I quit after a couple minutes and consulted Grok...
To measure exactly 45 minutes using two ropes that each take one hour to burn completely, despite non-uniform burning rates, follow these steps:
- Start by lighting both ends of the first rope and one end of the second rope simultaneously.
- When the first rope has completely burned (which will take exactly 30 minutes, since lighting both ends causes it to burn in half the time), light the other end of the second rope.
- The second rope, which has been burning from one end for 30 minutes, now has the equivalent of 30 minutes of burning time left. Lighting the other end at this point causes the remaining portion to burn out in half the remaining time, which is 15 minutes.
- The total time from the start until the second rope burns out is 30 minutes (for the first rope to burn out) plus 15 minutes (for the second rope to complete burning), which equals 45 minutes.
Thus, you can measure exactly 45 minutes by timing from the initial lighting until the second rope is fully burned.
Nope. Different burn rates throughout the rope means no length of either rope can be said to provide any measure of time.
Even folding one rope does not mean the two halves of the rope finish burning at the same time. One half could burn in 5 minutes and the other in 55 minutes.
It is a puzzle with a wrong answer.