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You have 2 identical ropes which burn at a specific rate, and an unlimited supply of matches. When you light one end of a rope, the fire will take exactly 1 hour to travel to the other end of the rope.

You need to measure exactly 45 minutes. You must start by lighting one or both of the ropes. You can light or extinguish either end of either rope later, but you must only do this immediately after a rope has finished burning, as this is the only accurate way to measure elapsed time. You also must not light anywhere but the end of a rope, or any other form of guessing. You may finish with or without any remaining rope.

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    $\begingroup$ Do you have to light an end of the rope? I guess finding the 3/4 part of a rope and lighting it there isn't an option? $\endgroup$
    – Jerry
    Commented May 16, 2014 at 17:17
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    $\begingroup$ @Jerry No, I forgot to mention that. Lighting the middle of the rope requires guessing. (Folding the rope isn't relevant to this question.) $\endgroup$
    – Kendall Frey
    Commented May 16, 2014 at 17:20
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    $\begingroup$ When I first read this question, I thought it was missing information; I read "2 ropes which burn at a specific rate" and assumed each rope burned at a different rate. Then I saw 1 hour and assumed that was for the first rope only, and couldn't find the time for the second. I ended up figuring it out and am now actually really fascinated by this question, but I think it might be better to edit to clarify. Two ideas on how to do this: move the "specific rate" bit to the second sentence, or mention the 1 hour time for both ropes in the first? $\endgroup$
    – WendiKidd
    Commented May 17, 2014 at 1:46
  • $\begingroup$ @WendiKidd I've tried to clarify it. $\endgroup$
    – Kendall Frey
    Commented May 17, 2014 at 5:03
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    $\begingroup$ The question as I know it, has no specific rate at which ropes burn, but instead emphasizes the fact that they have none - it's only guaranteed that burning a rope takes one hour. So if a rope is 3 meters long, it may burn 2 meters in the first half hour and 1 meter in the second. $\endgroup$
    – SQB
    Commented May 21, 2014 at 12:25

6 Answers 6

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Hint: You can light both sides of one rope. Solution:

Light rope $A$ on both sides so that the rope will be gone in $30$ minutes. You need to light rope $B$ at the same time you light rope $A$. Once rope $A$ is gone, light the other side of rope $B$. Rope $B$ will be gone after another $15$ minutes. That will add up to $45$ minutes.

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My solution:

Fold one rope in half and then in half again. It will then be a quarter of the original length. Lay the folded rope parallel the other rope. Light the other end of the unfolded rope. When it reaches the folded rope then 45 minutes have passed.

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    $\begingroup$ Ropes are not guaranteed to burn evenly, or even that they are any given length. This incorrectly assumes both. $\endgroup$
    – Kendall Frey
    Commented Sep 24, 2014 at 11:23
  • $\begingroup$ @kendallfrey doesn’t the op say ‘burns at a constant rate’? $\endgroup$
    – David LeBauer
    Commented Feb 20, 2022 at 5:49
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Alternative solution (?)

Lay the ropes next to each other and light each rope at opposing ends. WHen the flames meet in the middle, 30mins has passed. Then move the ropes so the unburnt sections are next to each other, when the flames meet again, 15mins has passed - hence 45mins.

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    $\begingroup$ This assumes that the speed at which the flame travels is constant. The question states that the only accurate way to measure time is by how long it takes an entire rope to burn. $\endgroup$
    – Kendall Frey
    Commented May 20, 2014 at 11:37
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    $\begingroup$ Ah well, it was worth a try ! $\endgroup$
    – Pat Dobson
    Commented May 20, 2014 at 18:08
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My attempt at solving the puzzle

1. Burn one ropes at both ends. After finished burning 30 has been passed
2. Burn another rope at both end and the middle.
3. Wait until one of the segments has been burned off.
4. Burn the middle of the remaining segments
5. Repeat step 3 and 4 until all the ropes are burnt

This answer disproves that lighting the middle of the rope requires guessing, as this answer doesn't assume that the rope will burn exactly at middle, as long as you burn anywhere between the two ends, it will work

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  • $\begingroup$ I like this solution! 2 flames on a 1 hour rope will take 1/2 hour, and 4 flames will take 1/4 hour, as long as you keep 4 flames. $\endgroup$
    – Matthew Jensen
    Commented Oct 1, 2020 at 23:49
  • $\begingroup$ With this solution you can theoretically time any 1/n+1/mhours where n and m are positive integers $\endgroup$
    – Matthew Jensen
    Commented Oct 1, 2020 at 23:53
  • $\begingroup$ @MatthewJensen You need to be infinitesimally fast, though, as mentioned here. $\endgroup$
    – Paul Panzer
    Commented Oct 2, 2020 at 5:12
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Here's my solution:

Take one rope and light both ends at the same time while also lighting one end of the second rope. That should take 30 minutes to burn completely leaving 30 minutes of the second rope unburned. As soon as the first rope has burned completely light the second end of the second rope. The remaining 30 minutes of the second rope will be halved leaving you with 45 minutes total.

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    $\begingroup$ This has already been posted $\endgroup$
    – Yout Ried
    Commented Sep 7, 2018 at 2:20
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Single Rope Solution

1. Fold a rope in half
2. Light one end
3. When fire reaches fold, light other end
4. Fires will meet at 3/4 length of rope or 45min

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  • $\begingroup$ Nope. As per the question, "immediately after a rope has finished burning ... is the only accurate way to measure elapsed time." and also more. $\endgroup$
    – Kendall Frey
    Commented Jul 9, 2015 at 20:24
  • $\begingroup$ Now that I have looked back I see your assertion '... is the only accurate way to measure elapsed time' but I have to respectfully disagree. $\endgroup$
    – Stephen Donecker
    Commented Jul 9, 2015 at 20:39
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    $\begingroup$ What are you disagreeing with? I don't see any way your answer is a solution to the puzzle. $\endgroup$
    – Kendall Frey
    Commented Jul 9, 2015 at 21:05
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    $\begingroup$ I'm not sure you're allowed to disagree with the puzzle's premise.... $\endgroup$
    – Chris
    Commented Jul 9, 2015 at 21:52
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    $\begingroup$ The way I've always seen this puzzle is that different parts of the rope can burn at different rates, and it is only guaranteed that the rope takes an hour to burn in full. Folding a rope in half does not divide it into two 30-minute sections. Otherwise, you could fold the rope into quarters and just wait for three-quarters of it to burn. $\endgroup$
    – f''
    Commented Jul 10, 2015 at 0:07

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