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After reading everyone's responses last night I was convinced that burning them from both ends would not work due to the possible change in burn rate of the rope. However, after waking up this morning I was convinced otherwise... I believe my mental block last night was assuming the burns would meet in the middle. That doesn't have to be the case obviously, but I think it was holding me back last night. I am sure there is a mathematical proof of this if you assume a few things like, i.e. the burning of a rope from both ends doesn't create more heat that would burn up the rope quicker than the smaller heat from one burn. In any case I do believe the rope should burn up completely in 30 seconds if you start the rope burning from each end at the same time. Every infinitesimal segment of the rope burns at a given rate. The total burn time using one flame for a length of rope is 60 seconds. Using two flames will use the entire available fuel (rope) in 30 seconds as every infinitesimal segment will burn at the same rate it burned at with one flame. The flames could meet up in the middle if the burn rate of each segment is roughly similar, but that is not necessary and is not assumed. If it takes 58 seconds to burn one finite segment as knifemissle suggested than one flame will eat through all the other segments in 2 seconds while the other flame knaws on the one segment. The two flames will then knaw together for the remaining 28 seconds on the remaining segment. Not quite a mathematical proof, but I am convinced this would work in theory. (But not in practice as I do believe the added heat would increase the burn rate of the infinitesimal segments.)
Finding a way to measure 15 seconds would involve both ropes as Saltfish discovered the answer too.
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