Suppose we make a 14 sided die by taking a cube and slicing a triangular pyramid off each corner, such that there are now 6 square faces and 8 triangular faces, each with unit side length (in other words, a cuboctahedron).
Assuming that the die is an evenly weighted solid and rolled with plenty of energy, what is the probability of it landing on a square (and hence with a square face up)?
There are two values that I know of that might yield insight to this problem.
Centre of Mass: the die seems more likely to remain on a face which lowers its centre of mass.
Area: the die seems more likely to land on a face with greater area.
The square both has a larger area, and a lower centre of mass; indicating that surely it is more likely to be landed on. But exactly how much more likely - and could such a probability be calculated as a function of the centre of mass and/or area?