This is a Doubt I have for a classic brain teaser question.
Puzzle :
We have 2 ropes. Each of them takes one hour to burn.
Using the 2 ropes , how to measure 45 mins ?
The rope has different densities at different points, so there's no guarantee of consistency in the time it takes different sections within the rope to burn.
Puzzle Answer :
The answer is to light both ends of one rope and one end of the second rope. Then after the first one burns out, light the other end (with the first end still burning) of the second rope, then it will be 45 mins after the two ropes burn out.
My Doubt :
It looks intuitively right.
Proving that "If a rope takes 1 hour to burn, it takes 30 mins to burn if we light both ends of the rope simultaneously" is my confusion , considering that the densities are different among & along the ropes.
Thanks for your help
Intuitive way to Check (Proof Outline) :
Take a "real" Clock or "real" timer.
Burn 1 End of 1 rope , & using the Clock or timer , wait for 30 minutes.
At the moment , stop the burning. [[ Cover the rope with some metal sheet or Pour water or What-ever is necessary ]]
Sit back & think about this :
When you restart the burn at the middle (new End) , how long will it take to burn up the left over half ?
If it is less than 30 minutes , then Original claim is wrong : whole rope would have burnt out sooner than 60 minutes.
If it is more than 30 minutes , then Original claim is wrong : whole rope would have burnt out later than 60 minutes.
Hence it must be Exactly 30 minutes.
Instead of restarting at middle (new End) , when you burn the other end of the left over half , how long will it take to burn up ?
Both Cases , it must take Same time , because it is the Same left over half
Hence we get this Conclusion : Both Ends must take Exactly 30 minutes to burn up to some Common Point , at which Point , the whole rope is burnt up !