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You are given a single long rope of length L (ft) with radius R (in), a very sharp knife, and 2 matches. Your task is to completely burn the rope in the minimum amount of time.

Calculate the total time required to burn the rope.

Details:

  1. The rope is considered stiff with a minimum bend radius of 10*R
  2. The rope is tightly woven with many fibers and may be considered to have a solid cross section
  3. You may cut the rope into more than one piece
  4. The rope burns at R in/min
  5. A match will light the rope instantaneously and will burn out as soon as the rope is touched
  6. Assume gravity
  7. Ropes in contact can transfer fire immediately to one another at the contact point only, but will themselves continue to burn at the constant specified burn rate
  8. Static structures
  9. The matches dimensions are 4R x R/4 x R/4

Be creative

Hints:

  • Think about 3D structures
  • Due to gravity a structure should be able to hold it's shape
  • Determine the burn rate per unit volume of rope material
  • Consider the shape of the structure sub-elements and how they transfer fire to one another
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    $\begingroup$ how long does a match burn, and how long does an end of a rope take to light? $\endgroup$ – dfperry Jul 10 '15 at 18:30
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    $\begingroup$ Will putting an unlit piece of rope next to a burning piece of rope set the unlit piece on fire? $\endgroup$ – dzastergamer Jul 10 '15 at 18:31
  • $\begingroup$ ^ All good questions. $\endgroup$ – Quark Jul 10 '15 at 18:32
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    $\begingroup$ @dperry I just added the specification to your question $\endgroup$ – Stephen Donecker Jul 10 '15 at 19:00
  • $\begingroup$ @ dzastergamer Yes $\endgroup$ – Stephen Donecker Jul 10 '15 at 19:02
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Cut the rope into a large number of very thin, tapered strips of length $\sqrt[3]{\frac{9LR^2}{2}}$. Arrange these strips into two spheres, with the pointed end of each strip at the center of its sphere. Support the lower half of each sphere with a hemispherical bowl of the correct radius1. However, leave an infinitesimally thin cone open from the top of each sphere to its center. Simultaneously drop each match directly above the centers of their corresponding spheres, so that both spheres simultaneously burn from the inside out.

The total volume of the rope is $12L\cdot\pi R^2$ (the factor of $12$ comes from the fact that $L$ is in feet, while $R$ is in inches), so the volume of each sphere is $6\pi LR^2$ and the radius is $\sqrt[3]{\frac{9LR^2}{2}}$; thus the amount of time it takes for the two spheres to burn is:

$$ \sqrt[3]{\frac{9L}{2R}}~\text{minutes} $$

1: If we're not allowed to use a premade bowl, then slice off a large number of infinitesimally thin strips from the rope and use them to weave a bowl of the correct size and shape.

Alternatively, if your knife is very sharp, make use of Banach-Tarski to rearrange the rope into a pair of spheres of arbitrarily small radius, which will then burn in an arbitrarily short amount of time.

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  • $\begingroup$ @ 2012rcampion Your physical solution could be optimal except for a few of the problem statement specifications. One, we can assume gravity, and since you did not specify how your spheres were constructed, they may fall apart. Two how do you get the lit matches into the sphere and only touch it once at the center? $\endgroup$ – Stephen Donecker Jul 10 '15 at 19:47
  • $\begingroup$ @Stephen Better? $\endgroup$ – 2012rcampion Jul 10 '15 at 19:59
  • $\begingroup$ This question morphed so much from what it was originally posted as. I'm partial to questions posed with at least general answers in mind but when accepted solutions turn out like this, you really can't complain :) $\endgroup$ – Quark Jul 10 '15 at 22:58
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Here's my idea: A powder will have more surface area exposed to the flame, so it should burn much faster than a solid rope, so the strategy is:

1. Cut the rope up into as many tiny particles as possible, to the point where the knife cannot cut any piece any thinner.
2. Arrange the "rope powder" thinly and evenly in 2 circles.
3. Place one lit match in the center of each circle, and watch the "rope" burn from the inside of each circle to the perimeter.

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  • $\begingroup$ I was thinking something very similar, pretty sure there needs to be more specifications. $\endgroup$ – Quark Jul 10 '15 at 18:40
  • $\begingroup$ I was thinking almost the same thing, but arranging the powder into two spheres, which each burn from the center outward. $\endgroup$ – 2012rcampion Jul 10 '15 at 18:54
  • $\begingroup$ @2012rcampion Excellent idea, but... how would you arrange the powder into a sphere, and then, how would you get the match to the center? $\endgroup$ – JLee Jul 10 '15 at 18:55
  • $\begingroup$ @2012rcampion: How would you arrange powder into a sphere? (A zero-gravity environment with oxygen?) $\endgroup$ – Ry- Jul 10 '15 at 18:55
  • $\begingroup$ @JLee, minitech: It's a mathematical problem, such things are irrelevant... but I'd probably pack the powder into two hemispherical bowls for each sphere, and drill an infinitesimally thin hole to the center through which to drop the match down. $\endgroup$ – 2012rcampion Jul 10 '15 at 18:59
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I have a proposal to minimize the burning time.

