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This puzzle is an harder version of a very famous one. It has been asked many times on PSE, but this time the question is slightly different (1) (2). So please read this question twice before marking as duplicate.

You are given an unlimited set of matches and two different ropes, each rope has the following properties:

  • if you light the end of a rope it will burn completely in one hour.
  • the fire does not burn the ropes evenly.

That means (for example) that the fire can take only five minutes to burn the first half of a rope and then fifty-five minutes to burn the other half. You cannot make any assumption on the burning speed of the ropes.

You start at time $t=0$ by lighting one or more fires. Afterwards you can instantly light or shut any fire anytime if you want to; you can perform these actions multiple times or simultaneously in a very precise instant of time of your choice.

  1. Can you measure 45 minutes? (This was the original question).
  2. Can you measure 10 minutes?
  3. How many intervals of times can you measure and which are them? Intervals must have an integer duration in minutes.
  4. What if there were $N$ ropes instead of two?
  5. What if you were given only 100 matches instead an unlimited number?

(1) Burning ropes as timers
(2) Burning ropes as timers - How many time intervals can be measured?
In the second linked question they state an example of answer: "For example, when N = 1, the answer is 2". That's not valid for this question because they impose some constraints that are not valid here.

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  • $\begingroup$ Confused by your last comment. What is your answer for N=1? $\endgroup$ – hexomino Dec 10 '19 at 12:29
  • $\begingroup$ See @Jan-ivan answer $\endgroup$ – melfnt Dec 10 '19 at 12:37
  • $\begingroup$ Does this answer your question? Burning ropes as timers $\endgroup$ – Thomas Markov Dec 10 '19 at 13:31
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    $\begingroup$ No I stated it very clearly in my question, haven't I? $\endgroup$ – melfnt Dec 10 '19 at 13:35
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1):

For 45 mins just light both ropes - only one from both sides. After one burned, light both ends of second one.

2): Because "you can instantly light or shut any fire anytime if you want to"

So for example I can use 1 rope and keep it on fire on 6 different places to measure 10 minutes? When any part of rope would be used, I just instantly set on fire different part - if it would make 7 fires, I would instantly extinguish one fire and continue this for 10 mins. This could use nearly infinite number of matches.

3)

Anything 60/x, when given one rope, and x is number of fires, so infinite. For "integer duration in minutes" that would be for one rope 1,2,3,4,5,6,10,12,15,20,30 and 60.

4)

Anything 60/x + 60/y + … , and 60/x + 60/y1 + 60/y2 +… and some other combinations. For two ropes there are already lots of possibilites.

5)

You can't keep on all fires at any given time, because of unknown burning speed per part, so only burning from two ends of each rope - like original puzzle. So can't measure under 30 mins. After that you can measure something like 45 mins, 60,75 and so.

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  • $\begingroup$ 2) is correct. 3) and 4) are not precise. 5) is correct but what about the $N$ ropes? $\endgroup$ – melfnt Dec 10 '19 at 10:12
  • $\begingroup$ Sorry, about 3) and 4) i have updated the question $\endgroup$ – melfnt Dec 10 '19 at 10:19
  • $\begingroup$ I think you answered question 3) thinking about having one rope, not two... also a little bit of clarification on how can you obtain these intervals would be nice $\endgroup$ – melfnt Dec 10 '19 at 10:53
  • $\begingroup$ It seems to me that #2 requires not only that you be infinitely quick but that you be willing to do an infinite amount of work to time a finite interval. $\endgroup$ – Gareth McCaughan Dec 10 '19 at 11:58
  • $\begingroup$ You can indeed: "you can perform these actions multiple times" by multiple I meant "arbitrarily many" $\endgroup$ – melfnt Dec 10 '19 at 12:39

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