The Classic Modeling Problem of Merchants and Attendants

Looking at the two people in the classroom, Professor Liu Xiangping was slightly taken aback. Smiling, he asked, "Only two people? I heard there was another one."

"...I'll call and ask her." Wang Xiaodong looked embarrassed as he took his phone outside.

The teacher was in no hurry. He unscrewed his thermos, took a sip of hot water, then slowly walked over to Lu Zhou and asked with a smile, "What are the roles in your team?"

Lu Zhou put down his phone and answered with a smile, "I'm responsible for modeling, Wang Xiaodong handles programming, and Lin Yuxiang is responsible for writing the paper and the defense."

"Modeling, programming, writing—the standard lineup." Teacher Liu smiled and continued, "Come to think of it, you're Lu Zhou from the thirteenth cohort, right? I read your paper. It really was a good one."

Lu Zhou's eyes lit up. "You work in Mathematics too, Teacher?"

"More or less, though my research leans toward Physics, so I rarely publish mathematics papers." Teacher Liu narrowed his eyes, glanced toward the classroom door, and continued with a smile, "This female teammate of yours doesn't seem very reliable."

Lu Zhou gave an awkward yet polite smile.

Teacher Liu smiled as well and did not dwell on the matter. After a brief pause, he continued, "As for your team's arrangement, I suggest you make a few adjustments. One person should handle both modeling and the paper, while the other two handle programming. The advantage is that the person responsible for modeling will have a stronger grasp of the problem as a whole, making the writing more coherent. Of course, there is a downside: your burden will be heavy. The competition only lasts three days, which means that after finishing the model, you must immediately begin writing the paper."

Lu Zhou considered it. "...Two people handling programming?"

Teacher Liu took a sip of tea and said leisurely, "That's right."

Lu Zhou smiled but did not answer immediately. "I'll think about it some more. After all, I need to discuss it with my teammates."

Just then, hurried footsteps came from the hallway. Their final teammate had finally arrived.

"Sorry, I overslept a little. I'm really, really sorry!" Walking into the classroom with Wang Xiaodong, Lin Yuxiang bowed her head apologetically to the teacher.

"It's fine. Since everyone's here, let's get started." Teacher Liu Xiangping smiled amiably and did not blame her for being late. He turned and walked to the podium, inserted a USB drive into the computer, turned on the projector, and displayed his PPT.

"Before we begin, I want you to look at an example problem. The problem itself isn't difficult, but it can serve as a Mathematical Modeling case. Through this example, I hope you'll first understand what Mathematical Modeling really is."

As he spoke, he clicked the mouse and turned the PPT to the next page.

"Three merchants, each with one attendant, must cross a river by boat. A small boat can carry only two people. The attendants have secretly agreed that on either bank, once the number of attendants exceeds the number of merchants, they will kill the merchants and seize their goods. However, the merchants hold the authority over how the boat is used. How can the merchants cross the river safely?"

Indeed, the problem was not difficult at all.

Even without relying on the system, Lu Zhou quickly came up with the answer.

"First trip: two attendants cross, and one attendant returns."

"Second trip: another two attendants cross, and one attendant returns."

"Third trip: two merchants cross, then one attendant and one merchant return."

"Fourth trip: two merchants cross, and one attendant returns."

"Fifth trip: two attendants cross, and one attendant returns."

"Sixth trip: the final two attendants cross. Everyone makes it safely!"

Clap clap clap! Lin Yuxiang softly applauded, her face full of admiration.

Wang Xiaodong's expression did not change. He looked like some detached master.

In his view, the problem truly was not difficult. Although he had not bothered to work it out, he trusted his own intelligence. At most, he would need a little time to solve it himself.

"Completely correct." Teacher Liu smiled and continued, "Even without using any mathematical knowledge, this problem can be solved through logical analysis alone. But what if we extend the problem to N merchants?"

The problem was indeed somewhat difficult, but not because of the mathematics. The difficulty lay in how to abstract it into a mathematical problem and solve it.

Lu Zhou thought seriously for a while. A rough idea had already formed in his mind.

"May I use the blackboard?"

"Of course." Professor Liu Xiangping smiled and made an inviting gesture.

Lu Zhou walked forward, picked up a piece of chalk, and began writing on the blackboard.

"That's amazing..." Lin Yuxiang stared blankly at the writing on the blackboard. As she watched Lu Zhou walk down from the podium, she opened her mouth slightly and asked in surprise, "You didn't even need scratch work?"

"I did it in my head." Lu Zhou smiled.

Wang Xiaodong remained as silent as ever and said nothing.

However, judging by his expression, this proud top student had finally been completely convinced of Lu Zhou's mathematical ability.

Looking at the work on the blackboard, Professor Liu Xiangping smiled and nodded. "Completely correct! However, from the perspective of Mathematical Modeling, this only counts as half the work completed. After building the model, we still need a program to implement it. Of course, this model is simple. With Wang's ability, I'm sure a program of this level would be effortless, so we won't waste precious time on it here."

Wang Xiaodong adjusted his glasses, his expression still indifferent, as though that was only natural.

Although he was a little disappointed that he had no chance to demonstrate his programming skills on the spot.

After a brief pause, Teacher Liu continued, "Data modeling is essentially the process of simplifying a real-world problem into one that can be expressed through data and concise language, then solving it with mathematical tools."

"Therefore, the key is: what real-world problem is your team facing?"

"Modeling tests neither a team's programming ability nor its mathematical ability. It tests the ability to translate real-world problems into the language of data and solve them. You must remember that."

"As for modeling itself, I do not have much advice. Other than practicing more and broadening your knowledge, there are no shortcuts. As for programming, I can say a few things."

"There are four commonly used software packages: MATLAB, Mathematica, LINGO, and SAS. You do not need to master all of them. Mastering at least one is enough. The key is being able to use the software that suits you to solve real-world problems." As he said this, Professor Liu Xiangping looked at Wang Xiaodong.

Because those words were mainly meant for him.

Wang Xiaodong adjusted his glasses and nodded. "I've used MATLAB and SAS. It won't be a problem."

Professor Liu nodded. "Knowing those two software packages is basically enough. You'll need to learn these things on your own. All I can do is provide guidance when you don't understand something."

"Lastly, regarding reference books, I recommend reading books with Mathematical Modeling cases, such as Mathematical Models, Mathematical Modeling Algorithms and Applications, and Fundamentals of Mathematical Modeling."

"I also have some internal school lecture notes and materials here. You can copy them and take them back to read, but remember not to distribute them or upload them online. Understood?"

"Understood!" the three of them answered in unison.

Professor Liu smiled. "All right, that's it for today. I have nothing else to say. It mainly depends on yourselves. Those who need the materials can download them from me. Once you're done, hurry back and study English."