Math Club Problem: The Great Cupcake Heist

Three pastry-loving critters must coordinate a bakery raid with purr-fect timing to avoid the bakery cat. But this isn’t just about survival—it’s about style, flair, and bragging rights in the underground dessert heist scene!

🐾 THE RAIDERS

🚨 CORE CONSTRAINTS

All critters must reach their destination within 10 seconds (≈0.167 minutes) of each other.
Speeds MUST stay strictly within limits and must be constant for the entire raid.

🌟 RAID RATING SYSTEMS

The Pastry Bandits Guild judges raids using two systems - raiders are awarded whichever score is higher!

SYSTEM 1: HAZARD HUSTLE HEROES

"Why play it safe when danger tastes sweeter?"

Mimi: 100 − 200 × (0.75 − Speed)
Gains points by approaching her 0.75 m/min squeak limit. Each 0.1 m/min slower loses 20 points!
Milo: 80 − 160 × (Speed − 0.5)
Scores best at his dirt-concealing 0.5 m/min minimum. Each 0.1 m/min faster loses 16 points!
Greta: 40 − (40 − Climb Time)
Racks up points by stretching climbs toward the 40-minute fall risk. Each minute saved costs 0.5 points!

Rating: The sum of all three scores

SYSTEM 2: EFFICIENCY ELITE

"Swiftness is its own reward… and so are extra sprinkles!"

Total Mission Minutes = Mimi’s crawl time + Milo’s dig time + Greta’s climb time = (12.5/Speed) + (30/Speed) + (18/Speed)

Rating: 200 − Total Mission Minutes

🧁 Goal: Recipe for Success

To out-heist their rivals "The Scone Squad", the team needs to explore their scoring options. Craft two separate blueprints: one maximizing their rating in SYSTEM 1 and the other maximixing their rating in SYSTEM 2. The blueprints should detail speeds and start times maximizing these ratings. A score of 125 or better will ensure that they beat their rivals. Can they do it?

📝 Solution Requirements

All solutions must use minutes for time and meters/minute for speed and be rounded to four decimal digits.
State departure times as offsets from each character's earliest possible start.
Explain your methodology - how did you arrive at your solution.
Your solution may involve writing code. Explain in words what your code is doing and why it is or is not giving a gauranteed optimal solution for each of these two ratings.