We outline a systematic procedure for obtaining exact analytic expressions for the mean walklength (n) for a random walker transiting on a d = 2 square-planar lattice with a single deep trap subject to periodic boundary conditions. As demonstrated in our earlier work on hexagonal lattices [Chem. Phys. Lett. 371 (2003) 365], the procedure depends on generalizing Montroll's first invariance relation Sigma(n = 1) to nth order. The central result of this work is the following exact analytic expression for E(n) for square-planar lattices: Sigma(n)=4{[Sigma(i=0)(n-1) 3(i) ] N - 1/3! (5 (.) 3(n) + 12 (n-2)3(n-1) +3)}. (c) 2005 Elsevier B.V. All rights reserved.