Arithmetic as the Trailblazer?

Great Chu Kingdom, Liyao City, Xiaoxiang Academy.

Though Xiaoxiang Academy was a rare academy for women, its reputation had recently soared in Liyao City and even throughout the entire human race.

That was because Xiaoxiang Academy had been the first to discover the Mind-Fixing Divine Ability and had promptly reported it to the military. As a result, they had won the great victory at Two Worlds Pass.

With that victory, the Chicken and Rabbit in a Cage Divine Ability had also become famous among the hundred races.

From then on, arithmetic became another part of the military's examinations. Anyone who was bad at it would only drag the others down.

Then, when the divine ability was updated again, it threw open the curtain on the current craze of "getting hooked on solving problems to improve one's cultivation." Around that same time, the divine ability also began making its way into countless households.

As the academy that had possibly been the first to discover and promote the idea that "solving problems on a large scale can steadily improve one's cultivation," Xiaoxiang Academy became a frequent topic of conversation among the people, and their words were filled with gratitude.

In one of Xiaoxiang Academy's pavilions, the academy's dean, Sun Xuehong, was enjoying tea and discussing the Way with the academy's arithmetic master, Gu Qiuhui. The two were chatting happily.

At one point, Sun Xuehong suddenly furrowed her brow and asked:

"Qiuhui, do you think that Chicken and Rabbit in a Cage Divine Ability is really that difficult?

"I heard that this time, masters from three countries and the hundred schools of thought gathered together—even grandmasters—and still couldn't solve it."

Even now, Dean Sun Xuehong still couldn't understand how such a small arithmetic problem could stump so many people.

As a Confucian master, she hadn't specialized in arithmetic, but she had studied it before. Weren't those arithmetic problems quite simple?

How could everyone say they couldn't solve it in a short time?

Hearing this, Gu Qiuhui took a sip of tea. After a long pause, she said softly:

"Sister Xuehong, the problem itself isn't especially difficult. It's just that we've never paid much attention to arithmetic, and hardly anyone has made a dedicated study of it. That's why this problem seems impossible to solve.

"In fact, even now, we still don't have a general method for solving 'Chicken and Rabbit in a Cage.' Each method only works for a certain type of problem.

"For example, some methods only work when the total number of legs is even, while others require it to be a multiple of three."

"How could that be?"

Hearing that there was no general method, Dean Sun looked up in astonishment at Master Gu across from her.

"And you're saying that even the School of Mathematics doesn't study it? Then what are all those mathematical classics about?"

The School of Mathematics didn't study arithmetic?

That was certainly beyond Dean Sun's expectations. Wasn't arithmetic part of mathematics?

"Sister Xuehong, it's not that we don't study it at all. We just haven't studied it systematically.

"Mathematicians serve as astronomers, calendar officials, historians, and diviners.

"Did you know that the School of Mathematics is divided into six branches: the School of Astronomy, the School of Calendrics, the School of Yin-Yang and Five Elements, the School of Divination, the School of Miscellaneous Divination, and the School of Geomancy?"

After a pause, Master Gu continued:

"The School of Astronomy studies the stars, calculates the movements of the sun, moon, and stars, and observes the weather.

"The School of Calendrics focuses on dividing the year into four seasons, setting the solar terms, and calculating the movements of the sun, moon, and stars to keep track of time and dates. It also studies imperial chronologies.

"The School of Yin-Yang and Five Elements studies their changes and makes deductions. The School of Divination studies turtle-shell and yarrow-stalk divination.

"The School of Miscellaneous Divination reads omens of good and ill fortune from the signs of various things. Its practices include dream interpretation, praying for blessings, exorcising demons, and praying for rain.

"The School of Geomancy studies the surveying of terrain and geography, physiognomy, and similar arts."

"You see? There's no branch devoted specifically to arithmetic.

"As for the mathematical classics and tables that appeared in the past, those only came about because a few people happened to be interested in arithmetic.

"Or you could say that arithmetic has been passed down within our School of Mathematics only sporadically. It has never truly formed a unified branch.

"That's why we've been so passive this time.

"Take me, for example. I belong to the School of Astronomy.

"If you ask me about the movement of the stars, I'm fairly knowledgeable. But when it comes to arithmetic, I only know the basics."

Hearing this, Dean Sun let out a long sigh. Her hope of perfectly replicating the Chicken and Rabbit in a Cage Divine Ability in a short time had been thoroughly extinguished.

She hadn't expected that no one had studied this.

