You have a lighter and two fuses. Each fuse takes an hour to burn, but they don’t burn at a steady rate throughout the hour.
How do you use the two fuses to create a 45 minute timer?
Light both ends of the first fuse and one end of the second fuse. When the first fuse has completely burnt, light the other end of the second fuse. When the second fuse has completely burnt, 45 minutes has passed.
Proof
When both ends of a fuse are lit, the fuse will burn in half the time. This is true however unevenly the fuse burns.
The first fuse will burn out completely in 30 minutes. At that time the second fuse is now has 30 minutes remaining.
We can think of the second fuse now as 30 minute fuse. If we light the other side, it will burn out in half the time: 15 min. Thus 30 + 15 = 45 minutes will have passed.
Extensions
In this problem we measured out a time which was \(\frac{3}{4}\) of the total fuse length.
In general the times we can measure with some number of fuses which burn in a unit length of time are called Fusible numbers.