This puzzle is an harder version of a very famous one. It has been asked many times on PSE, but this time the question is slightly different (1) (2). So please read this question twice before marking as duplicate.
You are given an unlimited set of matches and two different ropes, each rope has the following properties:
- if you light the end of a rope it will burn completely in one hour.
- the fire does not burn the ropes evenly.
That means (for example) that the fire can take only five minutes to burn the first half of a rope and then fifty-five minutes to burn the other half. You cannot make any assumption on the burning speed of the ropes.
You start at time $t=0$ by lighting one or more fires. Afterwards you can instantly light or shut any fire anytime if you want to; you can perform these actions multiple times or simultaneously in a very precise instant of time of your choice.
- Can you measure 45 minutes? (This was the original question).
- Can you measure 10 minutes?
- How many intervals of times can you measure and which are them? Intervals must have an integer duration in minutes.
- What if there were $N$ ropes instead of two?
- What if you were given only 100 matches instead an unlimited number?
(1) Burning ropes as timers
(2) Burning ropes as timers - How many time intervals can be measured?
In the second linked question they state an example of answer: "For example, when N = 1, the answer is 2". That's not valid for this question because they impose some constraints that are not valid here.