You have 2 identical ropes which burn at a specific rate, and an unlimited supply of matches. When you light one end of a rope, the fire will take exactly 1 hour to travel to the other end of the rope.
You need to measure exactly 45 minutes. You must start by lighting one or both of the ropes. You can light or extinguish either end of either rope later, but you must only do this immediately after a rope has finished burning, as this is the only accurate way to measure elapsed time. You also must not light anywhere but the end of a rope, or any other form of guessing. You may finish with or without any remaining rope.
Hint: You can light both sides of one rope. Solution:
Light rope $A$ on both sides so that the rope will be gone in $30$ minutes. You need to light rope $B$ at the same time you light rope $A$. Once rope $A$ is gone, light the other side of rope $B$. Rope $B$ will be gone after another $15$ minutes. That will add up to $45$ minutes.
Fold one rope in half and then in half again. It will then be a quarter of the original length. Lay the folded rope parallel the other rope. Light the other end of the unfolded rope. When it reaches the folded rope then 45 minutes have passed.
Lay the ropes next to each other and light each rope at opposing ends. WHen the flames meet in the middle, 30mins has passed. Then move the ropes so the unburnt sections are next to each other, when the flames meet again, 15mins has passed - hence 45mins.
1. Burn one ropes at both ends. After finished burning 30 has been passed
2. Burn another rope at both end and the middle.
3. Wait until one of the segments has been burned off.
4. Burn the middle of the remaining segments
5. Repeat step 3 and 4 until all the ropes are burnt
This answer disproves that lighting the middle of the rope requires guessing, as this answer doesn't assume that the rope will burn exactly at middle, as long as you burn anywhere between the two ends, it will work
Take one rope and light both ends at the same time while also lighting one end of the second rope. That should take 30 minutes to burn completely leaving 30 minutes of the second rope unburned. As soon as the first rope has burned completely light the second end of the second rope. The remaining 30 minutes of the second rope will be halved leaving you with 45 minutes total.