Exploring Immortality Cultivation
Chapter 433 - 186: An Eternal Conundrum? Too Crazy [Another Triple Release, Seeking Monthly Votes and Recommendations]

"Demon Emperor Taiyi had cows in Fusang, divided into four colors: black, white, brown, and yellow, and they were differentiated by gender. Discussing the female cows, the number of white cows is equal to the number of brown cows plus half the number of black cows, further increased by a third of the number of black cows. The number of black cows is a quarter of the number of yellow cows, plus a fifth of the number of yellow cows, then added to the number of brown cows. The number of yellow cows is one-sixth of the number of white cows, further increased by one-seventh of the number of white cows, then added to the number of brown cows. Discussing the male cows, the number of white bulls is a third of the number of black bulls, plus a fourth of the number of black bulls. The number of black bulls is a quarter of the number of yellow bulls, plus a fifth of the number of yellow bulls. The number of brown bulls is a sixth of the number of white bulls, plus a seventh of the number of white bulls. Thus, I ask about the number of the Demon Emperor's cows, how many are there? Additionally, if the arrangement of black and white bulls exactly forms a rectangle, and the arrangement of brown and yellow bulls can form a triangle, then, how many are there of each color and gender?"

Sage Zhi Long slowly presented the problem and looked at Wang Qi with a hint of triumph. When he saw the dumbstruck expression on Wang Qi's face, he became somewhat worried.

He had high hopes for this junior, not only because he had waited fifty thousand years to find such a suitable successor but also because Wang Qi possessed the inheritance of the Mathematical Sect and the Mathematical God Ring.

This was fate, indescribable fate.

What if he could not answer it? Should he provide a hint?

As Sage Zhi Long thought this, Wang Qi only felt the absurdity.

Was this... an ancient riddle?

Alright, it was indeed a difficult problem. The problem of Archimedes' cattle, a challenging issue that even the great mathematician Archimedes had studied for a long time without solving.

However, even the most difficult problems are limited by their eras. Before the invention of calculus, irregular shapes could only be measured using crude approximate methods, a tedious and inaccurate practice; but with the advent of calculus, such world problems became mere regular exercises. Going back even earlier, during the era when the method of substitution had not been invented, a system of two equations was a problem that could make great mathematicians scratch their heads in frustration. There was once a mathematician on Earth who resented another for learning his method of solving equations and accused him of witchcraft at a religious tribunal.

Advancements in mathematical tools and 'Seeking Dao Artifacts' gradually reduced the difficulty of these problems.

The reason elementary school math contests can stump university professors is also due to restrictions on the use of mathematical tools, forbidding the use of equations or calculus, deliberately making a simple question difficult.

"The biggest issue with this problem is the amount of computation..." sighed Wang Qi, directly giving the answer, "This question has an infinite number of solutions, the smallest solution to the first question is 5,916,837,175,686 heads, and the smallest solution to the second question has more than 206,500 digits alone; it would take several hours to state the full answer verbally. Elder, we should probably write this down..."

"Impossible! This is impossible!" Zhi Long's expression turned to one of terror, as though he had seen the most inconceivable thing in the world.

He... he actually... he actually just gave the answer?

"Surely someone must have solved this during these fifty thousand years, and you memorized it, didn't you?" Zhi Long found an excuse, trying to calm himself. But Wang Qi sensed the elder was somewhat blustering. He laid out several equations on the ground: "To show the sage, during the Near Ancient times, my Wanfa Sect's Li Sect made a breakthrough and acquired a new study called the 'Heavenly Element Formation,' derived from the ancient Arithmeticians' elementary algebra, but advancing further..."

"Elementary algebra?" doubted Zhi Long.

"It refers to the algebra developed by Middle Ages Mathematicians, distinct from the new algebra continuously developed by generations of my Wanfa Sect predecessors, called elementary. By the way, Li Sect also absorbed this content." Wang Qi continued to explain, "Here we can set up a system of four equations of the Heavenly Element Group..."

