I will reuse something I already answered another question with. I think the problem you describe will likely come up in a circumstances when we, as teachers, state and prove something is impossible (usually, an equation has no solution). It is often difficult to have students understand how such a proof is relevant, and the best way I can think of is to design a problem which is very close to the one at hand, with no obvious solution, but yet with some solution. I am not sure a general list is so useful, so I'll just pick one example.
Imagine you discussed the irrationality of $\sqrt{2}$. In other words, you discussed the impossibility to solve $p^2=2q^2$ in integers. If some students feel it obvious that there could not be a solution, then propose them the equation (also to be solved in integers) $p^2=12 q^3$. It is not obvious to find a solution when one is not familiar with the decomposition into primes, but of course $(p,q)=(18,3)$ does the job.