So here's the logic I find in this If I were to roll the smaller circle on a line three times the length of the circumference of the smaller circle, the center of the circle travels the same length as the line. Three rotations. Now if I make the line into a circle, the center of the circle will travel 1 + 1/3 the circumference of the large circle, or 4/3. It's all about how far the center travels. 4 rotations.