Brainteasers & Puzzles

Burning Ropes

Two ropes each burn for exactly one hour, but at wildly uneven rates along their length, so half a rope need not take half an hour. Using only these ropes and a lighter, how do you time exactly 45 minutes?

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You have two ropes. Each takes exactly one hour to burn from end to end, but the burn rate varies along the rope, so a given length need not correspond to a proportional time. With nothing but the ropes and a way to light them, measure exactly 4545 minutes.

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#Burning both ends halves the time

Assign each point of a rope the time a single flame would take to reach it, so the whole rope carries 6060 minutes of burn content and a one-end burn finishes in 6060 minutes. Lighting both ends sends two flames inward, one consuming the content ahead of it and the other the content behind. When they meet the rope is gone, and each flame has burned for the same elapsed time TT. The two stretches they consumed hold TT and TT minutes of content, and together they are the whole rope, so

T+T=60  ⟹  T=30.(1)T + T = 60 \implies T = 30. \tag{1}

This holds for any density profile, since only the total content matters, not where the flames meet.

#The schedule

At time 00 light rope A at both ends and rope B at one end. Rope A burns from both ends and is gone at time 3030. At that instant rope B has burned from one end for 3030 minutes, so its remaining content is 60−30=3060 - 30 = 30 minutes. Now light rope B's other end. With both ends alight that remaining content burns in half the time,

302=15,(2)\tfrac{30}{2} = 15, \tag{2}

so rope B is gone at time 30+15=4530 + 15 = 45.

rope Arope B
03045
Light rope A at both ends and rope B at one end together. Rope A is spent at the 30 minute mark, the moment to light rope B's second end. The remaining half-hour of content in rope B then burns from both ends in 15 minutes, landing the total at exactly 45.

#Result

The interval from the first lighting to the moment rope B finishes is

30+15=45 minutes.(3)30 + 15 = 45 \text{ minutes}. \tag{3}