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Abstract The 4 N {4N} carpets are a class of infinitely ramified selfsimilar fractals with a large group of symmetries. For a 4 N {4N} carpet F , let { F n } n ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact prefractal approximations with ⋂ n F n = F {\bigcap_{n}F_{n}=F} . On each F n {F_{n}} , let ℰ ( u , v ) = ∫ F N ∇ u ⋅ ∇ v d x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and u n {u_{n}} be the unique harmonic function on F n {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass,On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ = ρ ( N ) > 1 {\rho=\rho(N)>1} such that ℰ ( u n , u n ) ρ n {\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n . Such estimates have implications for the existence and scaling properties of Brownian motion on F .more » « less