A Variation on Euler's Seven Bridges?

July 3, morning, Second Teaching Building, Shuimu University.

The Second Teaching Building was an old structure, built before 1949.

Rumor had it that paranormal events often occurred there. People commonly said it was haunted.

The building was never open at night. The university claimed it was to save water and electricity.

An elite research institution like Shuimu University naturally championed science. In the face of its mighty power, demons and spirits could only be crushed. Since the founding of the country, nothing was allowed to become a spirit.

By this time of year, more than half the university students had finished their final exams, leaving the Second Teaching Building empty. It was the venue for this year's National Preliminary and National Finals of the High School Mathematics League.

The National Preliminary would begin at precisely nine o'clock this morning.

With half an hour to go, all 192 contestants from 32 provinces and cities were on standby in Rooms 201 and 202. Sixty of them, from ten teams, would qualify for the National Finals.

"Honestly, I'm already really happy just to be here at the national competition. I mean it," said Qi Jianhong, the youngest member of Nanyue Province's Math Olympiad team.

Shen Qi scolded him. "Qi Jianhong, what the hell do you know? If we don't make the National Finals, we might as well have never come to Shuimu University. You think you're on one of the five powerhouse teams—E, Xiang, Zhe, Jing, or Hu—where making the provincial team gets you a contract with Shuimu University or Yanjing University?"

Qi Jianhong was unconvinced. "I don't need a guaranteed admission. I can get into Shuimu or Yanda through the college entrance exam, all right?"

Shen Qi asked, "Are you good at your other subjects?"

Qi Jianhong said smugly, "Of course. At my high school, I've been number one overall in my grade for ages. If I'm not at least fifty points ahead of second place, I'll admit defeat. Math is just a hobby. My unbeatable all-around ability is what lets me dominate the entire grade."

"You're quite the little tough guy, aren't you? But I don't care how awesome you are at your school. You'd better give the National Preliminary everything you've got, or I'll beat you up," Shen Qi declared.

"You, you... How barbaric!" Qi Jianhong was only a little over five feet tall and couldn't fight to save his life. He stood no chance against Shen Qi, who was nearly six feet tall.

"So what if I beat you up? Ever since training began, I've kept telling you that we're a team, that we're from the same place and share the same dream. I can't beat all five of you by myself, but I've got a champion's heart. And you lot are all trash." After saying this, Shen Qi took his stationery set and left Room 202 without looking back.

"Shen Qi, you... You actually called us trash!" Qi Jianhong was furious and embarrassed. He turned to his four teammates. "Shen Qi called us trash. How awful!"

"Hmph. Shen Qi's such a show-off."

"We're not trash. We're strong!"

"Shen Qi, you bastard! I'll beat you one day!"

"I'll score higher than you in the National Preliminary!"

"Plus one!"

Qi Jianhong and the other four were united against Shen Qi. For the first time, an unprecedented sense of solidarity filled the team.

Shen Qi, Qi Jianhong, and the other four had been assigned to six different rooms. At five to nine, the proctors began reading out the rules: "The rules are simple. You have three hours to complete the National Preliminary paper. Don't look around while answering. Raise your hand if you need anything. That concludes the rules. We'll now hand out the papers and scratch paper."

Shen Qi was taking the exam in Room 101. He got his National Preliminary paper and quickly scanned the three questions. Each was worth seven points, for a total of 21.

From the provincial preliminaries to the provincial semifinals and then the National Preliminary, the total marks kept decreasing, but the difficulty kept increasing.

The first question had a diagram on the paper.

Two islands floated in a river, with bridges connecting the islands to each other and to the riverbanks.

There was one river, two islands, and eight bridges in all.

The question asked: How could a pedestrian cross all eight bridges exactly once, without missing or repeating any, and end up back at the starting point?

"Hey, who wrote this question? Does Euler allow this?"

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

Euler, a god-tier master of mathematics, had been full of energy in his youth. He loved math, and he loved women, too.

In his twenties, Euler fell in love with a beautiful, gentle art teacher. He pursued her passionately, and the two eventually married and had thirteen children... As you could see, Euler was not only a top scholar but also in fantastic shape.

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

Being late came naturally to women. He waited and waited, and an hour passed, but the art teacher still hadn't shown up.

Euler was bored, so he started studying mathematics. He noticed that two islands floated in a river in Königsberg's park, connected to the riverbanks by seven bridges. Visitors crossed the bridges to stroll on the islands and wander between them.

An idea suddenly came to Euler. He wondered whether there was a route that would let someone start anywhere, cross all seven bridges without missing or repeating any, and end up back where they started.

Euler later wrote up this idea in a paper and submitted it to the Saint Petersburg Academy of Sciences. The paper was titled "The Seven Bridges of Königsberg." Later generations also called it "Euler's Seven Bridges problem."

Later, Euler disproved his own conjecture, proving that no such route could exist.

To prove himself wrong, Euler devised a new method of proof and opened up a new branch of mathematics: geometric topology.

That was the outlook of a top mathematician: I'm unbeatable. I have no opponents. My only opponent is myself. To defeat myself, I'll create a new branch of mathematics.

Two or three hundred years later, Shen Qi faced a new problem: the Eight Bridges problem.

The original Euler's Seven Bridges problem had no solution. As for whether an equivalent route existed for the Eight Bridges, he'd have to work it out.

Shen Qi calculated this way and that, but half an hour passed without an answer!

Like the Seven Bridges, did the Eight Bridges have no route that could cross every bridge exactly once, without missing or repeating any, and return to the starting point?

"This is the national competition, after all. Topology is a tough subject. I can't find the route, and I can't prove it doesn't exist either."

Shen Qi put down his pen and ruler and pressed hard on his temples. This was a terrible start, a terrible start indeed.

Time ticked away, and Shen Qi still couldn't put pen to paper. He had a bit of a compulsion: he had to solve the first question before moving on to the other two.

"Euler, Seven Bridges, Eight Bridges... Right! Why do I have to use Euler's theory to solve a variation based on Euler's Seven Bridges? This is a trap, a dead end!"

Shen Qi suddenly understood. I've got it! Poincare's Network Theory!

If the two endpoints are connected just as they were before, that is an allowed topological operation.

Otherwise, it isn't allowed!

Exactly! The odd vertices in this Eight Bridges graph are at the two ends, so there's no way to connect them and cross every bridge exactly once, without missing or repeating any.

The answer was: IMPOSSIBLE!

Shen Qi hurriedly wrote out the proof. It took him just three minutes to finish, but he'd spent a full hour thinking it through.

"Whew... Seven points in the bag. On to the next question." Shen Qi let out a long breath. He'd burned through plenty of brain cells and was exhausted. But the battle had only just begun. He couldn't let up. He had to finish every question within the allotted time and make sure every answer was absolutely correct.

Even so, Shen Qi didn't know whether he'd ultimately reach his goal. He hoped those five useless teammates would make him proud!