is a woman's inherent trait. He waited and waited, and an hour passed, but the art teacher had still not arrived.

Euler was bored and began studying mathematics. He noticed a river in the Königsberg park with two islands, connected by seven bridges. Tourists walked across the bridges to relax on the islands and travel between them.

Euler suddenly had an inspiration. He proposed a hypothesis: could there be a path that allowed one to cross all seven bridges exactly once, without omission or repetition, and return to the starting point, regardless of where they began?

Later, Euler wrote this hypothesis into a paper and submitted it to the Saint Petersburg Academy of Sciences. The paper was titled "The Seven Bridges of Königsberg." Later generations also referred to it as the "Euler's Seven Bridges problem."

Even later, Euler himself refuted this hypothesis, proving that such a path could not exist.

To prove himself wrong, Euler invented a new method of proof, establishing a new branch of mathematics – Topology.

This is the caliber of a top mathematician: "I am invincible, I have no rivals; my only opponent is myself. To defeat myself, I founded a new branch of mathematics."

Two to three hundred years have passed, and Shen Qi is facing a new problem: the eight bridges.

The original Euler's Seven Bridges problem had no solution. As for whether an eight-bridge path existed, it required calculation.

Shen Qi calculated up, calculated down, calculated left, calculated right. Half an hour passed, and he couldn't figure it out!

Was the eight-bridge problem, like the seven-bridge problem, simply impossible to solve, with no path that could cross each bridge without omission or repetition and return to the starting point?

"After all, this is a national competition. Topology is very difficult to handle. I can't find this path, nor can I prove its non-existence."

Shen Qi put down his pen and ruler, pressing hard on his temples. A bad start, a bad start indeed.

Time ticked by, and Shen Qi couldn't write. He had a bit of a compulsion; he had to solve the first problem before tackling the other two.

"Euler, Seven Bridges, Eight Bridges... Right, why must I use Euler's theory to solve a variation of Euler's Seven Bridges problem? This is a trap, a vicious cycle!"

Shen Qi suddenly realized, I've got it, I've got it, Poincare's Network Theory!

If the two endpoints connect in exactly the same way as before, then this is a permissible topological operation.

Otherwise, it is not allowed!

That's right, the singularities of this eight-bridge graph are at the two ends, so this kind of connection doesn't exist, allowing for every bridge to be crossed without omission or repetition.

The answer to this question is: It! Does! Not! Exist!

Shen Qi furiously wrote down the proof process. He finished the proof in just 3 minutes, but his thinking process took a full hour.

"Hoo... 7 points secured, next question." Shen Qi let out a long sigh, having burned through a lot of brain cells. He was exhausted. But the battle had just begun; he couldn't slack off. He had to complete all the questions within the time limit and ensure absolute correctness.

Even so, Shen Qi didn't know if his goal could ultimately be achieved. I hope those five pig teammates can pull their weight!