  The $4N$ carpets are a class of infinitely ramified self-similar fractals
with a large group of symmetries. For a $4N$-carpet $F$, let $\{F_n\}_{n \geq
0}$ be the natural decreasing sequence of compact pre-fractal approximations
with $\cap_nF_n=F$. On each $F_n$, let $\mathcal{E}(u, v) = \int_{F_N} \nabla u
\cdot \nabla v \, dx$ be the classical Dirichlet form and $u_n$ be the unique
harmonic function on $F_n$ satisfying a mixed boundary value problem
corresponding to assigning a constant potential between two specific subsets of
the boundary. Using a method introduced by Barlow and Bass (1990), we prove a
resistance estimate of the following form: there is $\rho=\rho(N) > 1$ such
that $\mathcal{E}(u_n, u_n)\rho^{n}$ is bounded above and below by positive
constants independent of $n$. Such estimates have implications for the
existence and scaling properties of Dirichlet forms on $F$.
