“Morning, sleepyhead,” émile called over his shoulder, slipping the second slice of toast onto the egg to create Isabelle’s favorite breakfast sandwich. “Egg toast sandwich, as requested. What’s on the agenda today?”
Isabelle flopped onto a chair at the dining table, rubbing her eyes. “Tough luck, old man. Mr. Kotomine’s trying to kill us with math homework. That’s my agenda.”
émile chuckled, bringing her plate over and setting it in front of her. “Ah, math. The great equalizer. You’ll thank him someday, you know.”
“Sure, maybe after I rise from the grave he’s digging for me,” Isabelle muttered, biting into her sandwich. “What kind of teacher assigns derivatives to teenagers? We were barely surviving Algebra last year.”
“Ah, les joies de l'éducation,” émile teased, his French accent playfully exaggerated. “Eat up, Izzy. You’ll need your strength.”
Isabelle rolled her eyes but smiled faintly as Azzy hopped down from her shoulder to the table, sniffing curiously at the sandwich. “Sorry, Azzy. This one’s mine. You’ve got PokéPuffs in the cupboard.”
After breakfast, Isabelle spread her books and papers across the kitchen table, determined to tackle the easier problems first. Azzy sat nearby, tail bobbing as she played with a stray pencil. The early questions were manageable—basic algebraic equations and straightforward problem-solving.
“Alright, Azzy, time to knock out these problems,” she said, cracking her knuckles like a prizefighter stepping into the ring. Azzy squeaked encouragingly, though her attention seemed more on her reflection in Isabelle’s water glass than the math problems.
Problem 1: Simplifying an Algebraic Expression
Question: Simplify:
5x+2x?8+4
Isabelle barely looked at the problem before her pen flew across the page.
Steps Taken:
- Combine 5x+2x=7x
- Combine ?8+4=?4
- Final answer: 7x?4
“Piece of cake,” Isabelle muttered, leaning back in her chair. Azzy chirped approvingly. “Too easy, right?”
Problem 2: Solving for x
Question: Solve:
3x+9=21
Isabelle narrowed her eyes at the problem, confident she could finish it in seconds.
Steps Taken:
- Subtract 9 from both sides: 3x=12
- Divide both sides by 3: x=4
“Next,” she said, scribbling the answer. She paused briefly to draw a tiny Azzy doodle in the corner of the page, grinning to herself.
Problem 3: Evaluating a Polynomial
Question: Evaluate f(x)=x2+2x+1 when x=3.
Isabelle sighed. “Okay, polynomial time. Easy enough.”
Steps Taken:
- Substitute x=3 into the polynomial: f(3)=(3)2+2(3)+1
- Calculate step by step: f(3)= 9+6+1
- Final answer: f(3)= 16
She gave Azzy a thumbs-up. “Look at that! We’re on fire!”
Problem 4: Factoring a Quadratic
Question: Factor: x2+7x+10
Isabelle tapped her pen against her chin. “Let’s see… two numbers that multiply to 10 and add up to 7…”
Steps taken:
- Identity the numbers 5 and 2 because: 5x2=10, 5+2=7
- Rewrite the quadratic: x2+7x+10=(x+5)(x+2)
- Final answer: (x+5)(x+2)
“Ha! Math nerds love these, don’t they?” Isabelle said, twirling her pen. Azzy blinked at her, unimpressed.
Problem 5: Finding the Slope of a Line
Question: Find the slope of the line passing through the points (2,5) (6,9).
Isabelle groaned. “Okay, fine. Rise over run. I can do this.”
Steps Taken:
- Use the slope formula: m=x2??x1/?y2??y1??
Substitute: m= 9-5 / 6-2
- Simplify: m= 4/4=1
- Final answer: m=1
“Wow, such an exciting slope. One whole unit. Groundbreaking,” Isabelle muttered. Azzy squeaked like she agreed.
Problem 6: Simplifying a Rational Expression
Question: Simplify: 6x2+12x/3x
Isabelle squinted at the problem. “Alright, cancel stuff out…I think.”
Steps taken:
- Factor the numerator: 6x2+12x=3x(2x+4)
- Cancel the common factor 3x: 3x(2x+4)/3x= 2x+4
- Final answer: 2x+4
“Boom!” Isabelle declared, tossing her pen onto the desk like a mic drop. “Six down, and I’m still standing.”
