I've been thinking about these problems for a long time but I really can't wrap my mind around them. Please share your insights! Consider a circle of radius R/8 internally tangent (inside) a circle of radius R. How many rotations does it take for small circle to return to the same position? Does it matter whether or not the big circle is fixed in place? I got that only 7 revolutions are required from a solution to a problem that is slightly different. What about if the smaller circle was travelling on the outside of the larger circle, so that it's externally tangent? Again, does it matter whether or not the big circle is fixed in place? Thank you!