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SOLUTION TO PROBLEM 2:
Let A and B be the endpoints of one rope. Note that if you light end A of rope AB, then after 1/2 hour the flame will be at a point C between A and B. The remainder of the rope, CB, will take 1/2 hour to burn. So if we were to start by lighting both ends of rope AB, after 1/2 hour the flames would meet at the same point C and then extinguish each other. This is how we can measure 1/2 hour. To measure 3/4 hour, light both ends of rope AB and exactly one end of the other rope. When AB burns up, we light the other end of the other rope. The other rope has 1/2 left to burn, but with flames at both ends it burns in 1/4 hour. Thus we measure 3/4 hour.
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