Azzy hopped excitedly, chirping as if to cheer her on. Isabelle leaned back in her chair, flexing her fingers like a pianist after a long set. “Maybe Kotomine isn’t as evil as I thought.”
Isabelle glanced at the next question and froze. Her confidence wavered as she scanned the unfamiliar terms and symbols. “Oh no. No, no, no. What is this nonsense?” Her mental projection popped up, trapped under a mountain of notebooks labeled Impossible Math Stuff.
“Azzy,” Isabelle whimpered, slumping onto the desk, “I think Mr. Kotomine is trying to kill us.” Azzy wagged her tail, smacking Isabelle on the head like a scolding parent.
She read the problem aloud, trying to steady her nerves:
Find the derivative of f(x)= 5x^4-3x^3+7x-4.
“Okay, Isabelle,” she muttered, planting her pencil firmly on the paper. “You’ve got this- just take it one step at a time. Whatever a derivative even is.”
Her mental projection materialized beside her, wearing a hardhat and goggles, holding a jackhammer labeled “Math Solutions.” “Alright, soldier!” it barked. “We’re going in!”
Isabelle’s Solo Attempt:
- Look at the problem and panic a little.
Isabelle tapped the page nervously. “Okay, so…derivatives are…slope things, right? Like…finding how fast something changes? Sure. Slope things.” She scribbled: f′(x)=?
- Write down random pieces of the problem.
Isabelle glanced at the 5x^4 term. “I think…you multiply the number? Or…divide it? Maybe both?” She hesitated, then wrote: f′(x)=5(4x^3)?3(3x^2)+7x
- Try to simplify. Badly.
Isabelle stared at the numbers like they might rearrange themselves. “Okay, uh, just simplify. Easy.” She scratched out parts of the equation with her eraser, muttering incoherently before scribbling again: f′(x)=20x^3?9x^2+7x?4
- Second guess everything.
“Wait, what do you do with the constant?” Isabelle frowned. ”Do you add it? Subtract it? Or just…pretend it’s not even there?”
She circled the -4, drew an arrow on it, and wrote: “Does not exist?”
- Break down into despair.
Isabelle stared at the mess of calculations, then at Azzy, who chirped encouragingly. Her shoulders slumped. “I can’t do this,” she groaned, letting her head drop to the table. Tears pooled dangerously close to the edge of her paper.
Azzy’s cheeks puffed up indignantly. With a determined chirp, she spun her tail and delivered a sharp thwap to Isabelle’s head.
“Hey!” Isabelle yelped, rubbing her scalp. “What was that for?!”
Azzy chirped sternly, pointing her tail at the homework like a miniature schoolteacher.
émile stepped into the living room, drawn by the sound of his daughter’s misery. He carried a mug of coffee, his expression a mix of amusement and concern. “Mon trésor,” he began, his tone teasing, “what is this symphony of despair I hear? Are you auditioning for a tragedy?”
Isabelle looked up from her work, a pencil dangling precariously from her lips as her expression twisted into one of exaggerated agony. “You don’t understand, Dad. This isn’t just math. This is evil. Mr. Kotomine doesn’t want us to learn—he wants us to suffer!” She waved her homework at him like a flag of surrender.
émile chuckled, setting his mug down on the coffee table. “Come now, it can’t be that bad. Surely my brilliant daughter can conquer a few equations.”
“Brilliant? Ha!” Isabelle dropped her head dramatically onto the table, narrowly missing a stack of papers. “I’d rather fight a pack of Gyarados in the wild. At least then I’d have Azzy to help me.”
Azzy chirped from the couch, where she was nestled among the cushions like royalty, her tail bobbing in agreement.
“Besides,” Isabelle continued, her voice muffled by the tabletop, “I’ve got other important plans today.” She popped her head up, a grin spreading across her face. “Like watching Rosewood High Secrets.”
émile raised an eyebrow. “Rosewood High Secrets? What’s that? Some kind of documentary?”
