Burning Ropes

Question

You have two ropes and a lighter. Each rope burns completely in exactly 60 minutes, but burns non-uniformly (half the rope may take 5 minutes or 55). How do you measure exactly 45 minutes?

Accepted Answer

The key device: a rope lit at both ends finishes in exactly 30 minutes: the two flames consume the whole rope together, so total burn time halves regardless of non-uniformity (they meet wherever 30 minutes of rope remains on each side). Protocol: 1. At t=0: light rope A at both ends and rope B at one end. 2. Rope A finishes at t = 30. At that instant, rope B has exactly 30 minutes of burning left, now light B's other end. 3. B's remaining 30 minutes burns from both ends → finishes in 15 more minutes, at t = mathbf45. Why non-uniformity doesn't matter: "burns in 60 minutes" defines total burn time, not length; two flames always consume burn-time at combined double rate. The interview follow-through is recognizing the reusable primitive, k simultaneous flames divide remaining time by... well, two flames on one rope = rate 2. With clever relights you can construct many durations (e.g. 22.5 by another halving).

Burning Ropes

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