A variant of the "given a finite simplicial complex, is it the 5-sphere?" problem is the "given a finite simplicial complex, is it it a 6-manifold?".
I find this attractive because, because manifolds are such a basic and fundamental concept, you'd expect we'd be able to recognize one, but in fact we cannot.
This was pointed out by an answer to the question: http://mathoverflow.net/questions/71400/when-are-finite-simplicial-complexes-smooth-manifolds