  We have studied self-avoiding walks contained within an $L \times L$ square
whose end-points can lie anywhere within, or on, the boundaries of the square.
We prove that such walks behave, asymptotically, as walks crossing a square
(WCAS), being those walks whose end-points lie at the south-east and north-west
corners of the square.
  We provide numerical data, enumerating all such walks, and analyse the
sequence of coefficients in order to estimate the asymptotic behaviour. We also
studied a subset of these walks, those that must contain at least one edge on
all four boundaries of the square. We provide compelling evidence that these
two classes of walks grow identically.
  From our analysis we conjecture that the number of such walks $C_L$, for both
problems, behaves as $$ C_L \sim \lambda^{L^2+bL+c}\cdot L^g,$$ where $\lambda=
1.7445498 \pm 0.0000012,$ $b=-0.04354 \pm 0.0005,$ $c=-1.35 \pm 0.45,$ and
$g=3.9 \pm 0.1.$
  Finally, we also studied the equivalent problem for self-avoiding polygons,
also known as cycles in a square grid. The asymptotic behaviour of cycles has
the same form as walks, but with different values of the parameters $c$, and
$g$. Our numerical analysis shows that $\lambda$ and $b$ have the same values
as for WCAS and that $c=1.776 \pm 0.002$ while $g=-0.500\pm 0.005$ and hence
probably equals $-\frac12$.
