The problem
Two ropes. Each burns for exactly 60 minutes end to end, but unevenly - half the rope might burn in 5 minutes or 55. You have a lighter. Measure exactly 45 minutes.
The key realisation
You cannot measure a distance along a rope, because the burn rate is unknown and variable. But you can control how many ends are lit.
Light a rope at both ends and it burns out in exactly 30 minutes, whatever the density profile - the two flames consume the total remaining burn-time at twice the rate, and they meet when 60 minutes of rope-time has been consumed.
That is the entire trick. Everything else follows.
The solution
At t = 0:
- Light rope A at both ends.
- Light rope B at one end.
At t = 30: rope A is gone. Rope B has 30 minutes of burn-time left, though you have no idea where the flame is physically.
Immediately light rope B's other end. Its remaining 30 minutes now burns from two ends, so it finishes in 15.
At t = 45: rope B is gone. Done.
What else you can measure
With two ropes you can get 15, 30, 45, 60, 75, 90, 105 and 120 minutes - any multiple of 15.
The 15-minute unit comes from being able to halve twice. With three ropes you can reach multiples of 7.5, and with n ropes the granularity keeps halving.
Why this is asked
It is a test of whether you look for the available operations rather than the obvious measurements. Candidates who fixate on estimating positions along the rope never get there; candidates who ask "what actions can I take?" find it quickly.
That reframe - enumerate the operations, not the observations - transfers to a lot of constraint puzzles.
More in brainteasers.