http://www.georgehart.com/Puzzle2.html is not a genuine polyhedron at all. It contains several vertices where three hexagons meet. But the angle of a regular hexagon is 120º, and three of those add up to 360º. That means that three hexagons can only meet in that way as part of a flat surface. Yet the picture shows them as forming part of a convex solid. That in turn means that the hexagons must be distorted in some way so as to give the appearance of convexity.
In a genuine convex polyhedron, the sum of the angles at each vertex must be strictly less than 360º. The greatest number of faces possible in a polyhedron consisting of regular pentagons and hexagons is 32 (12 pentagons and 20 hexagons), in a truncated icosahedron.