We study the statistical mechanics of the travelling salesman on a Sierpinski gasket in which the bond lengths { λi } are quenched random variables. The problem of finding the shortest closed path which visits all N sites is tractable if all the| λi — 1 | are less than (2 N + 1)-1. For a particular choice of the bond-length probability distribution and at low temperatures, the system behaves like a set of non-interacting Ising spins in a quenched random magnetic field. The relevance of one of our results to collapsed polymer chains in random media is also discussed.