Les Reid’s advanced problem archive
Problem : A solid is formed by placing congruent regular square pyramids on each face of a cube as shown in the figure below.
What is the maximum value of V/S, where V is the volume and S the surface area of the solid ?
Can one fill space with congruent copies of the solid that attains this maximum ?
Source : Hungarian National Olympiad
With this nice problem, the rhombic dodecahedron appears like an optimal shape.
The rhombic dodecahedron is a beautiful solid that is not a Platonic solid since its faces are rhombi (which are not regular polygons) unlike the regular dodecahedron whose faces are regular pentagons.
Filling space with copies of the rhombic dodecahedron as seen in my solution given below.
I made a nice paper calendar with the help of my daughters following this link
You can choose the rhombic or the regular dodecahedron.