Day 5 · Burning Ropes¶
Edited by Charles
Brainteaser · Timing · Invariants
Original question on OpenQuant
Problem¶
You have two ropes and a lighter. Each rope takes exactly 60 minutes to burn from one end, but its burning speed varies along its length. How can you measure exactly 45 minutes?
Use the usual puzzle model: a segment has the same burn-time contribution from either direction, and separate flames do not alter one another's burn rates before they meet.
Core idea
Lighting both ends consumes the remaining single-ended burn time twice as fast, even when physical burning speed varies.
Short solution¶
1. Start three flames simultaneously. Light both ends of rope A and one end of rope B.
2. Wait until rope A finishes. Its two flames jointly consume 60 minutes of single-ended burn time in
At that moment, rope B has 30 minutes of single-ended burn time remaining.
3. Light the other end of rope B. Its remaining portion now burns from both ends, finishing after
Answer
Watch out
Half the physical length need not represent half the burn time. The flames do not need to meet at the physical midpoint.
Why uneven burning does not change the timing
After \(t\) minutes, each flame has consumed a portion requiring \(t\) minutes to burn from its respective end. Until the flames meet, these portions do not overlap. Together they exhaust a 60-minute rope when \(t+t=60\). The same reasoning applies to the 30-minute remainder of rope B.