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A ring puzzle. Imagine the object above in the figure to the left made from perfectly elastic materical. 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?
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