A ship has its planks replaced one at a time until none of the original material remains. The puzzle asks whether it is still the same ship, and the question generalises.
The puzzle has a second half
Suppose the removed planks are kept and reassembled into a complete ship. There are now two candidates, and each has a serious claim.
The repaired vessel has continuity of use, location and function. The reassembled one has the original material throughout.
The second half is what makes the problem difficult, since either criterion alone seems reasonable until it is set against the other.
Two different criteria are in play
One criterion is material composition: a thing is the same thing if it is made of the same stuff. This matches many ordinary intuitions.
The other is continuity of form and function through gradual change, which matches how we treat rivers, institutions and living bodies.
Both are used constantly and usually agree. The puzzle is constructed to be a case where they come apart, which is why it feels artificial and still bites.
Living things make the problem urgent
The material of a body is replaced continuously, so a person is not composed of what composed them years earlier, yet they are treated as the same person.
If material composition were the criterion for identity, this would be a serious result. It is generally taken instead as evidence that composition is not the criterion.
The alternative, continuity, then has to be specified, and the specification turns out to be difficult in exactly the way the ship case shows.
Institutions inherit the same structure
Organisations replace every member, change premises and revise their purposes while retaining names, obligations and legal identity.
Disputes over whether a body is genuinely the continuation of an earlier one are common in law, politics and religious communities, and they have real consequences.
These arguments rehearse the same two criteria, usually without reference to the philosophical puzzle that isolates them.
Why no answer settles it
The disagreement is not about the facts of the case, which are fully specified. It is about which criterion of identity is being used.
Different purposes select different criteria, and a customs officer, an insurer and a shipwright may reasonably reach different verdicts about the same vessel.
That is the durable result. Identity over time is not a single relation waiting to be discovered but a family of relations serving different practical ends.