“Documentary?” Isabelle snorted. “Not even close. It’s a drama. A really juicy one. You know, girl stuff.” She waved a hand vaguely. “It’s about a group of high schoolers trying to figure out who’s sending them creepy messages while dealing with a missing friend and, like, all these secrets and lies. There’s this anonymous sender called ‘Shadow,’ and they know everything about the girls—like, things they thought were buried forever!”
Unlawfully taken from Royal Road, this story should be reported if seen on Amazon.
“Sounds... cheerful,” émile said, his tone dry but amused.
“It’s so good, though!” Isabelle said, leaning forward, her enthusiasm spilling over. “The first episode had everything—mystery, betrayal, a love triangle. I mean, I’ve only just started it, but it’s addicting. I bet Amélie would love it. Maybe I’ll bring it up next time we hang out. Instant conversation starter, right?”
émile smirked. “And you think that will help you master derivatives?”
“Details, details,” Isabelle said, brushing him off with a wave of her hand. “Anyway, I’m just saying, Rosewood High Secrets is the perfect break after all this torture.” She gestured at the math spread across the table.
émile crossed his arms, giving her a pointed look. “And yet here you are, still sitting with this ‘torture.’ Now, mon trésor, let’s focus on conquering one battle at a time.”
Isabelle groaned but picked up her pencil, glaring at the math problem as though it had personally offended her. “Fine. But if Shadow sends me a cryptic message about solving derivatives, I’m blaming you.”
“Duly noted,” émile said, sitting beside her with a small laugh. He picked up her sheet, scanning it. “Alright, where are we stuck? Let’s start there.”
émile sighed, sitting down beside Isabelle, and gesturing for her pencil. Isabelle crossed her arms and leaned back in her chair, her lips quirking up in a sarcastic grin. “Oh, please, enlighten me, O wise mathematician. Show me the mystical ways of numbers.”
“You’re in luck,” émile replied smoothly, ignoring her tone. “This happens to be my area of expertise.”
Isabelle blinked, her grin faltering. “Wait, seriously? Like, for real? This is your thing?”
émile chuckled, reaching for her sheet of homework. “Majored in it at university. And it’s part of my job. Statistics, analytics, projections—it all relies on understanding math at this level. So yes, Isabelle, I can help.”
She groaned, dropping her head onto the table. “Great. Now I feel even dumber.”
“You’re not dumb,” émile said firmly. “You’re overwhelmed. Now, let’s break this down.” He picked up her pencil and flipped to a clean section of the page. “Alright. We’re dealing with derivatives. Let me introduce you to a very handy tool: the Power Rule.”
He wrote clearly and slowly:
d(xn)/dx = nxn-1
“See?” émile explained. “You take the exponent, multiply it by the coefficient, and subtract one from the exponent. It simplifies the process of finding how functions change.”
“Think of it like this,” émile said, tapping the pencil on the table. “Imagine you’re driving a car. The position of the car is like the original function. The speed you’re going is the first derivative—it tells you how fast the position is changing. The acceleration? That’s the second derivative—it tells you how fast the speed is changing. Derivatives are about measuring change.”
Isabelle stared at him, her lips slowly curving into a wry smile. “That’s actually... kind of cool. But I’m not a car.”
“Yet,” émile replied dryly. “Let’s see if you can apply this to your problem.”
Isabelle stared at the formula, her brain whirring. “So... if I use that on 5x?, it’s... 20x3?”
émile blinked. “Yes. That’s correct.”
“And for -3x3, it’s... -9x2?”
“Yes,” émile said slowly, watching her scribble out the rest of the problem with alarming speed.
f′(x)=20x^3?9x^2+7
Isabelle beamed, holding up her work. “Ta-da! See? Easy.”
émile blinked, mildly surprised. “That’s correct. Well done, Isabelle.”
“Obviously,” she said with a smug grin, but her tone carried more relief than arrogance.
“Alright,” émile said, setting the sheet down and pointing to the next problem. “Let’s see if you can keep that streak going.”
Problem: 3x^5?4x^4+2x^2?x+8
Isabelle got to work, mumbling to herself. “Okay, so… 3x^5 becomes 15x4… then -4x^4 becomes -16x3… and 2x^2 is 4x…”
émile watched her closely, his brows furrowing slightly as she scribbled down her solution. She skipped writing intermediate steps entirely, occasionally muttered numbers under her breath, and didn’t bother to double-check her coefficients.
