Last week I posted my first puzzle. I only received one response this week; it was from Calden, whose solution was quite a bit different from mine but which also made plenty of sense. I’ll cover his solution later on; for now, here is the intended solution.

As you recall, we have two one-dimensional ropes at our disposal that each take one hour to burn. The key thing to note is that if you light both ends of a rope, the two flame fronts will bisect the remaining burn-time of the rope — e.g. for a one-hour burn-time rope, the two flame fronts will meet thirty minutes later. From here it is a short step to realize that if you had lit one end of the other rope at the same time, then it would have thirty minutes of burn-time left once the first rope burns all the way through. At this point, you light the second end of the second rope. The two flame fronts bisect the remaining burn-time, counting out another fifteen minutes.

Alternative solution

Calden’s solution was quite a bit different. He assumed you had something tall — like a tree — to hang the rope from, as well as something heavy — like a rock — to hang from the rope. Suspending the rock from the tree using the rope, he formed a pendulum. Setting the (ideal) pendulum oscillating, he then lit one end of the other rope, counting the number of oscillations of the first rope in one hour. To count out forty-five minutes, he simply let the first rope go for three quarters of the oscillations as were present in one hour.