The strength of a Star Scholar primarily stems from three aspects: innate talent, star arrays, and various calculation abilities honed through training.
While square roots and exponents are among the simplest calculations—hardly challenging for professionals who make their living through mathematics—mentally calculating the square root of 1369 (which is 37 squared) requires more than just familiarity with numbers; there are no real shortcuts.
To young Star Scholars, someone who memorizes 37 squared (Note 1) would naturally retain more basic operation results, forming the foundation for enhanced computational abilities. Even in the 21st century, prodigious mental calculators first commit numerous common formulas and numerical results to memory before demonstrating astonishing calculation skills.
For a twelve-year-old in the 16th century, Cheng Jinzhou's performance clearly exceeded his tutors' expectations. Little did they know, he was deducing the third digit using Pythagorean triples—had he not recently studied Euclid's Elements, his numerical sensitivity would have paled compared to mathematics students preparing for exams.
Regardless, Liu Kuang developed considerable fondness for the slender, frail-looking youth who displayed both the "bearing" of a Star Scholar and commendable "diligence." His voice softened as he remarked, "You solved that quickly. That calculation kit truly deserves to be your gift."
With Cheng's genuine ability confirmed, the gift was naturally bestowed. Mrs. Cheng's face lit up instantly.
"Let me ask you another question," Liu Kuang continued. With a casual wave of his hand, he drew a perfect parabola on paper several meters away. "Have you seen this shape before?"
Cheng marveled at the Star Scholar's miraculous skill while facing a dilemma. In an era of widespread mathematical literacy, any large inverted-U shape would immediately register as a parabola rather than, say, a Mitsubishi-modified letter U. But in this time period, recognizing parabolas was genuinely uncommon.
After a pause, Cheng answered hesitantly, "I have."
"You've clearly read extensively—aristocratic youths are indeed different." Though Liu Kuang's words were measured, everyone heard the undisguised admiration. The Cheng family did possess a respectable library, though its mathematics and physics sections were notably sparse.
"We call this a parabola. If we add coordinate axes here at (0,0)..." To everyone's surprise, Liu Kuang launched into an explanation of coordinate systems, speaking rapidly but keeping the content elementary.
For this world, coordinate systems represented cutting-edge knowledge. Mastering them signaled the emergence of analytic geometry, which could solve problems traditional geometry couldn't—particularly in quantifying solutions, often more crucial than qualitative analysis in practical applications.
Perhaps due to Star Scholars' needs, their development in this area surpassed Cheng's expectations. Even low-level apprentices seemed proficient with coordinates.
After three minutes, Liu Kuang turned to Cheng: "Given this, can you find where this line intersecting the vertex coordinates meets the parabola's focus?"
A basic question, yet one requiring equation knowledge—in Cheng's original world, even linear equations took centuries to develop.
Several Star Scholar disciples watched intently. Beginners facing coordinate problems typically became utterly confused, much like Cheng's ancestors who, despite basic education, never grasped such concepts.
Feeling the peculiar satisfaction of being underestimated despite superior knowledge, Cheng theatrically tapped the paper before casually answering, "25, 25." Only upon speaking did he realize the old man's casually drawn parabola was remarkably precise.
The Cheng family exchanged astonished glances. The sickly child they remembered bore little resemblance to the boy before them—though time travel never crossed their minds.
Liu Kuang continued questioning impassively, soon touching upon simple functions—all child's play for Cheng. Elementary mathematics was second nature to a 21st-century PhD candidate; he hadn't felt this confident since undergraduate days.
Their exchange gradually shifted from examination to discussion. Initially overexcited and unfamiliar with the Daxia Dynasty's mathematical progress, Cheng made several startling remarks—though within acceptable bounds, given Star Scholars' broad standards.
Time flew by, including a meal break. By nightfall, the Chengs again invited Liu Kuang to dinner, which he declined. Despite evident satisfaction and abundant praise, he neither accepted Cheng as a disciple nor made any valuable offers—dashing Cheng's hopes of replicating typical transmigrator success stories.
Even Mrs. Cheng couldn't hide her disappointment, repeatedly saying, "Don't worry, you're still young"—though twelve was hardly young for aspiring Star Scholars, given their selective apprenticeship practices.
After dining in the main hall with mixed emotions, everyone retired. Under the hazy glow of red lanterns along the stone path, Mr. Cheng walked beside his son, hands clasped behind his back—unusually companionable.
Pleased more than surprised by the afternoon's events, he broke the silence: "I hadn't realized your interest in mathematics. It's a viable path—if you wish, I could arrange for colleagues to send you to Xiangcheng or the capital." (Xiangcheng being the south's premier city.)
Though tempted, Cheng quickly declined: "A filial child doesn't travel far while parents live. Perhaps later." His mathematical knowledge had little connection to this era's education anyway.
Perhaps disarmed by the classical quote, Mr. Cheng ignored his son's lazy tone and nodded: "Starting now, report to my study every three days for evaluation."
Cheng's face fell as if beaten. "But your official duties—"
"Decided," Mr. Cheng declared with patriarchal finality. Seeing his mother couldn't intervene either, Cheng compromised: "I need provisions for experiments."
"Mathematics is proper study; leave other matters be." Mr. Cheng shifted topics: "No waste. Discuss specifics with your mother."
Like traditional Chinese scholars, Mr. Cheng cared little for finances beyond basic needs.
After consideration, Mrs. Cheng said, "From now on, our country estate will deliver an extra 50 dan of grain monthly at midmonth. Inform the steward of your requirements beforehand."
"Fine." After a long day, Cheng couldn't decide whether to feel disappointed or satisfied.
Note 1: 37 is a prime number—primes' squares and cubes are commonly memorized.