As we can twist the rope into circles of radius $10R$, I don't think we need to make any cuts. Instead, we turn it into a circle of the minimum radius and then overlap it on top as we complete a circle. As we keep doing this, we'll end up with a circular spiral of rope. Now we burn the rope at two points farthest to each other i.e. at one point on the bottom circle and its diametrically opposite point on the top circle, and let the fire burn across the spiral.

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  • $\begingroup$ @mmking I used the word long to imply a typical rope to define the domain where L >> 10R. $\endgroup$ – Stephen Donecker Jul 10 '15 at 19:30
  • $\begingroup$ @mmking Can you think of a way to minimize the burn time by making modifications to the rope? You certainly demonstrated that we can make 3D structures. $\endgroup$ – Stephen Donecker Jul 10 '15 at 19:31
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This is a new and different answer than the one I posted previously and I felt it warranted its own spot. My apologies if this is bad form.

Solution: Cut the rope into X+1 pieces, where X is the number of atoms in the rope. Upon splitting an atom, all of the pieces of the rope, along with the two matches and the person doing the cutting will be consumed in nuclear fire and destroyed in an atomic explosion. This burns the rope in an infinitely tiny fraction of a second, but sadly destroys the evidence. Such is the price of efficiency.

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    $\begingroup$ an atom-splitting knife! i gotta get me one of those. $\endgroup$ – JLee Jul 10 '15 at 21:34
  • $\begingroup$ Sadly the prototype was destroyed. Good news for the one selling them, because they're single-use! Bad news for everyone else though. $\endgroup$ – Kingrames Jul 10 '15 at 21:36
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    $\begingroup$ Warning: atom-splitting knife can accidentally cause nuclear explosions if the cutting edge is not contained within a vacuum. Vacuum carrying-case sold separately. $\endgroup$ – DoubleDouble Jul 10 '15 at 21:47
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An atypical idea..

Burn time: (Speed of your hands)*L /(Number of cuts)

Cut the rope into 2 (or more) equal pieces and place them side by side (or stacked for >2). Then slide one rope along such that it is just touching the first rope at one end. Then light the point where both ends are touching, and push one rope along the other rope as fast as possible. This will burn the other rope and your rope at that same speed (which will be as fast as you can move the rope pieces together).

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  • $\begingroup$ This is a very interesting idea. Could you calculate the minimum burn time for this solution? I must admit that my intention was to ask for solutions to static structures. Perhaps this could be a second phase question. $\endgroup$ – Stephen Donecker Jul 10 '15 at 19:55
  • $\begingroup$ @StephenDonecker The burn time would be dependent on how fast you can move your hands (which should be much faster than R in/min) ;-P $\endgroup$ – Mark N Jul 10 '15 at 19:57
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Assuming rope contact doesn't cause the other ropes to burn and you can only light one rope with one match:

Cut the rope in half then light the two ropes.

Otherwise, if you can (a) make other ropes burn each other, then see 2012rcampion's sphere comment, and (b) if you can light more than one rope with a match but not have the ropes light each other, then you'd have to arrange strips of rope that come together at a point (probably in the shape of a sphere somehow), but at this point the solution is too convoluted to make sense anyway.

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Well, the OP stated in the comments of my other answer that the surface area (and thus the oxygen available to the flame) will not affect the burn rate in the fantasy universe of this puzzle, so, there is no need to cut the rope into powder anymore...

The structure that will burn fastest then becomes a tightly-packed sphere, since it minimizes the distance from the center to the perimeter.

Therefore, we must cut the rope in half and construct two tight spheres with each half by winding it, with a small hole in the bottom of each sphere, just large enough for the match to be lit and then inserted vertically up into. When the flame hits the center of each sphere, the tightly-packed rope balls would burn up.

I think this is optimal, given the conditions in the puzzle.

However, I realize that the OP probably wants to know exactly HOW the spheres were constructed. Moreover, how is the rope winded, and how is it cut beforehand in order to ensure that there is NO AIR in between the pieces of rope.

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  • $\begingroup$ You are correct that a densely packed solution is optimal, and a self standing structure depends on the sphere's construction. I didn't specify the rope winding or limit your cuts however, you could consider it solid unless you cut it to powder. I am looking for a formulaic solution to the problem based on the structure geometry. $\endgroup$ – Stephen Donecker Jul 10 '15 at 22:06
  • $\begingroup$ By the way you are on the right track. $\endgroup$ – Stephen Donecker Jul 10 '15 at 22:08
  • $\begingroup$ None of my ideas use a formula. I thought of cutting a piece from each rope the length of the radius of the spheres to be created. Then, I would use a bowl of some sort, maybe a spherical fish bowl, to help construct the spheres. then, light the ends of each of the 2 pieces of rope i cut out, and insert the burning rope down into the holes, thus accomplishing the goal. HOWEVER, if you want me to go into HOW I packed the rope where there is no air in between, I don't think I am capable of mathematically approaching that. $\endgroup$ – JLee Jul 10 '15 at 22:20
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If each rope is immediately lit up when it comes in contact with fire,

Chop it up so that it has minimum length $l$ inches. Arrange each piece side by side such that it forms a rectangle. Light up a corner of the rectangle. The fire will spread across the ends of the rope immediately, so basically all ropes start burning at the same time. It will take $l/R$ minutes to burn one piece, so it will take $l/R$ minutes to burn every single piece.

So now it boils down to how small you can chop each piece. When $l\to0$, $l/R\to0$.

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