A student writes down the chain N ≤ P < Q ≤ N and claims all three relations hold together. What is the correct assessment?

The chain starts and ends at N, so compose it right round: N ≤ P, P < Q and Q ≤ N give N ≤ P < Q ≤ N, that is N < N, because a run containing at least one strict link composes to a strict result. No number is less than itself, so the three relations cannot hold together. 'Nothing is wrong; it forces N = P = Q' is exactly what would be true of N ≤ P ≤ Q ≤ N, and it is the answer of a solver who overlooks the single strict link — that link is the whole difference. 'Impossible only if the two ≤ signs are read as <' gets it backwards, since the one < already present is enough. 'It gives no relation between P and Q' ignores the stated P < Q.