To measure 45 minutes using two identical wires (of unknown length) and a lighter, you can follow these steps:

Light both ends of the first wire at the same time.

Light one end of the second wire.

After the first wire burns out (30 minutes have passed), light the other end of the second wire.

When the second wire is completely burned out (15 more minutes have passed), 45 minutes will have elapsed in total.

Here's the explanation:

The first wire takes 30 minutes to burn completely (since both ends are lit).

When the first wire has burned for 30 minutes, this is also the same time it will take for the second wire (with one end lit) to burn completely.

At this point, the second wire has been burning for 30 minutes. By lighting the other end of the second wire, it will continue to burn for an additional 15 minutes, making a total of 45 minutes.

This puzzle utilizes the fact that the wires are identical but can be burned from either end. By using one wire to measure 30 minutes and the other to measure 15 minutes, you can combine their burn times to achieve a total of 45 minutes.