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Topic: NONHOMOGENOUS ROPE BURNING (Read 28848 times) |
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Evan
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Solution: Light both ends of one rope and one end of the other rope on fire, when the first rope is done burning, light the other end of the second rope. The first rope when lit at both ends only take 30 minutes to burn. After 30 minutes of burning from one end on the second rope, you are left with a 30 minute rope. When you light this rope on both ends it becomes a 15 minute rope allowing you to time 45 minutes.
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| « Last Edit: Dec 10th, 2002, 12:18pm by william wu » |
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cdsmith
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Re: EASY:NONHOMOGENOUS ROPE BURNING
« Reply #1 on: Jul 25th, 2002, 6:21pm » |
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Actually, under the terms of this question, you can time any rational fraction of one hour using only one rope. You homogenize the rope. Use your teeth, or your pocket knife, to break it up into small, uniform pieces. Make a big pile, and (in this case) divide it into fourths. Burn three of them one after the other, and you have 45 minutes.
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Flarb
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Re: EASY:NONHOMOGENOUS ROPE BURNING
« Reply #2 on: Aug 25th, 2002, 11:28pm » |
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It's likely the ropes are non-homogenous because they're made of multiple different materials, some of which burn faster or slower than others... I doubt you could rely on homogenizing it yourself.
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ggkrause
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Re: EASY:NONHOMOGENOUS ROPE BURNING
« Reply #3 on: Aug 26th, 2002, 10:24am » |
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The idea of separating the rope into small pieces has one serious drawback: it will change the overall burning time. By splitting it up, you give it a larger surface area, exposing more of the rope to the open air, and thus causing it to burn faster.
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-gk
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Chris (Hitchhiker)
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Re: EASY:NONHOMOGENOUS ROPE BURNING
« Reply #4 on: Nov 24th, 2002, 8:14pm » |
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I like Evan's result. Very elegant - and it works, if not "easily" than at least without any gotchas.
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wave
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #5 on: Mar 18th, 2003, 8:19am » |
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I like the kind of awnser Evan gave, but i don't agree it is 100% correct because after the first rope finishes burning you can't say rope 2 that remains is a 30 min rope, since it is non homegenous.
I don't know the correct awnser though.
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Icarus
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #6 on: Mar 18th, 2003, 7:02pm » |
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It has to be a thirty minute rope, no matter the inhomogeneity. You know that it takes an hour to burn the whole rope, and half that time is gone, so the remainder of the rope has to take an additional 30 minutes to burn.
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wave
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #7 on: Mar 19th, 2003, 12:36am » |
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You're right, i'd better be more careful next time... 
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NickH
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #8 on: Oct 3rd, 2003, 2:22pm » |
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Quote:| You know that it takes an hour to burn the whole rope, and half that time is gone, so the remainder of the rope has to take an additional 30 minutes to burn. |
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I disagree.
This conclusion is not purely deductive, but rests upon an assumption about the physical properties of the rope: namely that each section of the rope burns at the same rate in both directions. Since the rope is non-homogenous, and possibly asymmetrical, it's quite plausible that sections of it will burn faster in one direction than the other.
Nick
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Icarus
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #9 on: Oct 3rd, 2003, 4:54pm » |
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It's true that this does assume that the rope at each place burns just as fast in one direction as it does in the other. However, the puzzle statement explicitly states that the inhomogeneity in burn time is a result of varying thickness. That being the case, it should burn at the same rate for either direction - assuming that the thickness varies in a continuous fashion.
If the variance is not continuous, it may at least be assumed that the changes have some symmetry. After all, we are told it is a 1 hour rope, not a "1 hour from one end" rope.
So do you have a method of for timing 45 minutes with such a rope? I believe you need more information about the rope before you can proceed, but since you have it before you start, any measurements needed can be made then.
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NickH
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #10 on: Oct 4th, 2003, 1:40am » |
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OK, I'll grant you symmetry. We have a "1 hour rope" -- light it at either end, and it takes 1 hour to burn through. However, it has the curious property that, in either direction, the first half of the rope (by length) burns in only 15 minutes. Ergo, lit simultaneously at both ends, the rope is completely consumed in 15 minutes!