Her final answer:
f′(x)=15x4?16x3+4x?1
émile scanned it, his lips pressing into a line. It was right. The answer was correct. But the way she’d gotten there was... unorthodox at best.
“Isabelle,” he said carefully, “can you walk me through your steps?”
“Uh...” Isabelle paused, chewing the eraser of her pencil. “I just... you know. Did the thing.”
“‘The thing,’” émile repeated, raising an eyebrow.
“Yeah. I used your Power Rule. Multiplied, subtracted, simplified. It just... clicked.” She gave him a sheepish smile.
émile leaned back, crossing his arms as he watched her tackle the next few problems. Each time, her process became sloppier—skipping entire coefficients, jumping directly to the solution without showing intermediate work—but the answers were always correct.
When she finished the last problem, she dropped her pencil triumphantly. “Done. Genius level unlocked.”
émile remained silent for a moment, studying the sheet. Then, he smiled softly and reached over to ruffle her hair. “You’re something else, Izzy. Keep being you.”
Isabelle batted his hand away, laughing. “Hey! I’m a certified math genius now. Treat me with respect.”
émile chuckled, standing and gathering the papers. “One more question for you,” he said, his tone careful. “Would you like to try something... a little more challenging?”
Isabelle looked up, suspicious. “Define ‘challenging.’”
“Just a problem I’ve been working on for... let’s say, a project.” émile smiled, his expression betraying nothing. “Interested?”
Isabelle’s eyes narrowed, but her curiosity sparked. “Fine. Bring it on. But if I die, I’m haunting you forever.”
émile chuckled as he turned toward his office, leaving Isabelle and Azzy to their momentary triumph.
The sudden weight of a thick binder being set on the table in front of her made her freeze. She turned slowly, staring at the intimidating stack of papers émile had just placed between her and the TV.
“Uh, what is that?” Isabelle asked, raising an eyebrow.
émile hesitated, rubbing the back of his neck. “It’s... something from work. A project my team’s been stuck on for a while now.”
“Why does it look like a legal textbook?” Isabelle muttered, flipping open the first page and immediately being greeted by columns of equations, diagrams, and notes in various handwritings. “Oh. Math. Of course.”
“You don’t have to look at it if you don’t want to,” émile said quickly, though his tone betrayed a flicker of hope. “I was just... curious. You seem to have a knack for solving things in your own way.”
“My own way,” Isabelle repeated with a smirk. “Thanks, Dad. But isn’t this, like, top-secret classified stuff? Should I even be looking at this?”
“Probably not,” émile admitted, a sheepish grin creeping onto his face. “But let’s call it... an experiment. It’s all theoretical anyway, and we’re stumped. Who knows? Maybe you’ll see something we’ve missed.”
Isabelle stared at the thick binder in front of her, flipping through the pages with a mix of dread and curiosity. Columns of equations, diagrams, and notes scrawled in varying handwriting filled the sheets, each one more complex than the last. The bold heading on one page caught her eye: Proposed Linear Progression Model for Dynamic Energy Grids.
"Linear Progression Model?" Isabelle read aloud, her voice dripping with skepticism. “Sounds... fancy.”
“It was our most promising lead,” émile admitted, sitting across from her and watching carefully. “The idea was to streamline the energy fluctuations Pokémon provide into a more uniform output.”
“But?” Isabelle pressed, raising an eyebrow.
“But it hasn’t worked,” émile said with a sigh. “We’ve been stumped for weeks. Deadlines are looming, and we’re out of ideas. Maybe you can spot something we’ve missed.”
Isabelle gave him a flat look. “Dad, I’m sixteen. This binder looks like it belongs in a university archive.”
“You don’t have to solve it,” émile said, a hopeful smile tugging at his lips. “Just... take a look.”
“Fine,” Isabelle muttered, grabbing her pencil. “But I’m calling it now: whoever came up with this linear thing wasted everyone’s time.”
As Isabelle flipped through the binder, she skimmed the different contributions from émile’s colleagues. Each idea was marked with initials, notes in the margins, and sometimes entire sections crossed out in frustration.
- Key Formula:
y= mx + b adapted for energy output
E(t) = k1t + k2
Where: - E(t): Energy output over time.
