Topological Escapology
The occasional teasers I post on here seem to go down quite well so I thought I’d try this one on you. I recently found it in an old book on the topic of topology, a fascinating field that finds many applications in physics, including several in my own field of cosmology.
It’s probably best not to ask why, but the two gentlemen in the picture, A and B, are tied together in the following way. One end of a piece of rope is tied about A’s right wrist, the other about his left wrist. A second rope is passed around the first and its ends are tied to B’s wrists.
Can A and B free each other without cutting either rope, performing amputations, or untying the knots at either person’s wrists?
If so, how?
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In the Dark 
June 13, 2011 at 1:41 pm
I know this isn’t what you are aiming for. But because you said that you only can’t cut the rope or untie the knots, I am going with it is possible by cutting off one of their arms.
June 13, 2011 at 2:02 pm
I might have known someone would think of that. 😀
I’ve now edited the post to make it clear amputations aren’t allowed either…
June 13, 2011 at 3:00 pm
I think (k)not.
That is, assuming the knots are sufficiently tight that they can’t simply withdraw their hands from rope. And assuming that they are not already amputees. And also that “free” is understood in such a way that it is impossible to be truly “free” while being tied to another person with bits of rope (discuss).
But with those caveats, I’m going for “no”.
June 13, 2011 at 3:19 pm
One could wait for the other to die and rot, but that is obviously not what the question is about. Topologically they are like two interlinked circles, and I don’t think any amount of twisting and turning will change that.
June 13, 2011 at 3:55 pm
If they hold left hands, then person B does a somersault through the loop…? Or right hands, with person A somersaulting… possibly.
June 13, 2011 at 3:59 pm
Gah, where by “left hands” I mean “person B’s left hand holding Person A’s right hand” 😉
June 13, 2011 at 4:03 pm
Are the knots loose enough that the rope could pass around person’s wrist? In that case it is possible, as B could pass the rope under the link and around A’s hand.
Otherwise, as someone already said, the situation is isotopic to two linked circles and is therefore impossible to unlink.
June 13, 2011 at 5:53 pm
It works!
June 13, 2011 at 8:30 pm
I found an image of what I was trying to say:
June 14, 2011 at 9:53 am
Two linked circles, as provided by each man’s arms and chest and rope, obviously canot be unlinked. But if you are allowed to ‘cheat’, as in the real world, by stuffing rope between somebody’s wrist and the loop of rope around that wrist, then mmaluff’s link shows how to do it – although it might be clearer if the first movement after the zoom-in were to be redrawn, to show that the movement of the blue rope down the arm is also a wrist-grazer.
June 13, 2011 at 4:49 pm
I could mention that this is a standard trick for escapologists. I’ve seen it done myself, and it doesn’t involve somersaults.
June 14, 2011 at 10:15 am
OK I think it’s time to wrap this up, as mmaluff has got it.
Here is the trick in words.
One of the geezers, say B, takes up a small loop near the middle of his own rope, passes it under the loop around A’s right wrist – on the inside of the wrist and in the direction from elbow to hand – and slips it over A’s hand. He then passes it again under the loop around A’s wrist – this time on the outside of the wrist and in the direction from hand to elbow. Behold, his own rope can now be pulled free.
The point is that the situation is not equivalent to two interlocking circles – the connection is not simple because of the loops at the wrists.
June 14, 2011 at 1:41 pm
But the picture suggests that the loops are tightly tied around the wrists.
June 14, 2011 at 1:50 pm
Escapologists usually suggest that too…