Because the apartment only had two bedrooms, Tony's mom unceremoniously kicked him out of his own room.
"Girls need privacy," she said. "Lina's still young—she shouldn't have to sleep in the living room."
And just like that, Tony found himself banished to the couch.
With a quilt covering his legs and his laptop balanced on his lap, Tony sat on the sofa later that night, quietly typing away. He had just installed a fresh LaTeX environment, and now, he was finally getting started on his next project: writing a research paper.
Since the paper was going to be submitted to an international journal, there was no reason to write it in his native language. He dove directly into English.
The title at the top of the document read:
"A Dichotomy for the Weierstrass-Type Functions."
This was a real paper—a paper that, in the timeline of the life simulator, would be published in Inventiones Mathematicae many years from now by another researcher. But here Tony was, drafting it years in advance.
Abstract.
For a real analytic periodic function φ: ℝ → ℝ, an integer b ≥ 2, and λ ∈ (1/b, 1)…
As Tony typed with laser focus, the sound of soft footsteps approached. Lina, now freshly showered and in pajama shorts, leaned her head over his shoulder and squinted at the screen.
"All in English?" she muttered. "What's an… abstract? And… weier-something function?"
She could read well for a first-year high school student, but this was beyond her.
Tony kept typing without looking up. "It's a mathematics paper."
"But you're a biology major, aren't you? Why are you writing a math paper?" Lina asked, half-suspicious. It almost looked like he was making stuff up just to sound impressive.
"You don't have to be a math major to do math," Tony said calmly. "And besides, biology needs a lot of math too."
Lina frowned at the word "dichotomy" in the title. "What does 'dichotomy' mean?"
"It means a division into two parts," Tony explained, finally pausing to glance at her. "It's a pretty common word. Haven't you learned that yet?"
"I'm in ninth grade, genius," Lina said. "We're still doing sets and simple functions. We're not solving evil alien equations yet."
Tony chuckled. "Fair point."
He paused for a moment and then added, "The Weierstrass function is something famous in mathematical analysis. It's a kind of function that's continuous everywhere but nowhere differentiable."
"Huh?"
"In simple terms," he said, switching to a gentler tone, "it's smooth with no breaks, but you can't draw a straight tangent line anywhere on it. Think of it like a squiggly line that never settles down."
Lina stared at him blankly.
"Never mind," Tony said. "I forgot how early you are in high school."
She nodded solemnly. "Yeah, next time I'll just stay in the bedroom and scroll on my phone."
Tony smiled, then returned to his screen. He had just finished the abstract and was beginning the introduction.
Lina tilted her head. "You sure you're still my cousin?"
Tony looked up again. "What?"
"You weren't like this last year. When you came home last Christmas, you were still bumbling your way through CET-4. Now you're writing research papers in English? What happened?"
Tony gave a half-smile. "People change. You'll see."
Then, as if remembering something, he added, "Don't worry—tomorrow I'm taking you out for a proper tour. No laptops. Just food, museums, and fun."
Just then, their mom—technically his mom—finished cleaning up in the kitchen and called from behind them.
"Tony, don't bury your face in that computer all day. Your cousin is here—spend some time with her. You're always goofing off when it matters and pretending to be busy when you shouldn't be."
Tony didn't even respond. He was long used to these "accusations." The truth was, this wasn't "pretending to be busy"—the paper he was working on might end up in a T0-tier mathematics journal.
Even so, he had to face reality. Writing a research paper wasn't easy—not even with the help of future memories. Tony still needed to consult dozens of references to reconstruct the proof structure properly. He had barely scratched the surface, and after hours of work, the introduction still wasn't done.
The next day, true to his word, Tony took Lina out for the whole day. They visited two museums, walked a dozen city blocks, and ate out for both lunch and dinner. Lina looked completely delighted.
Tony's wallet, however, was not. He'd made some money tutoring earlier in the semester, but after treating Lina like a princess for two meals, he could already feel the burn.
On the third day, it rained.
Going out wasn't an option, so they stayed inside. Tony curled up with his laptop again, tapping away at the next section of his paper.
"Hey, how do you solve this?" Lina asked, wandering over with her workbook.
Tony looked up.
"Again?"
"Yes," she said defensively. "It's not like I can go anywhere. And you're using the laptop, so I might as well get my homework done."
She handed him her notebook.
Given the function f(x) = 2|x + 1| + ax, x ∈ ℝ.
Prove:
(1) f(x) is increasing on ℝ when a > 2.
(2) If f(x) has two zeros, find the range of a.
Tony skimmed the problem, then raised an eyebrow. "You're actually stuck on this?"
"I really am," Lina said with wide eyes.
Tony flipped back a few pages in her notebook. A few multiple-choice questions. Some fill-in-the-blanks. He frowned.
"You got two of the five multiple-choice questions wrong. Two out of five blanks wrong. One blank you didn't even try. Who are you saving it for—me?"
"I didn't know that one," Lina pouted. She looked genuinely ashamed now, the usual fire drained from her tone.
Tony sighed. He'd done some tutoring before, and he could see the signs—Lina wasn't lazy, just overwhelmed. Her confidence was fragile.
He softened his tone. "Math is all about practice. Do enough problems, and you'll start to see patterns. But you've got to focus—and stop making careless mistakes."
He pointed to a question in her book. "Like this one. It's asking for the relationship between two circles:
x² + y² - 1 = 0 and
x² + y² - 4x + 2y - 4 = 0.
You just need to complete the squares to find the center and radius of both circles."
He walked her through the problem step by step, simplifying the expressions, comparing distances, explaining intersections.
Lina nodded, eyes wide. This was the Tony she remembered—smart, patient, and strangely kind when it came to explaining difficult concepts.
When they first reunited, she had teased him—"What if I don't know anything at all?" At the time, Tony had laughed it off.
Now, looking at her workbook, he realized she hadn't been kidding. At least in math, Lina was dangerously close to not understanding anything.