Suppose you have two identical lengths of fuse that are 50 feet and would take exactly one hour to burn from end to end. The fuses are peculiar in that each one has a section of unknown length that burns at one rate, while the rest of the fuse burns at another rate. These rates and lengths are unknown. One possibility is that the first 10 feet of fuse take 50 minutes to burn and the remaining 40 feet take 10 minutes to burn. How can you use the two pieces of fuse and a lighter to determine when 45 minutes is up?

Answer: Light any three of the four ends of the two fuses. The fuse that has two ends burning will burn out twice as fast, that is, in 30 minutes. As soon as that happens, light the remaining end of the fuse. Since a half hour has passed, the fuse originally lit on only one end is “half burned,” and would take another 30 minutes to burn completely. But, since we are lighting the other end, it, too, will burn twice as fast, that is, in 15 minutes. The total time to burn the fuses in this fashion is the required 45 minutes.