Measuring 45 Minutes with Two Non-Uniform Ropes

Scenario

You are given two ropes of arbitrary, non-uniform density and thickness. Each rope takes exactly 60 minutes to burn completely from one end to the other, but the burning rate is irregular (e.g., half the rope might burn in 10 minutes and the other half in 50 minutes).

Your Task

1. Explain the physical invariance of lighting a non-uniform rope from both ends simultaneously.
2. Devise a protocol using only the two ropes and a box of matches to measure an elapsed time of exactly 45 minutes.
3. Prove mathematically why non-uniform density does not affect the exact timing of the protocol.

Constraints

• You cannot cut, measure, or fold the ropes.
• Burning rates along each rope are non-uniform and unknown.

Examples Burning a 60-minute rope from both ends takes exactly $60/2 = 30$ minutes regardless of density variation.