2 ropes and a matchbook

The scenario:

You’re given 2 ropes and a matchbook. You know a priori that these two ropes each take an hour to burn, but they don’t burn at an even rate. For instance, the first rope might take 1 minute to burn through the first half and 59 minutes to burn the second, and the other rope might be completely different.

Can you measure 45 minutes using only these two ropes and the matchbook?

What you know:

  1. You have two ropes and a matchbook.
  2. The ropes take an hour to burn completely.
  3. The rate of burn is inconsistent.

Hints (click to unblur)

  1. Can you measure 30 minutes?

  2. You may have to do two things at once

A solution

A description (and strategy)

You can measure 60 minutes easily by just burning a rope. You can measure 30 minutes by burning both ends of a rope. So you can measure 45 minutes by lighting two ends of one rope and one end of the other, then when the first rope burns out, light the other end of the second rope. It’ll be 45 minutes because the first rope takes 30 minutes to burn, so there will be 30 minutes left on the second rope when you light the second end, thereby turning the 30 minutes of rope into 15 minutes.