I can assume that all of the resistors will be very similar because the method of their construction is designed to make sure that they are.
I have no problem with the special case that Bob rightly points out as having many solutions. Let's take the simple case I posted where there are effectively two parallel arms with a resistor across the middle. Removing that middle resistor by making it infinite gives multiple solutions, so mathematically you are right to argue that the circuit as drawn has multiple solutions.
What I'm saying is that the bridging resistor will have a value, in fact all of the resistors will have quite similar values in practice, so there is an additional constraint that forbids any resistor taking on an infinite value. Once you take that constraint into account, I'm suggesting that there is only one unique solution.