Here is a circular coin with diameter D
From its starting position the small coin goes completely around a bigger circular body of diameter 4D without slipping, always in contact with the bigger circle, rotating around its own center point and returns to its original position. Both bodies are in a horizontal plane.
In this case the coin does the same thing EXCEPT the small coin is in a vertical plane (standing up as shown, perpandicular to the big circle). The big circle is lying in a horizontal plane. The small coin goes around the bigger circle without slipping, keeping contact with the bigger circle and rotating. It returns to its original position.
In this case the small coin goes around a SQUARE shape with each side = 4D. Again it does not lose contact with square, does not slip and is rotating around its own center. It returns to its original position
The small coin is placed INSIDE a circular shape with diameter 5D. Again it goes around (inside) without slipping, keeping contact with the bigger circle all the time and rotating around its own center point. It returns to its original position
In which case the small coin makes more rotations?
In other words in which case the rotational distance travelled by any point on the circumference of thee small coin is greater than the other cases or are they all the same?
As far as I know one of these cases may have been addressed before. So please no computers.