Rope-Burning and Measuring Time

The setup: you're given one or more ropes, each of which takes exactly 60 minutes to burn from one end to the other if lit — but the rope burns unevenly along its length, so lighting it at the halfway mark does not guarantee 30 minutes remain. You can only measure time by watching a rope burn out completely, and you can light a rope at either end, or both ends at once. The question is how to measure some other interval, like 45 minutes.

The key trick is that lighting a rope at both ends simultaneously makes it burn out in exactly 30 minutes, regardless of the uneven burn rate — because the two flame fronts are together consuming the whole rope's fixed total burn-time in half the time, no matter where they happen to meet. To measure 45 minutes with two identical ropes: light rope A at both ends and rope B at one end, at the same moment. Rope A burns out after 30 minutes. At that instant, light the other end of rope B too — whatever length remains on B, now burning from both ends, takes half of its remaining time, which is exactly 15 more minutes. Total elapsed: 30 + 15 = 45 minutes.

The generalizable idea interviewers are testing is recognizing that "burns unevenly" only breaks proportional reasoning about a single lit end — lighting both ends still gives you an exact, rate-independent halving of whatever time was left.