It seemed she would have to agree to the government's request.

She had come to see Gu Qiuhui, thinking this master of mathematics could lower the cost of replicating the divine ability and help Xiaoxiang Academy steal the spotlight once more.

But from what Gu Qiuhui had said, no one in the world could solve this problem in a short time. She had no choice but to abandon the idea.

Perhaps Anonymous could do it.

But that person was too well hidden, and they couldn't find him, so they could only act as though he didn't exist.

As for whether the problem would ever be solved, that would depend on how much effort went into arithmetic.

Sun then told Gu Qiuhui all of this, so she wouldn't keep wearing herself out trying to crack the Chicken and Rabbit in a Cage problem.

"By the way, Qiuhui, do you think we should train some students specifically to study arithmetic?"

Actually, Dean Sun had first had this idea half a month ago. With the arithmetic divine ability growing popular, arithmetic was becoming more and more important.

But back then, she hadn't known that the School of Mathematics had no branch devoted specifically to arithmetic. She had assumed students could simply study it under the existing schools, so she hadn't been in much of a hurry.

But judging by what Gu Qiuhui had just said, hardly anyone in the School of Mathematics studied arithmetic. So although Sun Xuehong thought arithmetic must be important, she didn't know whether vigorously developing it was the right thing to do.

After she finished speaking, Dean Sun looked anxiously at Gu Qiuhui across from her, hoping to hear her thoughts. This was a major decision, and she couldn't make up her mind on her own. She needed the opinion of a specialist like Gu Qiuhui.

"Are you saying you want to cultivate a new school?"

Gu Qiuhui understood what Sun Xuehong meant. A group devoted to studying arithmetic would amount to a new branch of the School of Mathematics.

"Mm."

Sun Xuehong nodded, then furrowed her brow. "But I don't know whether doing this might ruin the students' prospects. No one has ever done it before, so everything is uncertain.

"And we don't even know whether a new school could survive."

Hearing this, Gu Qiuhui didn't rush to answer. The concerns Sun Xuehong had raised were real.

With a new path, no one knew whether it was a road full of thorns, a smooth highway, or a dead end.

If it succeeded, its pioneers would be honored and admired by countless people. But if they failed, they could ruin someone's whole life.

Xiaoxiang Academy couldn't treat its students' futures as a gamble.

This really did call for caution, caution, and still more caution.

After weighing it over for a long while, Gu Qiuhui finally spoke:

"How about this? I'll first arrange a few more arithmetic classes with Xia Qing and the others. That way, we can find the students who are especially interested in arithmetic.

"As for whether they want to make a dedicated study of it afterward, we'll leave that up to them. We won't interfere too much.

"Of course, we'll explain the pros and cons clearly and let them choose for themselves."

As she said this, a young woman named Jiang Zichun came to Gu Qiuhui's mind.

She had heard that Jiang Zichun used to be especially fond of practicing martial arts, but recently she had become fascinated with arithmetic and hadn't missed a single arithmetic class.

In Gu Qiuhui's eyes, Jiang Zichun would be an excellent candidate. She just didn't know whether she would choose to specialize in arithmetic.

"Of course. We definitely can't force the students."

Sun Xuehong quietly weighed Gu Qiuhui's suggestion for a while. It seemed like a good approach, and not a reckless one, so she agreed.

"Mm, then let's start with that. We really can't rush this. Whether they stick with it will depend on what they want."

A moment later, Sun Xuehong asked, "By the way, Qiuhui, how are things going with the other arithmetic problems you've been studying?"

As soon as she finished speaking, Gu Qiuhui smiled. "Sister Xuehong, we've already made progress.

"We've found solutions to the 'Hundred Chickens Problem,' also called 'A Hundred Coins for a Hundred Birds,' as well as a similar problem called 'A Hundred Monks Share a Hundred Buns.' Once we have enough problems, we can upload them directly in a couple of days."

In fact, Gu Qiuhui's team could already upload the Hundred Chickens Problem. They just didn't have enough problems yet, so they needed to wait a few more days.

They needed a large pool of problems to steadily improve cultivation over a long period—and to sell more.

"That quickly?"

Hearing that Gu Qiuhui and the others had already solved one problem, Sun Xuehong was filled with surprise. She'd been told they were difficult to solve, so how had they finished this one so quickly?

Recently, she had been keeping an eye on the Heavenly Dao Virtual Network every day. Other arithmetic divine abilities had indeed appeared, but there were very few problems for each one—only a few dozen or so, which was hardly useful.