I admit it's hard to imagine the physical basis for such a huge variation in directional burn rate. However, I think smaller variations would be quite likely in an inhomogenous rope. Burning is a messy process.
Having said that, I don't have a better solution than the one Evan originally presented. I just think the puzzle is slightly flawed. Either the hypothesis of directional symmetry should be clearly stated (too big a clue?), or we must admit there is no neat, deductive solution.
Nick
<added>
For versions of this puzzle in which you have no control over and limited knowledge of the ropes...
A plausible reason for a rope to burn at different rates in different directions: gravity. Suppose both ends of the rope are suspended at equal height, so that the rope falls into a U-shape. Then it may realistically take 45 minutes to burn to the bottom, and then only 15 minutes to burn to the top! Clearly this is true even of a homogenous rope.
</added>
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| « Last Edit: Oct 4th, 2003, 6:54am by NickH » |
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NickH
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #11 on: Oct 10th, 2003, 11:25am » |
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Here is a method to measure 30 minutes using a '60 minute' rope, without assuming that the burn rate is independent of direction. An obvious extension of the method allows us to solve the full puzzle.
Chop the rope in half and simultaneously ignite both left ends. (Without reorienting either section, of course.) If they both burn out at the same time, we have measured 30 minutes. If not, instantaneously chop the unburnt portion of rope in half and ignite the non-burning left end. And so on, ad infinitum...
Since, at any given time, we have two pieces of rope burning in the same direction, once the entire rope burns out, we have measured 30 minutes!
If we treat this as a logic problem, and conceptualise the burning rope as a line segment with point(s) moving along it, this approach offers some advantages over Evan's solution. Firstly, as already mentioned, it does not implicitly assume a unidirectional burn rate. This appeals (to me) aesthetically. Secondly, it can immediately be generalised to yield a way of measuring any integer division of 60 minutes.
As a solution to a physical problem, it may or may not be more accurate than Evan's solution, depending upon the nature of the rope.
Nick
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william wu
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #12 on: Oct 10th, 2003, 1:23pm » |
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Your solution reminds me of a variant:
Given a nonhomogeneous rope that burns up in 60 minutes, time 60/n minutes, where n is an integer.
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| « Last Edit: Oct 10th, 2003, 1:23pm by william wu » |
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Zeke the Geke
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #13 on: Oct 20th, 2003, 10:44am » |
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To take a totally different slant on solving the problem, what if you focus on the lighter instead of the rope? Many (most?) lighters are designed so that you can see the butane inside. So mark the top of the butane on the lighter, then light the rope. Keep the lighter burning while the rope is burning for 1 hour. Mark the bottom of the butane. Then, using the rope if necessary, determine the height of butane corresponding to 45 minutes and make another mark at that point. Burn the lighter until the butane is lowered to that mark.
I realize I'm making assumptions about the style of lighter, total amount of butane available, etc...
Any other ideas in this vein?
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aero_guy
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #14 on: Oct 20th, 2003, 12:16pm » |
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From a used-to-be smoker, there ain't no lighter that is lastin you an hour.
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aero_guy
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #15 on: Oct 20th, 2003, 12:26pm » |
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Willy, the answer to your problem would seem to be that you need to keep n fires burning at all times, when they all go out you have timed it.
So, cut the rope into n/2 segments and light both ends of each. When one burns out light the middle of the longest remaining segment ad infinitum. For odd n you can have one segment lit at only one end. If that one goes out you will need to chop one of your other segments and light only one of the newly cut ends. Of course this gets real messy towards the end of the time as n gets large and there is probably a better implementation.
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| « Last Edit: Oct 20th, 2003, 1:57pm by william wu » |
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redPEPPER
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #16 on: Oct 23rd, 2003, 3:56pm » |
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Or you can simply cut the rope into n segments, whether n is even or odd, and light only one end, which also solves the directional problem. Just keep n segments at all time: if one runs out, cut another one (any other one) in the middle or anywhere else for that matters, and lit its end. The problem is that it's going to be very hectic in the last few minutes or seconds. If you want infinite precision you'll very likely have to make an infinite number of cuts before 60/n minutes are elapsed.
