Calculating 25 minutes by burning 2 inconsistently burning 1-hour ropes

You are given two ropes and are told that they burn at inconsistent rates, but will always take 1 hour to completely burn up. This means that cutting one of the ropes perfectly in half will not give you two smaller 30-minute ropes.

You are told that you need to approximately calculate 25 minutes by burning these ropes in some fashion.

How do you accomplish this?


Hint:

first ½ of hint

I carefully chose approximately instead of precisely

second ½ of hint

because to calculate it precisely would require that you ignite an infinite number of flames

  • @ascallion
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    21 year ago

    Fold both ropes to find 5/12ths of each length of rope. Cut off those lengths and burn both of the 5/12 portions? Sounds like it’d work.

    • stravanasu
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      1 year ago

      Burn the four portions cut this way from the ropes, starting to burn them simultaneously. The 5/12 should be taken starting from the end of the ropes. When the second of these four pieces has burnt fully, say it’s approx 25 min.

    • MrMusAddictOPM
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      11 year ago

      Good suggestion! That would work well if the ropes burned consistently, but it’s possible that those 5/12 portions are more/less volatile than the remainders, meaning they could burn for 25 minutes, but also they could burn for 10 minutes, or 50.

      • @ascallion
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        1 year ago

        Ah, very tricky. I’m not very good at math, so my next idea is to brute force it by unraveling both ropes into their individual threads, counting up 5/12ths of each ropes’ threads and re-tying and burning those threads. Each thread would be the same length as the original rope and would have the same inconsistencies, you’d just be left with a skinnier rope that hopefully has 25min of material left to burn between the two.

        Hopefully somebody figures out the real answer and can chime in, ::: I’m curious how lighting multiple fires helps :::

        • MrMusAddictOPM
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          21 year ago

          Waited 24 hours in case anyone could figure it out. I’ve posted the solution as it’s own top comment.

  • MrMusAddictOPM
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    21 year ago

    Answer context: if you light both ends of one rope, the two flames will always meet each other and fizzle out in exactly 30 minutes.

    Key takeaway:

    • 1 flame on a rope = 1 hour
    • 2 flames on a rope = ½ hour

    So you may (correctly) think that 3 flames = ⅓ hour. But there’s trick to it. Those 3 flames should always be; 2 on the end, and 1 arbitrarily in the middle.

    Doing this will actually cause 1 rope segment with 3 flames to turn into 2 rope segments with 4 flames. Since that middle flame was placed arbitrarily, we can expect one of the segments to burn up before the other. When this happens, we just place a new flame arbitrarily in the middle of the remaining segment, splitting it back into 2.

    As long as we keep adding flames so that we always have 2 segments, the total rope will ultimately burn in 20 minutes (⅓ hour).

    This is generalizable; to achieve any fraction of 1/x hours, we need to always maintain x-1 segments (min 1). So for ½ hour, we need to maintain 1 segment. For ⅐ hour, we need to maintain 6 segments, etc.

    Final answer: 25 minutes can be expressed as ⅙+¼ hours. Therefore we need to burn the first rope such that we always maintain 5 segments. Then once that’s fully burnt, we just burn the second rope such that we always maintain 3 segments.