A circle with radius 1 rolls without slipping once around a circle with radius 3. How many revolutions does the smaller circle make?

Does it matter if the smaller circle rolls on the inside or outside of the larger circle?

In astronomy, rotations are distinguished from revolutions, and in this case it would be said that only one revolution takes place--that is the small circle only went around the larger one once.  But that argument would indeed place this puzzle into the category of Tricks.

So we assume that "revolutions" is taken to be synonymous with "rotations".

Let's say the small circle begins with one particular point on its circumference touching the topmost point on the larger circle, and is externally tangent.  Let's also say it rolls around to the right (clockwise).  Our chosen point, originally in contact with the circle, will go toward the top of the small circle, then pass it and be on the right of the small circle, and then, when the small circle is 1/3 of the way around its path, the chosen point will again be tangent to the larger circle.  When it does so, it will be at the 10-o'clock position on the smaller circle--it will already have passed through its original 6-o'clock position and gone another 1/3 of the way around the smaller circle.  It will then advance another extra 1/3 by the time the smaller circle gets to the 8-o'clock position on the larger circle.  And in the final leaf of its 3-lobed figure, the point will have advanced another 1/3 of a rotation.  So even though it touches the larger circle 3 times per orbit, it has rotated about the center of the smaller circle 4 times.

If, however, the smaller circle is inside the larger one, starting again with the point of tangency at the top of the larger circle, this time the point of tangency is also at the top of the smaller circle.  As the smaller circle rolls clockwise inside the larger, it itself will be rotating counterclockwise.  The chosen point will start going downward, reaching another point of tangency when the smaller circle as a whole is at the 4-o'clock position on the larger; and this time, our point is also at the 4-o'clock position on the little circle. It has not even gotten back to the 12-o'clock position where it started, so it has gone 1/3 less than one rotation.  So by the time it returns to its original spot it will have lost one full rotation, and therefore having completed only 2 rotations. 

Posted by Charlie on 2004-06-27 15:06:38