My 10-year old son asked me this simple question, and I've been unable to answer it.
Suppose we start with two (unlinked) circular wire loops (maybe different sizes), and allow both to be continuously deformed in three-dimensional space, the only constraint being that different parts of the loop mustn't touch. The two loops can be deformed in different ways.
Is it possible to do so in such a way that neither of the loops (now considered rigid) can be passed through the other by rigid motion? Consider the wire to be one-dimensional; the definition of 'passing through' permits the two loops to touch (as is obviously necessary in the case of two identical circles), but not to cross each other.
For a formal definition of 'passing through', suppose the two loops are A (fixed) and B (moving). If there is a point on B that traces a path C through space as we move B, such that C is a closed loop that is linked with A, then we say B has passed through A.
Edit Joseph O'Rourke has given an interesting example where the wires are 'interlocked'. So I'd now like to add a condition to exclude this situation, and say that, prior to the 'passing through', the wires must be separated (occupying different sides of some plane).
Edit Experimenting with garden wire, inspired by some of the comments below, I came up with a configuration that may work: But even with this I am not certain. Ideally I'd like a simple provable example. In particular, could there be an example where both loops are planar?