Find all $n$ for which a polyhedron with $n$ edges exists
Any $n \geq 6$, except $n = 7$, will work.
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For even values of $n$, simply consider the pyramid with $(n/2)$-gon as its base.
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For odd values of $n$ with $n \geq 9$, start with the pyramid with $n-3$ edges. Cut a small corner around one of the base's vertices. This will generate $3$ extra edges, bringing the total number of edges to $n$.
