Thread: Burn ropes View Single Post
P: 657
Burn ropes

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