Decidable problems for which no concrete decision procedure is known I am looking for an example of decidable problems the decision procedures of which are unknown. I believe someone mentioned one to me once, and I also have read somewhere, but my memory is corrupted. I suppose the decidability of these problems are proved non-constructively, so that they have this kind of counter-intuitive property.

I think that in graph theory, there are a number of problems in this trait. I am wondering if you are aware of any of these.
Here is a trivial example of such a problem: consider any yes-no question with an unknown answer, e.g. $P=NP$. For any $n \in \mathbb{N}$, let $A(n)$ be true iff $P=NP$. Then $A$ is decidable by one of two decision procedures: 


Always return "true".
Always return "false".


But you need to solve $P=NP$ to know which one.