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You are given three ropes and a lighter. The time required to burn through rope $1$ with a single flame is $20$ minutes. The time required to burn through rope $2$ with a single flame is $30$ minutes. The time required to burn through rope $3$ with a single flame is $60$ minutes. The burn rates in different sections of the rope are unknown and vary. For how many integers $m$ in the set $\{5, 10, 15, \ldots, 100, 105, 110\}$ is it NOT possible to measure out exactly $m$ minutes using the three ropes and the lighter?


I know that $20, 30, 50, 60, 80, 90, 110$ are all possible by just lighting the ropes after one finishes after the other (to get $110$, for example, light the $20$ minute rope, when it finishes, light the $30$ minute rope, when it finishes light the $60$ minute rope). However, I'm wondering if it's possible to get any intermediate values. I've seen a variant of the puzzle before, and I'm thinking it might be possible to light the other end of the rope when one of the shorter rope finishes?

I'm not completely sure if it's possible here though, because I'm given the assumption that "the burn rates in different sections of the rope are unknown and vary", and I'm not fully sure about how to interpret this

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    $\begingroup$ Do we have prep time? As in, is it ok if I can measure 5 minutes, but only after a 20 minute wait? $\endgroup$ – Bass Aug 22 at 21:00
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    $\begingroup$ The classic version of this kind of puzzle is Burning Ropes As Timers. The remark about burn rates is in order to avoid solutions that involve folding/cutting the ropes into halves and quarters, because those are not guaranteed to burn up in half/quarter the time. Regardless of that, a rope is guaranteed to burn in half the time if you light both ends. $\endgroup$ – Jaap Scherphuis Aug 22 at 21:30
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All except 5 and 105 minutes can be done. If we get prep time, 5 minutes is also possible.

5 min (prep needed):

start 20 and 30. When the 20 finishes, extinguish the 30. Now you have a 10 minute rope, which you can light at both ends. (Prep time can be halved to 10 min by lighting both ropes at both ends.)

10 min:

light both ends of the 20 min rope

15 min:

Light both ends of the 30 min rope.

20 min:

Use the 20 min rope.

25 min:

Light both the 20 and 30. When the 20 finishes, light the other end of the 30 too.

30 min:

Use the 30 min rope

35 min:

20+15

40 min:

30 + 10

45 min:

Do 15, then light both ends of the 60.

50 min:

20 + 30

55:

light both ends of the 60. when it finishes, do 25 from above.

60:

There's a rope for that.

65:

20 + 45

70:

60 + 10

75:

do 60, then 15.

80:

60 + 20

85:

60 + 25

90:

60+30

95:

60 + 20 + 15

100:

60+30+10

105:

Impossible, even with prep

110:

60 + 30 + 20

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If you have infinitesimally good hands the 5 min can be done without prep:

Light the 20 min at both ends and in the middle. The instant either half has burned up light what remains of the other half in the middle and start over. If you are quick enough the rope will at all times burn at four ends and be consumed after precisely 5 min. Things may get a bit frantic towards the end, though.

Note that by a similar if even slightly more mechanically demanding method you could also measure thirds: Light one end and the middle and proceed similar to the quarter method, only in the unlikely event that the single end burning bit finishes before the other you have to light the other one in the middle and extinguish one of the four burning ends.

And if that wasn't silly enough for your liking you can, in principle, measure any n-th part of a rope's denomination by simply (well, I say simply) making sure that it is at all times burning at n ends.

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    $\begingroup$ Now try to measure some irrational number of minutes. $\endgroup$ – WhatsUp Aug 23 at 2:28

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