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All convex ten-faced polyhedra meet this criteria, and perhaps other polyhedra as well. There are several solutions, including: the polyhedron the half truncated cube: it has 24 edges.
The octahedral prism has 24 edges.
The square anti-prism 16 edges.
The pentagonal bipyramid has 15 edges.
The nonagonal pyramid has 18 edges.

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