  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 ${\mathbb Z}^2$. 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 {\em 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 ${\mathbb
Z}^2$. 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
$\tau < \lambda.$
