I was introduced to the two circles puzzle by a mathematician friend. This puzzle had the diameter of the large one 3 times that of the small one. Now you need to understand that I am no sort of a mathematician, so my friend was surprised when I said "4 revolutions" within five minutes. My reasoning was as follows the circumference of the small circle is 1/3 of the large circle. Therefore if one "walks" the small circle round 1/4 of the circumference of the large circle it will have rotated 3/4 of one revolution around its own axis. But it will have rotated through 1/4 of that of the large circle. It will thus have completed one complete revolution. Therefore to "walk" around all 4 quadrants of the large circle will will require 4 rotations of the small.one. One may derive a general rule by noting that the centre of the small circle travels around a circle described the the sum of the two radii.R = r1 + r2