Thirty minutes from one uneven rope

Question

You have one rope and a box of matches. The rope takes exactly 60 minutes to burn from one end to the other, but it burns unevenly: half its length might burn in five minutes and the other half in fifty-five. You cannot cut it, mark it, or measure it.

How do you measure exactly 30 minutes? Explain why your method is exact even though the rope burns unevenly.

Accepted Answer

Light both ends at the same time. When the two flames meet, exactly 30 minutes have passed. Why it is exact. Think of the rope as a sequence of tiny pieces, each with its own burn time, adding up to 60 minutes. Lit from one end, the flame works through every piece in order. Lit from both ends, two flames work through the pieces from opposite sides, and at every instant both flames are burning. When they meet, every piece has been burned exactly once, and the two flames shared that work. Since they burned simultaneously for the whole time, each flame was alight for half of the total 60 minutes…

Thirty minutes from one uneven rope

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