Sunday, May 15, 2011

Counting 45 minutes

There are two pieces of identical ropes. If one burns a rope from one end, it takes an hour to completely burn the whole rope. However, the materials that make up the rope are not even. In other words, if one cut the rope into two equal halves, it does not necessarily takes half an hour to completely burn one of the halves. 

Now, you are asked to count 45 minutes, but you do not have any clock or watch. How would you do so by using these two pieces of ropes and matches?


  1. catcat8:02 AM

    Any tips, please?

  2. catcat: Try to think about the ways you can burn the rope :P

  3. catcat12:36 PM

    step 1)
    Burn both ends of rope 1 & one end of rope 2.

    step 2)
    when rope 1 is completely burnt, that's the 30mins mark, light up the another end of rope 2.

    Once rope 2 is completely burnt, then, that's the 45mins mark :)

  4. catcat: Hehe yup that's the correct answer :)

  5. As a follow-up puzzle, suppose someone follows catcat's solution and finds, to their consternation, that when rope 2 is completely burnt out only a little over 44 minutes have elapsed! How can this be?

  6. Nice: Because there's is some delay in igniting the rope?

  7. No, there is almost no delay in igniting the ropes, and yet the procedure measures about 44 minutes. (There's no special significance in 44 minutes: it could be 43 minutes, or 46 or 47 and a half minutes.) The puzzle is: how is this logically possible?

  8. Nick: Hehe I can't figure out how this is logically possible. Would like to hear your answer :)

  9. OK, consider a rope that, because of its asymmetrical construction, burns slightly faster in one direction than the other. What could cause this? Well, the rope might have loose strands that stick out in one direction only, meaning that a flame can progress a little faster by igniting those strands. Whatever the reason, let's say this rope takes 28 minutes to burn from end A to end B, and 32 minutes from end B to end A.

    Rope 1 happens to be made of two such ropes, with the "B" ends joined together. So rope 1 takes 28 minutes to burn from either end to the join, then a further 32 minutes to burn from the join to the other end. Thus rope 1 takes 60 minutes to burn from either end to the other end, as we require.

    But now consider what happens when we ignite boh ends of rope 1 simultaneously. The flames meet at the join after only 28 minutes!

    Rope 2 is just a "regular" rope, without any asymmetry, so it will measure a further (60 - 28)/2 = 16 minutes, giving a total elapsed time of 44 minutes.

    That's one logical possibility. All it takes is a slightly unusual rope. :)

  10. Nick: This is an interesting observation!

    Basically, what you're arguing that it doesn't necessarily take 30 minutes to completely burn a rope by igniting both ends if it takes 1 hour to completely burn it by igniting one end, which is the assumption of the answer. If this assumption is incorrect, the answer will fall as you describe.

    I'm not very good at physics so I'm not sure how you can make such rope with the property that you describe. But I'd guess that it's possible...

    So, the question probably should state that though the materials can be uneven, the burning rate for any material in the rope are the same from either direction. That'd make the question more complete :P

  11. Yes, you could add that condition, although it might provide too big a hint! Or you could ad something like, "the burning rate at any point on a rope is dependent only upon the thickness of the rope at that point." That condition is unnecessarily restrictive, but at least it wouldn't give the game away!