Gu Qiuhui said they could upload theirs in a few days. By then, they would have at least eight or ten thousand problems. So Master Gu's speed really took her by surprise.

Still, it was a pleasant surprise.

But Gu Qiuhui's expression turned awkward. "Actually, we still haven't found a quick way to solve this problem ourselves.

"As for how we did it so quickly, we have to thank Master Anonymous.

"The Hundred Chickens Problem is a classic example of a problem with multiple solutions.

"And recently, that master came up with some problems that also have multiple solutions. There's even a function for generating problems, and it provides the answers too. So we used that function to crack the Hundred Chickens Problem."

"You used the problem-generation function to crack it? Aren't those two different problems?"

Sun Xuehong was confused.

Yes, they both involved problems with multiple solutions, but they were clearly different problems. How could you use one to solve the other?

Had she fallen behind?

Seeing Sun Xuehong's confusion, Gu Qiuhui patiently explained:

"Sister Xuehong, let me give you an example. Then you'll understand."

"Let's use the classic Hundred Chickens Problem as an example."

"Suppose one rooster costs five coins, one hen costs three coins, and three chicks cost one coin. With a hundred coins, you must buy a hundred chickens. How many roosters, hens, and chicks should you buy?"

"This problem has three solutions. They are:"

"Four roosters, costing twenty coins; eighteen hens, costing fifty-four coins; and seventy-eight chicks, costing twenty-six coins."

"Eight roosters, costing forty coins; eleven hens, costing thirty-three coins; and eighty-one chicks, costing twenty-seven coins."

"Twelve roosters, costing sixty coins; four hens, costing twelve coins; and eighty-four chicks, costing twenty-eight coins."

"Now, let's use the Chicken and Rabbit in a Cage Divine Ability's problem-generation function to solve it."

"First, we replace roosters, hens, and chicks with three kinds of beasts: Beast A, Beast B, and Beast C. Then we treat the number of chickens as 'heads' and the price of each chicken as 'legs.' That gives us:"

"Suppose there are beasts of Type A, each with one head and five legs; beasts of Type B, each with one head and three legs; and beasts of Type C, each with one head and one-third of a leg. There are one hundred heads above and one hundred legs below. How many beasts of each type are there?"

"Since fractions appear here, we multiply everything by three to get rid of them. That gives us:"

"Suppose there are beasts of Type A, each with three heads and fifteen legs; beasts of Type B, each with three heads and nine legs; and beasts of Type C, each with three heads and one leg. There are now three hundred heads above and three hundred legs below. How many beasts of each type are there?"

"Now we can solve it using the divine ability. The answers we get in the end are exactly the same.

"We tried several other problems as well, plugging in the answers and checking them. They all worked."

Hearing this, Sun Xuehong's eyes suddenly lit up. You could do it this way?

And it was so simple?

She then came up with several problems of her own and tested them, only to discover that it really worked.

So the Chicken and Rabbit in a Cage problem and the Hundred Chickens Problem were actually one and the same?

That was truly beyond her expectations.

"No wonder you said you'd have ten thousand problems ready in a few days."

"I only discovered it by chance. If we hadn't done this, and had relied on solving them ourselves, we wouldn't have been able to produce enough problems, never mind whether our answers were correct or complete."

"Exactly! These problems with multiple solutions really take a lot of effort. You can't come up with more than a few in a day. It looks like Xiaoxiang Academy is going to take the lead again."

The two exchanged a smile. They had no doubt about it.

They had both been at Xiaoxiang Academy for decades and had long since come to see it as a second home. The more famous and powerful the academy became, the happier and prouder they were.

A moment later, Sun Xuehong seemed to think of something. She asked excitedly, "Then could other problems, like 'The Unknown Quantity,' be solved this way too?"

If that were possible, Xiaoxiang Academy would have what it took to establish an entirely new school.

Only a wealthy household could afford to talk big.

But Gu Qiuhui shook her head. "That one's a different type of problem. I haven't figured out what to substitute for it yet.

"Besides, 'The Unknown Quantity' currently has infinitely many solutions. I don't even know how to write out the answers, so there's no hope for it for now."

"Infinitely many solutions? Then it really isn't very suitable."

Dean Sun narrowed her eyes. This really was quite different from the Hundred Chickens Problem.

The Hundred Chickens Problem had a finite number of solutions, while the Unknown Quantity had infinitely many.

How were they supposed to write down infinitely many answers?