This is a classic puzzle that has been around for at least 250 years. It is very challenging, but it does give students a chance to get students up and moving. It depends on lateral thinking and topology (rubber geomettry or the mathematics of distortion). It can also make a good party game (for adults too). 

The Set-Up

For this puzzle you need two people, some rope and some empty space to do the puzzle in. Each person will need a piece of rope with a loop tied in both ends, so it can be worn as handcuffs. The rope should be reasonably long, so that the person wearing it can easily step over it if they want. 
 

Each person puts on a complete set of handcuffs. Before putting them on, they loop their handcuffs around each other so they are tied together. Each person should wear a complete set of handcuffs. They then have to get themselves apart while following these rules:

The Handcuff Puzzle Rules 

The handcuffs cannot be removed. 

Do not break, cut, saw through, bite through or in any other way damage the rope. Damaging each other is probably a bad idea too.

If you are doing this puzzle with a class, make certain you tell them they need to be able to show you their solution. Otherwise it won't be so enjoyable. 

The Solution

Imagine you are one of the people doing the puzzle. Start by moving the other persons rope along yours until it is lying on your arm. Make sure that the other person's rope is not wrapped around your rope, it should only be touching your arm. You can see now that if you could pull the rope through your arm, the puzzle would be solved. However, this would be very messy and not have a lot of repeat potential. Instead, you need to move it around your arm.  Reach in through your handcuff with a thumb and finger; and grab the other person's rope. Now pull it through your handcuff and over your hand. It should now be on the other side of your arm. Now let it go back through the handcuff and you are separated.

Tips

If you are showing this one to a class, some students will say that the puzzle is easy and they know how to do it. Don't worry, they don't. The first thing they will do is either step over the other person's rope or duck under it. This will not work. 

People very rarely solve this one without assistance. Some hints that may help is to get them to stand within arms reach of each other, then tell them that they do not need to move their feet. Alternatively, explain to them that it is impossible to unlink two linked rings, but they are not two linked rings, there is a gap at each wrist. 

If you are working with a single pair, you can lead them to the solution using a rubber band. 

First have one student take off a handcuff, put the rubber band on their arm and get them to put the handcuff back on. Ask them to take the rubber band off their wrist without getting it caught on the rope. They will take it under the handcuff. 

Next put the rubber band back on their arm but this time hold onto one point on the band. They will take it under their handcuff again, but this time they will have to stretch it over their hand and let it go back again. 

Now use their partner's rope in the place of the rubber band. 

Further Information

If there was a gap in one handcuff or in your arm, the puzzle is easy to solve. You just need to move the other person's rope through the gap. To a topologist, this is no different to doing it when your wrist is in the loop. There is a gap between your rope and you; and you move the other persons rope through that gap. The only practical difference is that with the rope on your wrist there is a hand-shaped obstacle you need to move the rope around. 

Further Activities

A single person version is to tie a knot in the rope, then put the handcuff on. They need to untie it, following the same rules. 

Another single person version is tying someone to a tall pole or something similar. The knot for this one is important. You need to take a loop of rope and wrap it around the pole. Then stick one handcuff through the loop before putting the handcuffs on. (See image)

Tied to a Pole

You can also use more than two people.