Designate the resistance of any of the wires (connecting two adjacent points such as 1 and 2) as ρ so that both r and R are fractions of ρ. I will solve this problem in two steps. The first step is technically not necessary, as it merely explains a factor of 2 needed to find the numerical values of these fractions. Nevertheless, I think it is helpful to include it for completeness.
Step 1: Find a formula for the resistance RAB between any two nodes A and B (with a finite separation between them) in terms of the currents and potentials in the lattice net.
To accomplish this step, consider three different situations. In situation 1, connect the positive terminal of an emf ε to node A so that we inject current I into it. (The negative terminal of the power supply can be imagined to be...