Fifteen minutes with a single rope

Question

You have a single rope that burns end to end in exactly 60 minutes, unevenly, and a very large box of matches. You cannot cut or mark the rope.

Lighting both ends gives you 30 minutes. A friend claims that, in principle, the same rope can also measure exactly 15 minutes, but that no finite number of matches is guaranteed to be enough.

Explain the method and why it needs an unbounded number of matches.

Accepted Answer

The method. Light the rope at both ends and also at some point in the middle. The middle match creates two flames, one heading each way, so there are now four flames burning. Four flames together eat through the rope's 60 minutes of burn four times as fast, so if you could keep exactly four flames alive the whole time, the rope would be gone after 15 minutes. The catch. The rope now consists of two burning segments, each with a flame at each end. Whenever two flames meet, both go out, and you drop to two flames, which is too slow. So the instant one segment is consumed, you must immediately l…

Fifteen minutes with a single rope

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