# topology puzzle - without cut the rope, separate two rings

hello I wonder whether this puzzle is possible to solve.

if possible, what kind of thing should I learn to solve this?

the problem is make left one to right one without cut the rope only stretch and bending are allowed I found out this puzzle here->(www.ocf.berkeley.edu/~wwu/riddles/hard.shtml/)

I wish this problem lead me to learn math intuitively.

• This puzzle can be found here. It's labeled "topological rings". Nov 15, 2012 at 11:47
• The puzzle description: "Imagine the object above in the figure to the left made from perfectly elastic material. Can you transform it so as to unlink the two rings as in the figure on the right? One possible way is to cut one ring, move the other ring through the gap, and rejoin the the first ring exactly as it was. That would be a legitimate topological transformation. However, it is also possible to transform the first shape into the second without any cutting, simply by manipulating the objects in the appropriate manner (stretching, bending, but not breaking). Can you see how to do it?" Nov 15, 2012 at 11:52
• math.hmc.edu/funfacts/ffiles/20007.7.shtml Nov 15, 2012 at 12:31
• There's a nice diagram showing how this is done in Keith Devlin's Mathematics: A New Golden Age. Nov 15, 2012 at 15:20

Here's one way to think of it.

Call the loops $L_1$ and $L_2$, joined by the stem which attaches to $L_1$ at $S_1$ and $L_2$ at $S_2$. You will notice there is a point $A$ where $L_1$ crosses and is above $L_2$ and another point $B$ where $L_1$ crosses and is below $L_2$.

Now contract the stem until it vanishes so that $S_1$ and $S_2$ become a single point $S$ and you will see that one of the points $A$ or $B$ will also move to $S$ under this deformation.

If you try to draw the two rings now joined at the single point $S$ you will see they are no longer linked - there is only one crossing point now so you simply have two rings sitting on top of one another joined at a point.

All you need to do now is stretch out the stem again and the rings will look like the second picture.

This is a classical topological problem and magic trick.

The concept is fairly simple and the diagram is misleading. If you manufacture latex or plastic loops as per the diagram, it is an impossible task.

However, if you take a string then to create the loops, you need to tie 2 knots. That's the key. The idea is basically to pass one loop through the other's knot.