In the shortest common superstring problem (SCS) one is given a set s ₁,..., s_n of n strings and the goal is to find a shortest string containing each s_i as a substring. While many approximation algorithms for this problem have been developed, it is still not known whether it can be solved exactly in fewer than 2ⁿ steps. In this paper we present an algorithm that solves the special case when all of the input strings have length 3 in time 3^n/3 and polynomial space. The algorithm generates a combination of a de Bruijn graph and an overlap graph, such that a SCS is then a shortest directed rural postman path (DRPP) on this graph. We show that there exists at least one optimal DRPP satisfying some natural properties. The algorithm works basically by exhaustive search, but on the reduced search space of such paths of size 3^n/3.