- k1, k2?: Constants determined by Pokémon types.
- Notes:
- "Assumes energy output is stable enough to follow a straight-line trajectory."
- "Oversimplifies fluctuations from Pokémon with inconsistent energy, e.g., Electric types during thunderstorms or Fire types in heatwaves."
“Wow. It’s literally a straight line. Who decided Pokémon energy was a constant? That’s like saying Azurill’s tail bounce is predictable—it’s not. Hats off to whoever wasted their time twisting a ‘constant’ into this mess.”
émile coughed into his hand, suppressing a chuckle. Internally, he remembered how his entire team had thought this was their best shot.
- Key Formula:
P(x) = anxn + an-1xn-1 + +?+a0?
Adapted for cumulative energy contributions:
E(x) = c3?x3 + c1x + c0 - Notes:
- "Designed to account for cumulative contributions over time, using Pokémon type interactions as coefficients."
- "Complicated to calculate in real-time; relies heavily on accurate type-match predictions."
“So this one’s just a math flex. I mean, sure, if we want to calculate cumulative energy for one battle scenario, great. But real life isn’t one battle—Pokémon are messy. This overcomplicates it.”
- Key Formula:
E(t) = Asin(Bt+C) + D
Where:
- A, B, C, D: Constants adjusted for amplitude and frequency of Pokémon energy spikes.
Notes:
- "Accounts for energy fluctuations with a sinusoidal model."
- "Too reactive; can’t predict sustained output. Feels chaotic."
“Oh, cool. Let’s make the grid a rollercoaster! ‘Whee, Pokémon spikes! Oh no, crashes!’ Who thought this was a good idea?” Isabelle’s take on it.
(“Strategic Harmonic Integration Theorem”)
It’s a nonsensical acronym she scrawled in frustration, muttering “holy shit, my brain is fried,” while her father stifled a laugh. Despite the chaos of her process, what emerged was a shockingly innovative theoretical model that managed to bring coherence to a problem stumping seasoned mathematicians.
Proposed Formula:
S(x, y) = ∑i=1n? (Ei?(t)?Di?(x,y)? / 1+k )
Ei (t): Energy input Pokémon i at time t.
Di (x, y): Distance weighting factor to redistribute power dynamically across grid coordinates.
K: variable resistance factor (adjustable for regional variances)
Isabelle’s notes were a chaotic mix of scribbles, sarcastic doodles, and arrows pointing to rewritten equations. Her adjustments emphasized dynamic balancing between real-time Pokémon energy surges and localized demand, allowing for smoother redistribution across grids without overload or delay.
émile leaned over the table, watching as Isabelle muttered to herself.
“Who even thought linear would work?” she grumbled, slashing through the formula and rewriting it. “Grid systems aren’t straight lines. They’re basically spiderwebs with temperamental spiders.”
Her father chuckled, recognizing the idea of a “temperamental spider” as an apt metaphor for overloaded energy grids. Isabelle’s pencil moved furiously, skipping steps as she jotted down connections only she seemed to understand. Arrows darted across the page, connecting unrelated equations in ways that looked more like a conspiracy theory than mathematical work.
“Are you—” émile started, pointing at one particularly wild scribble.
“Shh,” Isabelle said, her focus unbroken. “The brain cells are... braining.”
After an hour, Isabelle sat back, her hands smudged with graphite and her hair sticking out in random directions. She held up the paper triumphantly, the acronym SHIT scrawled across the top.
“Done!” she declared.
émile took the sheet, scanning it with disbelief. His eyes widened as he traced her work, noting how her equation addressed the flaws in every prior proposal. It was messy, unpolished, and utterly brilliant.
“Holy—” émile began, but Isabelle cut him off.
“Language, Dad.” She smirked, though her eyelids drooped from exhaustion. “Can I go watch Rosewood High Secrets now?”
émile set the sheet down carefully, like it might shatter. “Of course,” he said, ruffling her hair. “I’ll call you for dinner.”
As Isabelle shuffled off, Azzy bouncing happily at her heels, émile stared at the formula again. It wasn’t perfect, but it worked. He sighed, already dreading the reaction from his colleagues during the evening’s conference call.
“She really did it,” he murmured, a mix of awe and apprehension in his voice. “She solved it.”