 Fifth, the fact that the counterpart relation need not be an equivalence relation has also been used to diffuse certain puzzles. Consider an ordinary bicycle. Intuitively, such an object does not have all its parts essentially: the bike could have had a different chain, for instance. On the other hand, arguably it could not have been composed of entirely different parts. A bike made of entirely different parts would have been a different bike. But this creates a puzzle (see Chisholm 1967, Chandler 1976): if the bike had a different chain, it would still be an ordinary bike that does not have all its parts essentially; it could have had a different saddle (say). Iterating this line of thought, we can create a sequence that leads from the actual bike to a bike composed of entirely different parts. Each possible bike in the sequence could have been the next bike in the sequence. But didn’t we say that it is impossible that my bike is composed of entirely different parts? Counterpart theory offers a natural answer: a counterpart of a counterpart need not be a counterpart (see Lewis 1968, 28f; Lewis 1986e, 243–6; Ramachandran 2020).