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?
The implication of the problem would allow for only a few solutions. Lighting any non-endpoint of the two ropes would be arbitrary and unreliable, since the ropes burn at inconsistant rates. Hence, the only reliable place to burn the ropes is at the endpoints. And there are only 4 possible ends to burn. Since the only timing device is the rope, and the burn rate is arbitrary, the only reliable start/stop times are when particular ropes burn completely. Hence, there are certainly a managable finite number of possibilities. Let's go through them. 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
My solution is implementable. And it's efficiency is highly competitive with your eleven possibilities. I am also using only known to you assumptions. Ropes can be changed in a form and be cut. The light can be started and stopped. And I am using the same and the only timing device - rope.
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.
Rogerio already gift a solution there you don't even need to cut it, just burn the remaining part somewhere in the middle. It is also the way I meant.
Rogerio cut the rope in 3 pieces. What did he use? 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?
I see you don't like cuttings :) Lets begin to burn one rope from one side, and the second rope from two sides and 2 another places on the rope. This will cut the second rope in three pieces. And then, use Rogerio's solution to keep the second rope burning in 6 places. In this way the second rope burns out in 10 minutes. Now you have exactly 50 minutes to wait for the first rope to burn away.
Ahh, interesting solution! I certainly didn't think of it in terms of lighting and re-lighting at arbitrary points... The premise is that you keep rope Y burning in 3 pieces no matter what, each burning from both ends. Effectively, this means that you'll be able to time 1/6th of the rope's full burning time (10 minutes). When one of the 3 segments burns out, you're left with 2, so you had better cut one of the 2 remaining ones in half so that you have 3 again, and light the new endpoints! Of course this becomes *more* feasible without cutting, since simpliy lighting an arbitrary point on one of your remaining pieces is the same (in theory) to cutting and re-lighting. Of course, it's far fetched in a realistic sense since you'd spend the last minute frantically re-lighting 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
The operation "cutting and relighting" would have to be repeated an infinite number of times. That's why I said "not implementable".
True, but even in the standard 45 minute solution, it required infinite precision to light the fourth end at the right time. Such precision is "not implementable" either.
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 end-to-end to make six equal-length sections. Light one end of one section to measure ten minutes. As before, the full-length 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 1/2-length one to measure ten minutes. Or measure from each of the six burn-out times and take an average. The principle error here lies in how well a 10-piece end-to-end section replicates a single 1/6-length piece with average burn rate.