= Exact self-dual rectangle crossing probability
{title2=$\mathbb P_{1/2}(H_n)=1/2$}
For the lattice rectangle with <graph vertices> $\{0,\ldots,n\}\times\{0,\ldots,n-1\}$, let $H_n$ be an open left-to-right crossing. Its complementary event is a closed dual top-to-bottom crossing. The dual crossing rectangle has width $n-1$ and height $n$, so rotation and translation identify it with the original rectangle. Boundary <edges> along the starting and ending sides do not affect the crossing event. At $p=1/2$ the primal and dual <edge> states have identical laws, hence
$$
\mathbb P_{1/2}(H_n)=1-\mathbb P_{1/2}(H_n)=\frac12.
$$
The one-unit adjustment in the side lengths makes the symmetry exact rather than an informal assertion about a finite <graph vertex> square.
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