That's fun. I of course immediately selected 3 which means I could have a bright career in test preparation ahead of me.
Instead, I think's its easier to note that that the _center_ of a circle of radius r travels 2 * pi * r distance over one rotation. In the problem, the center of the smaller circle has to travel further than the circumference of the bigger circle - it traces a circle whose radius is the sum of the two radii.
So, if 3 * r_small = r_big, the center of the small circle has to travel 2 * pi * (r_s + r_b) = 4 * 2 * pi * r_s, then divide by 2 * pi * r_s per rotation to get 4 rotations.
Radius is always proportional to circumference, so a circle twice the size is twice as big around.
Take the case of two identical circles. To move a point on the first circle from 12 o'clock back to 12 o'clock, it only goes halfway around the other circle, which you can prove to yourself by imagining you've wrapped a string around the circle and marked it at 12 and 6 o'clock. If you unwrap half of the string and wrap it around the other circle, then the end of the rope is at 6 o'clock. To roll the string back up by moving the circle, the top of the circle will be pointing upward again when it reaches the bottom. 1 full revolution. Now wrap the other half of the string around the other side, 6 has to go back to the bottom again to roll the string back up. 2 revolutions.
Hold center of small circle still, rotate big circle once counter clock wise, small one rotates 3x clockwise.
Glue them together, rotate big circle once clockwise, small circle also rotates once clockwise.
Sum them together, 0 rotations for big circle, 4 for small. I'm not at all sure how to rigorously generalize.
Only by introducing a larger frame of reference, a grid or in the video a table, you gain an outside perspective. From this outside perspective you redefine a revolution according to some new orientation and end up with n+1 revolutions.
Or maybe the argument is backwards and I just try to justify answering 3.
Imagine the limiting case, as the inner circle approaches the size of the outer circle - the inner circle completes much less than one rotation per lap around the inside edge of the outer circle, and ‘seizes’ (if we’re imagining these as gears), completing zero rotations per lap when the circles are the same size. However, rolling around the outside, a circle of the same size completes two rotations.
In general the problem is like the old Spirograph toy (which I had to break out to convince myself)
I made a diagram that helped me think through it visually:
https://i.imgur.com/dospt2w.png
If circle A is rolling around the edge of circle B from within, it is actually revolving around a new, smaller circle C which has the Radius Circle B - Circle A.
The reason that this problem is tricky, and has a counterintuitive solution is that Circle A is rolling around Circle B, and so the 'radius; of the circular path it is following is the radius of Circle B + the radius of Circle A.
I'm no expert, but some quick math:
Circle B has a radius of 9, circumference then is 56.55 Circle A (1.3) has a radius of 3, circumference is 18.84
56.55 / 18.84 = 3. This suggests that you could "unwind" circle A (say it was made of pipecleaner), and you would need 3 circle As to fully ensconce Circle B
BUT that wasn't the question. The question states that Circle A is rolling AROUND circle B.
So the circumference of the circle we are now trying to 'ensconce' is Circle A Radius + Circle B Radius = 12, so the new circumference is 75.39, and divided by Circle A Circumference, we end up with 4, which makes sense, and matches the demonstration from the video.
HOWEVER, if we are 'rolling' around the inside of circle B, then I think you're right, the radius of the circular path Circle A will take is 9 -3 = 6, and the circumference of said circle is 38, so we it only will take two rolls. I think this is correct, i do not think it will be 4 rolls.
Think of it this way: when Circle A is rolling around the inside of Circle B, the way you're picturing it is circle A following the outside of Circle B, which is why I think intuitively it feels like the answer will still be 4 revolutions. BUT a better way to think of it is that Circle A is actually revolving around a new Circle, Circle C, which has a circumference of 38. Circle A is not revolving around Circle B, it's revolving around this new, smaller circle within circle b. Does that make sense?
I made a quick diagram to illustrate my point. I did it in a wire-framing tool that has snap to grid, so it's definitely not perfect:
https://i.imgur.com/dospt2w.png
You can play around with what I made here:
Without the diagram (which I didn't see until watching the video) this is ambiguous. In my visualization, the plane of the small circle (A) was perpendicular to the plane of the larger circle (B). (Think of circle B drawn on paper, while circle A is a coin on its edge)
With that interpretation of "rolls one trip around", 3 is indeed is the correct answer.