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Burn ropes 
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#1
Nov1607, 03:01 AM

P: 30

You have got two ropes. Every rope is burning in a space of one hour. Burning speed of these ropes is not proportional. I mean, half rope not necessarily will burn in a space of half hour.
How can you measure out 50 minutes, using these ropes and fire only? 


#2
Nov1607, 06:43 AM

P: 2,179




#3
Nov1607, 06:48 AM

P: 30

I know you can do it if I mean 45 minutes, but I mean 50 minutes.



#4
Nov1607, 10:24 AM

P: 657

Burn ropes
Let's call them rope 1 (endpoints A & B) and rope 2 (endpoints C & D). Possibility I  Start by lighting A. When rope 1 burns out, 1 hour has elapsed, and we can move on to either lighting C or both C & D (note lighting just D is equivalent to just lighting C). Lighting C alone allows us to measure 2 hours total. Lighting C and D allows us to measure 1.5 hours total. Admittedly, we could also choose not to light C or D, and avoid using rope 2 entirely, with the result of 1 hour. Possibility II  Start by lighting A and B. When rope 1 burns out, 0.5 hours have elapsed, and we can move on to either lighting C or both C & D. Lighting C alone allows us to measure 1.5 hours total. Lighting C and D allows us to measure 1 hour total. Again, we could avoid using rope 2 at all, with the result of 0.5 hours. Possibility III  Start by lighting A and C. Unfortunately, the only measurable point after this is when both ropes burn out, which is after 1 hour, and there aren't any further ropes to burn. Possibility IV  Start by lighting A, B and C. We now have the option of lighting D when rope 1 burns out (after 0.5 hours), or not lighting it at all. If we light D after rope 1 burns out, we can measure 0.75 hours. If we do not light D at all, the only remaining measurement is 1 hour, which is when rope 2 burns out. Possibility V  Start by lighting A, B, C, and D. Again, we have no further options after we make this decision, and are forced into measuring exactly 0.5 hours, which is when both ropes burn out. Possibility VI  The empty set. Burn neither rope 1 nor rope 2, and we can measure 0 hours. And, that's it. 11 possibilities, where we can measure 0 hours, 0.5 hours, 0.75 hours, 1 hour, 1.5 hours, or 2 hours. Since none of these are 50 minutes, your solution must therefore be unreliable (aka arbitrary), or you're making futher assumptions that you're not telling us about the ropes, the fire, or one's ability to keep time. Hence, the best solution would be to measure out 0.75 hours (the closest to 50 minutes without going over), and then take your best guess as to when 5 minutes had elapsed beyond the 45 minutes. DaveE 


#5
Nov1607, 10:57 AM

P: 403




#6
Nov1707, 06:49 AM

P: 30




#7
Nov1707, 07:52 AM

P: 403

Two ropes , X & Y.
Cut rope Y in 3 pieces, Y1,Y2 & Y3 Light rope X (one endpoint) , and all the endpoints of Y1 , Y2 & Y3. At the time a piece Yn burns out, cut one of the remaining Y pieces, and light the 2 new endpoints. When all the pieces Yn burns out, start the clock. When X burns out, stop the clock. 50 minutes elapsed !  However, this solution is not implementable. 


#8
Nov1707, 09:03 AM

P: 2,179




#9
Nov1707, 09:16 AM

P: 30




#10
Nov1707, 09:18 AM

P: 30




#11
Nov1707, 09:23 AM

P: 2,179

Rogerio's solution is no solution. What if two of the three pieces burn up in an instant and the third piece takes the entire hour? 


#12
Nov1707, 10:03 AM

P: 30




#13
Nov1707, 10:18 AM

P: 657

Ahh, interesting solution! I certainly didn't think of it in terms of lighting and relighting at arbitrary points...
Of course, it's far fetched in a realistic sense since you'd spend the last minute frantically relighting as you got infinitesimally closer to the 10 minute mark, but the theory works. So you could in theory get a large degree of granularity out of the method. If you had the fastest hands imaginable, you could keep it burning in (say) 30 sections and time exactly 1 minute. DaveE 


#14
Nov1707, 10:37 AM

P: 2,179

Now I see. Clever.



#15
Nov1707, 10:24 PM

P: 403

That's why I said "not implementable". 


#16
Nov1807, 03:16 AM

P: 2,179




#17
Nov1907, 05:26 PM

P: 228

The underlying principle in Rogerio's solution, as you all know, is that the rope will burn six times faster with six fires. Here is an alternative approach that tries to match the burning times of six parts. It avoids the need to cut rope and start fires during the burn. This still has "granulation" problems. I'm certainly not saying it's better than Rogerio's.
Begin by cutting one of the ropes into very short lengths, say 60 pieces. Sort these randomly (or [1, 7, 13, etc.], [2, 8, 14, etc.], ... [6, ..., 60]) into six sets. Place the ten pieces in each set endtoend to make six equallength sections. Light one end of one section to measure ten minutes. As before, the fulllength rope is lit at one end at the start time. For better accuracy: Start burning one end of all sections simultaneously (the six sections can be arranged like spokes so that lighting the center will light all at once). When one burns out, find the one remaining that is closest to 1/2 the length of the longest remaining. Use the 1/2length one to measure ten minutes. Or measure from each of the six burnout times and take an average. The principle error here lies in how well a 10piece endtoend section replicates a single 1/6length piece with average burn rate. 


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