The Burning Ropes
The puzzle
You have two lengths of rope and a box of matches. Each rope takes exactly sixty minutes to burn from one end to the other.
The ropes are not uniform. One might burn through its first half in five minutes and take fifty-five for the rest, and the two ropes are not alike. You cannot measure the ropes, mark them or cut them accurately.
Measure exactly forty-five minutes.
Stuck?
One at a time, smallest first. Open only as many as you need — the whole point of the puzzle is the part you work out yourself.
Hint 1
Nothing says you must light a rope at one end only.
Hint 2
If a rope has thirty minutes of burning left in it and you light the far end too, how long does it last?
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Light rope A at both ends and rope B at one end at the same moment. When A burns out, thirty minutes have passed; at that instant light B’s other end, and B burns out fifteen minutes later. Thirty plus fifteen is forty-five.
How to get there
At time zero, light both ends of rope A and one end of rope B.
Rope A now has two flames eating it from opposite directions. However unevenly it burns, the two flames between them consume a full hour of rope - so they meet after exactly thirty minutes. That moment is your thirty-minute mark, and rope B has exactly thirty minutes of burning left in it.
Light rope B’s other end now. The same argument applies to the remainder: two flames eating thirty minutes of rope finish it in fifteen. When B goes out, forty-five minutes have passed.
The technique worth keeping
Unevenness is what makes this look impossible, and lighting both ends is what makes it irrelevant - the argument never mentions where the rope is thick. When a quantity is unknown but a total is known, look for an operation that depends only on the total.