Day 19: The Power Grid
Act IV: The Final Takeover
Location: Hamburg Energy Control Center
With the Christmas markets operational, attention turns to the city’s festive lighting. But the entire Christmas lighting grid remains dark, controlled by GrinchTech’s power management algorithms.
You’ve been brought to Hamburg’s Energy Control Center. The Christmas lighting subsystem shows: OFFLINE - GRINCHTECH OVERRIDE ACTIVE.
The Grinch appears on the main display. He looks pensive—almost nostalgic:
System Message from CEO
“Day 19: Power Functions.
The city’s lighting grid isn’t controlled by simple switches. It’s governed by power functions—mathematical relationships where exponents determine how energy flows, how brightness scales, how systems behave.
Power functions are fundamental. They describe root relationships, growth rates, and behaviors that linear functions cannot capture.
Understand power functions, and I’ll restore Hamburg’s Christmas lights. The city will glow again.
I admit… I’d like to see that.”
– GR
The Challenge
Four problems analyzing power functions that control Hamburg’s lighting infrastructure.
Lock 1: “The Square Root Domain”
The main Christmas lighting generator’s energy output follows:
\[f(x) = x^{1/2} = \sqrt{x}\]
What is the domain of \(f(x) = \sqrt{x}\)?
Lock 2: “The Fractional Exponent Mystery”
A backup generator operates on \(g(x) = x^{2/3}\). Unlike \(f(x) = x^{1/2}\), this function CAN accept negative inputs.
Grinch’s Note: “Fractional exponents are just roots and powers in disguise. If you remember your radical rules from high school—and I use ‘remember’ generously—you’ll figure this out.”
Why can \(g(x) = x^{2/3}\) accept negative \(x\) values?
Lock 3: “The Growth Rate Comparison”
The main Christmas tree at Rathausmarkt has LED strands that follow \(h(x) = x^4\).
Grinch’s Note: “Compare \(x^4\) to \(x^2\). For large values of \(x\), which grows faster? This determines brightness scaling.”
For \(x > 1\), how does \(x^4\) compare to \(x^2\)?
Lock 4: “The Asymptote Analysis”
A voltage regulator operates on \(k(x) = x^{-1} = \frac{1}{x}\).
Grinch’s Note: “This function has asymptotes—lines the graph approaches but never touches. There’s one vertical and one horizontal. Identify the vertical one.”
What is the vertical asymptote of \(k(x) = \frac{1}{x}\)?
Enter the vertical asymptote (as x = ?):
Status Update
You submit your power function analysis:
- Domain of \(\sqrt{x}\): \(x \geq 0\) ✓
- Fractional exponent: Odd root (3) allows negative inputs ✓
- Growth comparison: \(x^4\) grows faster than \(x^2\) for large \(x\) ✓
- Asymptote: Vertical at \(x = 0\) ✓
The lighting control systems respond immediately. Throughout Hamburg, Christmas lights begin to flicker on—first the Rathausmarkt tree, then the Jungfernstieg promenade, then street by street.
From the control center windows, you watch Hamburg transform. The dark city becomes a glowing wonderland of festive lights reflecting off the Elbe.
The Grinch watches the same transformation from his office. When he speaks, his voice is quieter:
#lighting-restored
“Beautiful, isn’t it? I’d forgotten.
The lights are back. The markets are open. Hamburg is celebrating.
But I should warn you about tomorrow. Day 20 involves my exponential failsafe systems. I built them… aggressively. There may be some complications.
Four days left. Let’s finish this properly.”
– GR
Act IV Progress:
- ✓ Polynomial defenses cracked
- ✓ Power function grid restored
- ✓ Christmas markets operational
- ✓ City lighting fully active
- Exponential systems: Tomorrow (Warning issued)
- Three final systems: Days 21-23
- Master synthesis: Day 24
Major Achievement: Hamburg’s Christmas lights restored!
Time Remaining: 5 days until Christmas Eve