Remember those old chestnuts like, "You are given a 7-quart and a 9-quart pail; return from the well with exactly one quart"?
For a>b, give the set S in terms of a and b of all possible returns from the well. Design a parametrized production system (alphabet plus rewrite rules) or, equivalently, a Turing machine to derive and return any member of S. For extra credit, generalize to any (finite) number of pails a[i].
Does an infite number of pails add anything interesting??