Day 20: The Exponential Crisis
Act IV: The Final Takeover
Location: GrinchTech Headquarters - Climate Control Emergency
It’s 2 AM when the emergency alert wakes you: CRITICAL SYSTEM FAILURE - GRINCHTECH HEADQUARTERS.
You rush to Hafencity to find the GrinchTech building glowing an ominous red. Emergency vehicles surround it. Frau Weber meets you at the entrance, her face grim.
“The climate control has gone haywire,” she explains. “The backup facility—where all the presents are stored—is overheating. Exponentially.”
Inside the control room, you find the Grinch. He’s not on a video call. He’s actually here, in person, frantically working at his terminal. He’s sweating. His hands are shaking.
This is not the smug CEO you’ve been facing for 19 days.
EMERGENCY - DIRECT ADDRESS
“This isn’t me! I didn’t activate this!
It’s a failsafe I built months ago—an exponential escalation protocol designed to trigger if someone bypassed too many of my security systems. And you’ve bypassed everything.
The temperature is rising exponentially. The lights are failing exponentially. My entire infrastructure is in exponential collapse.
I can’t stop it alone. The override codes are mathematical—exponential equations and decay models. I need help.
This isn’t a test anymore. This is a crisis.
Please.”
– GR
The Challenge
Four exponential problems that must be solved to prevent catastrophic system collapse. The Grinch works alongside you, no longer an adversary.
Emergency 1: “Temperature Escalation”
The warehouse temperature \(T\) (in °C) is rising according to:
\[T(t) = 20 \cdot 2^{t/5}\]
where \(t\) is time in minutes since the crisis began (currently \(t = 0\)).
Grinch’s Note: “At 140°C, irreversible damage begins. I need to know when we hit critical. Calculate the temperature at 20 minutes. We need to know how much time we have.”
What is the temperature after 20 minutes?
Emergency 2: “Lighting Decay”
Christmas lights throughout the warehouse are failing according to exponential decay:
\[N(t) = 1000 \cdot (0.8)^t\]
where \(N\) is working lights and \(t\) is hours since crisis began.
Grinch’s Note: “How many lights will still work after 5 hours? If it drops below 300, the emergency systems lose visibility.”
How many lights remain after 5 hours? (Round to nearest whole number)
Enter the number of lights:
Emergency 3: “Backup Fund Access”
The Grinch reveals he’s been investing in a Christmas restoration fund (apparently feeling guilty). The account grows according to:
\[A(t) = 5000 \cdot e^{0.06t}\]
where \(t\) is years.
Grinch’s Note: “We need €5,985 for emergency repairs. After 3 years, is there enough? I genuinely don’t know—I haven’t checked this account in months.”
Is €5,985 available after 3 years? (Calculate \(A(3)\) using \(e^{0.18} \approx 1.197\))
Emergency 4: “The Fundamental Error”
The Grinch needs to understand why his original projections failed. His linear Christmas market model: \(f(x) = 100x\) stalls/year. His exponential tech projection: \(g(x) = 10 \cdot 2^x\) facilities/year.
Grinch’s Note: “After 8 years, which dominates? I need to understand my own mistake. I compared short-term data and made long-term conclusions.”
After 8 years, which value is larger?
Status Update
You input the calculations rapidly:
- Temperature at 20 minutes: 320°C (CRITICAL - must act within 15 minutes!) ✓
- Lights remaining at 5 hours: ~328 (above emergency threshold) ✓
- Backup funds: ~€5,985 available ✓
- Growth comparison: Exponential (2,560) far exceeds linear (800) ✓
The system accepts the codes. The temperature escalation STOPS at 75°C. Emergency cooling activates. The lighting failure stabilizes.
The Grinch collapses into a chair, physically exhausted. When he looks at you, there’s something different in his eyes. Gratitude. And something else—understanding.
DIRECT ADDRESS - G. RINCH
“You stopped it. Everything’s safe.
And you showed me my fundamental error. I was comparing short-term linear growth against long-term exponential potential. I was looking at year 3, not year 8.
If Hamburg had let Christmas markets grow for another 5 years, they would have exponentially exceeded my projections. I shut them down precisely when they were about to explode with growth.
I made a mathematical error. And nearly destroyed everything because of it.
Thank you. For the math, and for… for helping when you didn’t have to.
Three more systems remain. But I’m not testing you anymore. We’re working together now. Let’s finish this. Properly.”
– G. Rinch
Act IV Progress:
- ✓ Polynomial defenses cracked
- ✓ Power function grid restored
- ✓ Exponential crisis RESOLVED
- ✓ Grinch fundamental error identified
- Three final systems: Days 21-23
- Master synthesis: Day 24
Time Remaining: 4 days until Christmas Eve