Abstract: We study a restricted class of self-avoiding walks (SAW) which start at the origin (0, 0), end at (L, L), and are entirely contained in the square [0, L] \times [0, L] on the square lattice Z^2. The number of distinct walks is known to grow as $\lambda^{L^2+o(L^2)}$. We estimate \lambda = 1.744550 \pm 0.000005 as well as obtaining strict upper and lower bounds, 1.628 < \lambda < 1.782. We give exact results for the number of SAW of length 2L + 2K for K = 0, 1, 2 and asymptotic results for K = o(L^{1/3}). We also consider the model in which a weight or fugacity x is associated with each step of the walk. This gives rise to a canonical model of a phase transition. For x < 1/\mu the average length of a SAW grows as L, while for x > 1/\mu it grows as L^2. Here \mu is the growth constant of unconstrained SAW in Z^2. For x = 1/\mu we provide numerical evidence, but no proof, that the average walk length grows as L^{4/3}. We also consider Hamiltonian walks under the same restriction. They are known to grow as \tau^{L^2+o(L^2)} on the same L \times L lattice. We give precise estimates for \tau as well as upper and lower bounds, and prove that \tau < \lambda.