Fascinating, and utterly unintuitive. This is a question that appeared on the American SAT test until it was recently removed. Every student ever has gotten it wrong, and that's because the SAT writers got it wrong too. Can you get it right? Please Register or Log in to view the hidden image! Spoiler: The correct answer is not listed at all. I guarantee that, even knowing this, you will get it wrong too. I sure did, and I can hardly believe it even after having been shown the correct answer. (Hint: It is an exact number; nothing approximate or irrational).
I've seen this, and it did confuse me that the answer I came up with wasn't one of the options. For those of you who want to try to answer for yourself, read no further, otherwise go to the bottom of this post. . . . . . . . . . . . . . Here's how I look at it. Imagine an arrow affixed to the smaller circle and pointing towards the center of the larger one. Roll the smaller circle around the larger one until the arrow is pointing towards the center again. You will have rolled 1/3 the way around the larger circle. But the arrow be now pointing not directly to the right, but 120 degree clockwise of that. The smaller circle has gone through 1 1/3 rotations. It has to do this a total of three times to get back where it started and 3 x 1 1/3 = 4 In fact, no matter what the ratio of the larger to smaller circle is, the answer will always be 1 more than that ratio.
For astro buffs it's like the diff between sidereal day and Solar day. For fiction buffs, it's like what happened in Verne's Around the World in Eighty Days
Using a simple observation, I got the correct answer. All about POV, looking down on the drawing from above and noting where the center of circle A is. And yes, it is a bit like the sidereal v solar thing. I see now Janus had a similar approach.
I would think it's 3. Circumference is 2*pi*R. So the circumference of B is 3 times the circumference of A. Why am I wrong?
That is the intuitive answer, yes. But it's wrong. The narrator of the video (link to follow) even physically demonstrates this solution in cardboard by stretching the big circle's circumference out into a line segment, showing that it's 3 times longer than the small circle' circumference. And it's still the wrong answer. Hint: You live in New York and set out to travel around the world. You travel East around the Earth over the course of 79 days. When you arrive back in New York, how many sunrises have you witnessed?
Here is the video. It's a fascinating article. https://digg.com/digg-vids/link/sat-question-fail-coin-paradox?utm_source=digg For those with TLDRitis, skip ahead to about the 4 minute mark. But I recommend you don't.
Aha! I see it! https://plus.maths.org/content/circles-rolling-circles EDIT: By the way - it was seeing the 1 to 1 ratio, same size circles, that made it click.
Good question in the OP. It took your spoiler for me to actually bother to discard my initial intuitive answer, and, yeah, I got there, and like gmilam it was looking at a 1:1 circle that made me realise. Please Register or Log in to view the hidden image!
Here's a quick video that visualizes it a bit differently. Assume that the circles are merely rotating against each other without their centers moving like this: Please Register or Log in to view the hidden image! 3 rotations of the small circle to every one of the larger Then imagine that the whole video rotates clockwise so that the larger circle appears to not be rotating. you are adding
It is the center of A that is making a complete circle to return to its starting point, so the tangent point is a red herring. It's not the ratio of circumferences but the ratio of AB to rA = four. A fun puzzle.
