Measure 61 Minutes with Ropes

Here's how you can measure 61 minutes using the ropes:

1. Light both ends of the first rope (which burns in 48 minutes) at the same time. Since it's burning from both ends, it will burn up in 24 minutes.

2. At the same moment you light the first rope, light one end of the second rope (which burns in 72 minutes).

3. When the first rope has completely burned out (after 24 minutes), the second rope has been burning for 24 minutes, which means it has 48 minutes left to burn. At this moment, light the other end of the second rope. Now it's burning from both ends and will burn up in 24 more minutes, which is half of its remaining time.

4. Also, as soon as the first rope burns out, light both ends of the third rope (which burns in 128 minutes) at the same time. This rope will burn up in 64 minutes when lit from both ends.

5. After lighting the second end of the second rope, wait for 13 minutes. At this moment, the second rope will have completely burned out (since it burned for a total of 37 minutes: 24 minutes from one end and 13 minutes from both ends). The third rope will have been burning from both ends for 37 minutes, which means 27 minutes remain for it to burn completely.

6. Now, you have a rope that's burning from both ends with 27 minutes left, which will burn out in 13.5 minutes.

So, if you sum up the times: 24 minutes (from the first part of the procedure) + 13 minutes (from the second part) + 13.5 minutes (from the third rope burning completely) = 50.5 minutes.

I apologize, I made an error in my calculation. The method above only measures 50.5 minutes and not 61 minutes.

To correct this:

After the first rope burns out at 24 minutes, light the other end of the second rope, and at the same time light the third rope at one end.

When the second rope burns out completely, 48 minutes have passed in total (24 minutes for each rope). The third rope has been burning for 24 minutes and has 104 minutes left to burn.

Now, light the other end of the third rope. With both ends burning, it will take half of 104 minutes, which is 52 minutes, to burn completely. But you only need to wait for an additional 13 minutes to reach a total of 61 minutes (48 + 13 = 61).

So, the correct method is:

1. Light both ends of the first rope and one end of the second rope.
2. After 24 minutes, the first rope burns out. Light the other end of the second rope and one end of the third rope.
3. After 24 more minutes (48 minutes total), the second rope burns out. Light the other end of the third rope.
4. Wait 13 more minutes for a total of 61 minutes.