= Solution
{c}
A complete pairing of the six <polygon edges> leaves three edges and one face as <CW cells>, together with the vertex <equivalence classes>. If $V$ denotes their number, then $V\geq1$ and the <polygonal-schema Euler count> gives
$$
\boxed{\chi=V-3+1=V-2\geq-1.}
$$
A <closed orientable surface> of <genus> two has <Euler characteristic> $2-2\cdot2=-2$. Since the <Euler characteristic> is a <topological invariant>, \b[no complete pairing of a <hexagon> can give that <surface>]. Equivalently, the <polygonal-schema Euler count> bounds the <genus> of an <orientable> result by $2g\leq3$, so $g\leq1$.
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