Measure time with burning fuse puzzle

You have two lengths of fuse. Each will burn for exactly one hour.

But the fuses are not necessarily identical and do not burn at a constant rate.

If half the fuse is burnt, then it does not mean that it had burnt for 30 minutes.

There are fast-burning sections and slow-burning sections, on the whole length of the fuse.

How do you measure thirty minutes using only the fuses and a lighter?

How do you measure forty-five minutes using only the fuses and a lighter?

Solution:

Light either fuse at the end, and you won't know how much time has elapsed until the lighted fuse reaches the other end: sixty minutes. No good.

Notice that you can find the middle (by length) of either fuse without a yardstick. Just bend it in half.

But light either fuse in the middle, and you won't learn anything. Because the fuse burns unevenly, the burn will reach one end before the other. While the combined burn times of the two halves must sum to sixty minutes, that is unhelpful here.

To give an extreme case, it might be that the right half of the fuse is super-fast burning and will reach the right end in one minute. Then the left half would have to be super-slow burning and take fifty-nine minutes to reach the left end. That doesn't help you decide when thirty or forty-five minutes have passed.

Have we exhausted all the possibilities? No. A clever scheme is to make an X with the two fuses. Set them out so that they cross and touch in the middle of each fuse's length. Then if you light one corner of the X, the flame will burn to the middle and branch out in three directions at once. All this does is to light the crossing fuse in the middle (something we've already decided is useless) at an unknown time in the future (however long it takes for the lit end to burn to the crossing point). It's garbage in, garbage out.

Does this exhaust all the possibilities? No, you can also light a fuse at both ends. The burn rate of the two flames means nothing in itself, and there is no guarantee that the two flames meet in the middle. But they do meet, obviously. When that happens, two flames will have traversed the full sixty-minute length of fuse. That means that the time interval' must be half of sixty minutes, or thirty minutes.

Great! That solves the simpler version of the problem.

It also gives us a start on the forty-five-minute version. By burning one fuse at both ends, we measure thirty minutes. If we could measure fifteen minutes with the second fuse, the problem would be solved.

What we've established is that you can halve the burn time of any fuse by lighting it at both ends. Had we a thirty-minute fuse, we could light it at both ends the instant the sixty-minute fuse's two flames met. This would give us fifteen minutes more, for a total of forty-five minutes. We don't have a thirty-minute fuse.

We can make one by burning the second fuse, from one end, while we time thirty minutes with the first fuse. Here's the whole procedure: At time zero, light both ends of fuse A and one end of fuse B. The fuses must not touch each other. It takes thirty minutes for fuse A's two flames to meet. When they do, there is exactly thirty minutes left on fuse B. Instantly light the other end of (still-burning) fuse B. The two flames will now meet in fifteen minutes, for an elapsed time of forty-five minutes.

Source:

"How Would You Move Mount Fuji?"

Microsoft's Cult Of The Puzzle
By William Poundstone