Burning Ropes (Measure 45 Minutes): Puzzle and Step-by-Step Solution

Step 1: What you cannot do

Because the burn rate is uneven, cutting a rope in half or marking a point on it tells you nothing about time. Half the length might take five minutes or fifty-five. Any solution must use only the one fact you know: a full rope takes 60 minutes.

So the only measurable events are a rope burning out. The puzzle is about engineering those events to land at useful times.

Step 2: Lighting both ends halves the time

Light a rope at both ends. The two flames burn toward each other and meet somewhere, not necessarily in the middle. Wherever they meet, the time taken is the same for both flames, and together they have burned the whole rope.

If the left flame covers a part that would take t minutes alone and the right flame covers the rest, which would take 60 - t, then since both flames burn simultaneously for the same time, t = 60 - t, so t = 30. A rope lit at both ends burns out in exactly 30 minutes, regardless of its unevenness.

Step 3: The 45-minute schedule

At time 0, light rope A at both ends and rope B at one end. At 30 minutes, A burns out. At that moment B has burned for 30 minutes from one end, so whatever remains of B would take exactly 30 more minutes from that one flame.

Immediately light B's other end. The remaining piece now burns from both ends, so by the same argument it takes 15 minutes. B burns out at 30 + 15 = 45 minutes.

Step 4: Follow-ups

With one rope you can measure 30 minutes (light both ends) or 60. With two ropes you can mark 30, 45, 60 and 90 minutes, and the 15 minutes between the 30 and 45 marks. A common follow-up asks whether you can measure 20 minutes with two ropes; the answer is no, because every event happens at a multiple of 15 minutes, and saying why is the answer interviewers want.