There is a common brain teaser that goes like this:
You are given two ropes and a lighter. This is the only equipment you can use. You are told that each of the two ropes has the following property: if you light one end of the rope, it will take exactly one hour to burn all the way to the other end. But it doesn't have to burn at a uniform rate. In other words, half the rope may burn in the first five minutes, and then the other half would take 55 minutes. The rate at which the two ropes burn is not necessarily the same, so the second rope will also take an hour to burn from one end to the other, but may do it at some varying rate, which is not necessarily the same as the one for the first rope. Now you are asked to measure a period of 45 minutes. How will you do it?
Now I usually love brain teasers but this one frustrated me for a while because I could not prove that if a rope of non-uniform density is burned at both ends it burns in time $T/2$. I think I have sketched a proof by induction that shows that it's not actually true.
Given a rope of uniform density the burn rate at either end is equal so clearly it burns in time $T/2$. Now, consider a rope of non-uniform density, the total time T for this rope to burn is the linear combination of the times of the uniform density "chunks" to burn, i.e. $T = T_1 + T_2 + \ldots + T_n$. So consider, $T/2 = T_1/2+ T_2/2 + \ldots + T_n/2$. If we look at each $T_i/2$ this is precisely the time it takes to burn the uniform segment $T_i$ if lit at both ends. Therefore, in order to arrive at a rope that burns in time $T/2$, one would need to light each uniform segment on both ends, not simply the end of both ends of the total rope. What am I doing wrong?
Best Answer
You say "Therefore, in order to arrive at a rope that burns in time $T/2$, one would need to light each uniform segment on both ends, not simply the end of both ends of the total rope. What am I doing wrong?"
If I understand what you mean here, the idea is that you cut the rope into short segments, each one of (approximately) uniform density. Then you light the first one at both ends. As soon as it's done burning, you light the next one at both ends, and so forth.
That's certainly one way to arrive at a rope that burns in time $T/2$. Your mistake is thinking that's the only way -- i.e., instead of "one would need to," it's "one could." But here's another way to do it: Suppose you knew, or somehow guessed, exactly how far the rope would burn in time $T/2$ if lit from one end (say the left). If you imagine cutting the rope at exactly that point, the other portion will also burn in time $T/2$. But you don't actually have to make the cut in order to start burning the other portion -- you can light it from the other end. So you don't need to know where to make the cut; you just light both ends. When they're done burning, at time $T/2$, you'll know where the cut would have been.