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Proof of recursive formula for “fusible numbers”

There are N ropes given and

it is given that each ropes burn in 1 hour

You have to calculate 40 minutes using these. How can I do that?? I made 45 minutes, but 40 is not made by me. Help me please.

Thanks in advance.

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devsda
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  • Is there any other information about the lengths of the ropes, the type of rope, anything? How did you get 45 minutes? – Daryl Sep 15 '12 at 22:22
  • first burn first rope from both end and burn second rope from one end, when the first rope burns completely means 30 minutes passes. At this time, burn second rope from another end , it will burn completely in 15 minutes. Means 30 + 15 minutes = 45 minutes – devsda Sep 16 '12 at 05:08
  • Lighting ends of ropes and lighting other ends when some ropes have burnt down can time periods that are multiples of $\frac{1}{2^N}$ hours but some kind of cheat a finite number of ropes don't seem able to time $\frac{2}{3}$ of an hour. – Angela Pretorius Sep 16 '12 at 07:29
  • exactly, I also thought the same thing, but this is asked in interview to me, I don't know why interviewer ask wrong question in an interview – devsda Sep 16 '12 at 09:59
  • @jhamb: Often the interviewer will be interested in seeing how the candidate reacts to problems he cannot solve. In many situations this can be more relevant for predicting fit than just knowing which riddles the candidate has memoized well enough to solve on the spot. (For example, if the job involves figuring out not only how to do things but also whether they can be done profitably at all, it wouldn't do to hire someone who is not comfortable with declaring that he doesn't see a good way to do something). – hmakholm left over Monica Sep 16 '12 at 14:11
  • Voting to close as subproblem of Proof of recursive formula for "fusible numbers" – Gerry Myerson Sep 17 '12 at 06:03
  • Every fusible number is a rational number with denominator of the form $2^n$. So $\frac23$ is not a fusible number. – TonyK Jan 26 '16 at 18:47

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Assuming that the ropes burn at a uniform rate, just loop one rope into three coils of equal length, thereby dividing it into thirds, and mark the end of one loop. Light the end further from the mark; your $40$ minutes will be up when the rope burns down to the mark. If no other way to mark is available, tie one of the other ropes around your burning rope at the desired point.

Brian M. Scott
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    Usually in this kind of problem, one is not permitted to assume the ropes burn at a uniform rate, nor that one can find, say, exactly one-third of any given length. – Gerry Myerson Sep 17 '12 at 06:01
  • @Gerry: I’ve only ever seen such a problem once before, and that was here a while back, so I’ve no idea what the standard ground rules are. The OP’s $45$-minute solution requires some degree of uniformity, so I just tossed out the sort of idea that I might toss out in the interview setting. – Brian M. Scott Sep 17 '12 at 06:08