Slicing dice and cutting corners

[Sariaht]

A dice has 6 sides.
If you cut the corners of a dice you get 6 + 8 = 14 sides.
If you cut again, the minimum number of sides you get is 6 + 8 + 12.
If you cut n times, what is the minimum number of sides you get?

[NateTG]

A dice has 6 sides.
If you cut the corners of a dice you get 6 + 8 = 14 sides.
If you cut again, the minimum number of sides you get is 6 + 8 + 12.
If you cut n times, what is the minimum number of sides you get?

Perhaps you could try to figure this out for a tetrahedron (a triangular pyramid) first?
Can you predict how many vertices the resulting shape will have after each cut?
Does it matter how many edges meet at each vertex?

[Sariaht]

Good idea!