Zhi Long felt that things were beyond his expectations. He hurriedly interjected, "Wait, how could Li Sect absorb the essence of Wanfa?"

Algebra and number theory are naturally fundamental to the Li Sect.

Wang Qi had to explain, "There was once a predecessor who wished to unify the Li Sect and Wanfa, creating a way called the 'Heavenly Position Skill'..."

The so-called Heavenly Position Skill naturally refers to the basis of the Heavenly Position Technique, analytic geometry.

Analytic Geometry appeared very early, and Sage Zhi Long had no difficulty understanding it. However, because he could understand it, his amazement was even greater, "This... is unparalleled exquisite method! This is the Dao! This is the Dao!"

Seeing Sage Zhi Long dancing with excitement, Wang Qi felt somewhat distressed. The gap between the two was not one of personal skills, but a gap in civilizational development. Excluding the Near Ancient period of fifty thousand years, their difference spanned two thousand years in the history of mathematical development!

To confront the Wanfa Sect's millennia of talent single-handedly was difficult, difficult, difficult!

In a blink, Wang Qi listed several eight-variable linear equation systems. He then listed the formulas for square numbers and triangular numbers, solving two systems of quadratic equations.

"The Heavenly Position Skill is based on Euler's five postulates, requiring eight mutually perpendicular axes to calculate the eight-variable Heavenly Element Group..."

Sage Zhi Long was going crazy, "Eight... How could there possibly be eight straight lines that are mutually perpendicular and intersect at a point?"

"Oh, this is an extension called 'Xiangyu', by Euler in the 'Taiyu'. Ordinary space only has up-down, left-right, front-back, these three sets of directions, known as the 'Three Phases'. The 'Xiangyu', however, has infinitely many 'phases'..."

"What use is that?"

"Uh... Well, it's generally useless on a normal scale, but it's essential for studying the microscopic world or complex systems..."

The more questions Sage Zhi Long asked, the more frantic his expression became. But by the time Wang Qi solved the problem, he had calmed down.

The Great Middle Ages Mathematician sighed long, "The younger generation is truly fearsome, truly fearsome!"

"Great River... ahem, Great River's waves push the waves ahead, each wave stronger than the last," Wang Qi said earnestly, "That's a local expression from my hometown."

Sage Zhi Long laughed heartily, "What a 'each wave stronger than the last'! What a great statement! So flavorful!" Immediately after, he stared eagerly at Wang Qi, "I still have a few questions... no, not testing you. I know I can no longer test you. I just want to know if these problems have been solved in future generations."

Wang Qi nodded, "Alright."

"Using only compass and unmarked ruler construction, is it possible to solve the duplication of the cube, squaring the circle, and trisecting an angle?"

Compass and ruler constructions, also known as those using a compass and an unmarked ruler on Earth. Duplication of the cube means to construct a cube with a volume twice that of a given cube; squaring the circle means constructing a square with area equal to a given circle's area; trisecting an angle means dividing a given arbitrary angle into three equal parts.

Wang Qi shook his head, "They are all impossible. First, squaring the circle involves a number, π, the ratio of the circumference of a circle to its diameter. We can prove with arithmetic methods that π is a transcendental number—what a transcendental number is, you can ask later, but for now, I'll just say that it's absolutely not possible to do with just a compass and ruler... The solution for the cube's volume can be a bit hard to explain. First, we need to mention a few concepts. These concepts originate from the general methods of High Phase Celestial Element Form... This problem can be translated into a group theory problem..."

Galois Theory, one of the critical theories in group theory.

Sage Zhen Chan, who attended the course for over half a year more than Sage Zhi Long, found his receptivity less than that of this Middle Ages Mathematician. He found the whole scene somewhat absurd.

So much for the trial, it seemed... to have turned into a lecture?

And it was the trialist giving lessons to the trialer?

This... was too crazy! (To be continued. If you like this piece, please support it by voting for it at qidian.com, your support is my greatest motivation. Mobile users, please visit m.qidian.com to continue reading.)

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