Transition from 1D to 2D Problem

• Part 2 of candles-in-cake mathematical puzzle series
• Original problem: twigs (1D) → upgraded to 2D cake for aesthetics
• Aesthetic upgrade created unexpected mathematical consequences
• Challenge: placing random candle and random cut on 2D surface less obvious than 1D

Initial 2D Simulation Setup

• Used GeoGebra for square cake simulation (easier to draw)
• Random placement method:
  • Candles: random x,y coordinates
  • Cut: random x,y coordinates + random angle
• Initial simulation converged to ~1/3 probability (0.3 something)
• Presented findings at MathsJam annual weekend

Recognition of Random Distribution Problem

• Nagging mathematical concern about "random" definition
• Connection to Bertrand's paradox (random chord in circle)
• Multiple valid methods for randomly assigning cuts
• Problem similar to Grant's video discussion on random chords

Circular Cake Analysis Methods

• Candle placement: spray random x,y coordinates, reject if outside circle
• Multiple chord selection methods tested:

Method 1: Two Random Endpoints

• Pick two random points on circumference
• Creates chord connecting endpoints
• Results in ~0.16 probability after 10,000 attempts
• Distribution clusters toward edges, middle less sparse

Method 2: Random Midpoint

• Choose random point inside circle as chord midpoint
• Different distribution pattern
• Rarely cuts near middle
• Results in ~0.19 probability

Method 3: Random Radial Point

• Choose distance along radius + angle around center
• Use as midpoint for chord
• Different distribution again

Method 4: Random Point + Random Angle

• Pick random point and add random angle
• Results in 0.3 probability (close to 1/3)
• Most uniform-looking distribution
• 0.325 in some runs

Mathematical Conclusions

• None of the methods produced exact 1/3 probability
• Results ranged from 0.16 to 0.325
• Author admits: "I don't know what's going on"
• Real-world random distributions in 2D more complicated than expected
• Need precision in modeling real-world scenarios

Practical Demonstration Considerations

• Real-world cake cutting wouldn't be truly random
• People would likely cut from one side (not random)
• Alternative: double spinner approach for endpoints
• Used lazy Susan turntable for random angle demonstration
• Successfully cut between candles in live demonstration

Overall Lessons

• Simple problems become complex when dimensions increase
• Multiple valid interpretations of "random"
• Importance of being careful about probability definitions
• Glimpse into how mathematicians work: simple → complex
• Need theoretical work beyond Monte Carlo simulations