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You are given two ropes that when lit burn in one hour. Which one of the following time periods CANNOT be measured with your ropes? a) 50 min b) 30 min c) 25 min d) 35 min.

I know 30 minutes is definitely possible as we could burn the rope at both ends.

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    $\begingroup$ @Oray seems like so, but the question here is a bit weird. Seems like a, c, d are all wrong. $\endgroup$
    – justhalf
    Dec 4, 2022 at 11:45

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I would argue that:

All of the times can be measured

Here's why:

Assuming you are allowed to cut the rope and set as many portions as you would like on fire, you can measure a time of $\dfrac{60}n$ by cutting the rope into pieces and making sure that there are $n$ points lit at any given time. With two ropes, any times that can be written as one factor of 60 or a sum of two factors will work.
For a): $\dfrac{60}2 + \dfrac{60}3 = 30 + 20 = 50$ minutes

For b): $\dfrac{60}2 = 30$ minutes or $\dfrac{60}4 + \dfrac{60}4 = 15 + 15 = 30$ minutes

For c): $\dfrac{60}{12} + \dfrac{60}3 = 5 + 20 = 25$ minutes

For d): $\dfrac{60}{12} + \dfrac{60}2 = 5 + 30 = 35$ minutes

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    $\begingroup$ assuming that the rope is uniform. $\endgroup$
    – Oray
    Dec 4, 2022 at 20:33
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    $\begingroup$ Even if it is not, whenever one section burns out you can just light another area. Cutting the rope and keeping the right amount of sections lit could be difficult, but it is mathematically possible. $\endgroup$
    – PiGuy314
    Dec 4, 2022 at 20:35
  • $\begingroup$ How can you be sure that N points are lit at any given time, if you don't know the initial distribution of the rope burning rate ? Are you cutting the ropes as they are burning ? $\endgroup$
    – Alois Christen
    Dec 6, 2022 at 12:56
  • $\begingroup$ Possibly yes to the latter question. You could cut the rope into enough pieces that needing to cut in again would be extremely unlikely, but it would always be possible. $\endgroup$
    – PiGuy314
    Dec 6, 2022 at 15:00

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