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TimMann
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #17 on: Oct 25th, 2003, 9:55pm » |
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Here's a related puzzle. Like the rope puzzle, this one also has the unstated assumption that the burn rate at a given point is the same regardless of which direction the rope is burning. Source: http://www.g4g4.com/contentsmmpp.html
Shoelace Clock. You are given some matches, a shoelace, and a pair of scissors. The lace burns irregularly like a fuse and takes 60 minutes to burn from end to end. It has a symmetry property in that the burn rate a distance x from the left end is the same as the burn rate the same distance x from the right end. What is the minimum time interval you can measure?
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psychedelicwind
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #18 on: Nov 24th, 2003, 10:51pm » |
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Hi every one. here is my way
light Rope A on both ends, and Rope B on 1 end.
When rope A has burned complitly, 30 min. has passed, so on rope B there is still 30 mins to burn.
light the other side of rope B and it will finish burning in 15 minutes.
30+15=45 minutes.
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Brad711
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #19 on: Dec 17th, 2004, 8:46pm » |
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When you burn both ropes, it's 2 hours of burning, cut in half because your burning them at the same time. Thats one hour. Then when you have half of one left, you cut that half in half, so .75 hours or 45 minutes. 2 hrs. / 2 = 1 hour. 1/2 hour is plain and 1/2 hour is devided in half, so 1/4 hour. 1/2 + 1/4 = 3/4 = .75 = 45 minutes.
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SteelSoul
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #20 on: Jun 11th, 2005, 12:43pm » |
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ok, take a deep breath and check this out.
It after 30 minutes it is NOT that half the rope will be burnt but rather a section that is WORTH 30 minutes. That means that there will be 30 minutes left to burn NO MATTER the length of the rope left.
length of the rope is meaningless in this riddle and the point of it being NonHomogenous is only to stop you from getting smart with a knife and start carving it up. the first solution is 100% correct
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NickH
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #21 on: Jun 11th, 2005, 1:45pm » |
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Quote:| length of the rope is meaningless in this riddle and the point of it being NonHomogenous is only to stop you from getting smart with a knife and start carving it up. the first solution is 100% correct |
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As long as the rope is not so nonhomogenous and ill-behaved that sections of it burn at a different rate in either direction, in which case the first solution is incorrect!
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Ziad D. Jaber
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #22 on: Aug 4th, 2005, 4:52am » |
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I beleive that it might be easeier solution to fold and twist rope A into hahlf and to fold and twist rope B into quarter and tying them together at their ends thus we will have a 30 min burning rope plus a 15 min burning rope adding up to a 45 mins burning rope .
What do you think
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rmsgrey
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #23 on: Aug 4th, 2005, 6:58am » |
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on Aug 4th, 2005, 4:52am, Ziad D. Jaber wrote: I beleive that it might be easeier solution to fold and twist rope A into hahlf and to fold and twist rope B into quarter and tying them together at their ends thus we will have a 30 min burning rope plus a 15 min burning rope adding up to a 45 mins burning rope .
What do you think |
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The trouble with that solution is that it needs the rope to burn at a constant rate - if half rope A burns in the first 5 seconds, while the rest takes 55 seconds, then folding A in half will give a rope that ignites the other end in 5 seconds...
Also, twisting two strands together will usually produce something that burns at a different rate from either strand separately (not necessarily an intermediate rate either)
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javitxusan
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Re: NONHOMOGENOUS ROPE BURNING
« Reply #24 on: Oct 2nd, 2005, 5:54am » |
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First at all, sorry about my english (it´s quite hard to try to answer this in other language).
Well, my solution its to cut one of the ropes in 4 pieces and burn the five (this four pieces and the second rope) at the sime time. The only thing u need its take the time since the last of the four short ropes burn out till the long one does. This is aproximate. Since the rope its nonhomogeneous you cant do it more exact cuttin the rope in 100 pieces and distribute them in 4 groups without any special order. then do the same:
burn the large one and at same time start to burn the first piece of all 4 groups (then next as soon as the last piece burn out). When the last piece of the 100 burn out there is more probability to aproximate to 15 minutes. (Obviously more pieces are equal to more exactitude)
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