Self-avoiding walks contained within a square

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