There is an ascetic solemnity descending on this brainteaser. I like the answers I like because both will give an answer within tolerance to an average punter just looking for a method. Tristran's method answer is nice because it abstracts the essential part of the answer to the original classic teaser. vplanas answer is attractive because it works with the material allowed and shows that the questioner doesn't have the control they thought they had over the question.

Hello,Here is a solution using the two ropes without any cuttings (in case you don't have Scissors). Nevertheless it's using an infinite number of burnings :The point is, since we can't get one rope burning from three different points, let's make a ten minutes over each :Burn the first rope from the two sides an from two other different points in the middle. This gives you six different flames. Each time a piece burns completly you burn another one in the middle. Thus you have 10 minutes. You repeat the same thing for the other rope an there you are.

Here's another infinite burning solution, maybe a bit more practical than the first. It's like trisecting an angle by trial and error.Light both ropes at both ends, and one in the interior. The first thing that should happen is one end of the interior flames will meet an end flame. When that happens, light the other rope in the interior. The next event can be either (a) the first rope burning up completely or (b) an end of the interior flame in the second rope will meet one of the ends. If (a), split the second rope (which is easy because the middle is now ash) and light one of the interiors. If (b), light either rope in the interior. If you want to save yourself work, always choose the longest remaining segment to light the interior.Lighting simultaneously with two flames meeting is no more unrealistic than the usual assumption that you can light several rope ends simultaneously. And while this requires an infinite number of lights, that's only for infinite precision.

I crossed with Quantor, I think we have the same basic idea.

aaron is helpful