Of Ropes and Hills !
1) A tribe uses ropes to measure time. A rope takes 60 minutes to burn when lit at one end. How do you measure 30 minutes ? ( Hint aka answer - Light both ends !) Now, given an unlimited supply of these ropes, how do you measure 45 minutes ?
2) At 6 AM on a Monday morning you set out to climb a hill ( I know you won't but let's assume anyway;-). You vary your pace, take several breaks, and reach the summit at 5 PM. You spend a few days there, and start climbing down at 6 AM on Friday morning. You again vary your pace, take breaks as and when you feel like it , and reach the bottom late in the evening ( not necessarily at 5 PM). Prove that there is a certain time of the day, when you are at exactly the same spot when climbing up and down !
Note - I think someone will hopefully post the solution to the cofftea problem. There is an alternative very simple solution ! Do give it some thought. You could speculate on what the answer might be, even if you don't have a rigorous concrete proof :-) Do not assume that your ideas are silly or anything of that kind ! Others could well be having the same doubts :-)
Karthik, i love the early morning exercise my mind has been getting the past two days :). My gut feeling for the cofftea problem was that the quantities/proportions were the same, but I struggled with an elegant mathematical proof. Thanks to Anonymous for the detailed explanation!
ReplyDeleteRegarding the Rope problem:
1. Cut one rope into half its size. Light both ends (thanks for the hint!) - time it takes to burn = 15 min
2. Right after, light both ends of a full rope. Time taken to burn = 30 mins
Total 45? :)
For the hill problem I need to think more. I'm clueless right now :)
Neeraja,
ReplyDeleteYes, Anonymous gave a good explanation :-)
Rope problem - I forgot to specify that the ropes can't be cut into half ! Nevertheless, you are almost there!! Just needs a small modification since the rope can't be cut. Actually, there is a reason I combined the hill problem with this ! Now, that's a hint !:-)
OK.
ReplyDeleteBurn 2 ropes simultaneously.
Rope 1: Is lit on both sides
Rope 2: Is lit on 1 side.
When Rope 1 burns out we know we have half an hour left on rope 2. Now light the other end of rope 2 to get 15 mins. So you have 45 mins on the clock or should I say rope?
If you have 2 people instead of 1 and the 2 of them start their morning walks at 6 AM. One person starts to descend while the other climbs assuming they are taking the same path (Let's say the scenic route just so its beautiful) it is but natural they will cross each others path (Let's hope there is enough place for the two of them to cross each other without having to get into a Robin Hood style cudgel fight) at some point in time and place on that fateful day. So yeah same goes with the same man doing the trek on two different days. He will cross paths with his older self at some point in time and space. QED. (Did you want a specific time?)
Rafiki - Great answers to both questions ! Thanks for posting your answers :-)
ReplyDeleteWow, Rafiki :) Awesome :)
ReplyDeleteSince both questions have already been answered I'll just post an alternative answer for the rope problem.
ReplyDeleteIf I'm allowed to, I shall make a mark right at the middle of a rope. Light the rope at one end and wait till it burns till the mark. That takes 30 min. Then light it at the other end too. The remaining half will now burn out in 15 min, making it a total of 45 min.
Lalit,
ReplyDeleteThanks for posting this solution which involves only one rope! The two rope solution has the advantage that it works even when different sections of the ropes burn at different rates.