Circle revolutions rolling around another circle I just watched this video, and I'm a bit perplexed.

Problem:

The radius of Circle A is 1/3 the radius of Circle B.

Circle A rolls around Circle B one trip back to its starting point.

How many times will Circle A revolve in total?


The intuitive answer is 3, but the correct answer is 4. I understand the trick -- that the center of Circle A must travel a distance of $2\pi(r_B + r_A)$, not $2\pi r_B$ -- but I'm still confused on one item.

At the risk of sounding very un-mathematical, how do the (infinite set of) points on the circumference of each circle map to each other to accomplish this?

Consider Circle A rolling along a straight line the length of the circumference of Circle B. Then it will revolve 3 times. It's like the universe "knows" when to apply a different point mapping when you change the arrangement of matter.
Thank you everyone for your answers. They were all informative, but I wasn't able to intuitively understand them until I saw this visual:
https://www.geogebra.org/m/v3a437ux
The key for me was to see that there are two kinds of revolution happening:

Revolutions of Circle A with respect to Circle B, and
Revolutions of Circle A with respect to the (overhead) observer.

To the observer, the revolutions of the type (2) complete before revolutions of type (1). The revolutions of type (1) happen at the $r_B/r_A$ roots of unity, and the revolutions of type (2) happen at the $(r_B/r_A + 1)$ roots of unity. It was very helpful to see precisely where these revolutions occur, because it was mentally impossible to unify the two types of revolution without the visual. It's also useful to notice that at any given moment, with respect to the observer, the points of Circle A on the far side from Circle B are moving faster than the points of Circle A on the side that is touching Circle B.
David K's answer is great for understanding that the parametric mapping of time to point pairs is the same whether Circle A is rolling along a straight line or a circle, and that we are dealing with frames of reference. I simply didn't believe the mapping was the same until I saw the visual.
zoli and Hans Lundmark's answers are great for understanding that one extra revolution must occur at some point along the rolling path. The complete answer, of course, is that this last mysterious revolution does not happen all at once, but gradually along the entire roll.
The visual was discovered in the comments on this page, which is another discussion of the same problem:
https://plus.maths.org/content/circles-rolling-circles
I don't know why people overthink these simple problems. The circumference of the larger circle is 3x the circumference of the smaller circle. Provided the small circle is not slipping, it will take 3 turns to cover the distance around the larger circle. Nothing magical here.

The reason you were perplexed by the video is because the uploader makes a mistake at 1:34 to 1:42. He counts this as a full revolution of the smaller circle, which it clearly is not. If he marked both circles with points on their circumferences, you'd see that it would take 3 revolutions. It's simply conservation of length. The rolling motion is completely irrelevant to the stated problem.

By the way, the applet you linked to was for a 4:1 scale factor, this is why in the applet it takes four revolutions.