Comments on: Circular Puzzle

By: Anonymous

Anonymous — Sun, 18 Jan 2009 23:22:55 +0000

A convenient thought experiment is the following: suppose the inner circle had radius approaching zero. Then it is immediate that you need one rotation around the axis just to get back to position.

By: Anonymous

Anonymous — Tue, 30 Sep 2008 05:21:32 +0000

The question is how many times does the the smaller circle rotate around its OWN axis to cover the distance. Therefore, the answer is three. If the smaller circle did not rotate around its own axis but still traveled once around the bigger circle, it would have rotated once. However, the amount of times it rotated around its OWN axis would still be zero. The degrees of rotation of the smaller ball as it turns around its axis does not get further increased based on its position on the bigger ball.

By: ken

ken — Tue, 22 Jul 2008 05:37:29 +0000

agree with arvindn that it depends on the observer or the frame of reference. it’s a different answer if you are observing from outside of big/small circle, observing within the small circle or observing from within the big circle. Or if we claim that the observer is rolling together with the small circle, then the answer is 0? Or if the observer is rolling infinitely faster than the small circle, then the answer is infinity.Anyway, it’s a SAT math question and not a language question. While 4 or 2 should be correct for the math genius, it shouldn’t penalize the most common answer (2*pi*3)/(2*pi*1) = 3.

By: Ian Young

Ian Young — Fri, 20 Jun 2008 17:08:15 +0000

Actually an even simpler answer comes to mind: if you plot it on a line, the answer is three. But then consider that to get the endpoints to touch, you have to rotate one end through 360 degrees, adding a rotation.

By: Ian Young

Ian Young — Fri, 20 Jun 2008 17:03:23 +0000

since the circumference of the top circle (a) is effectively 1/3 the circumference of the lower (b), for the point of contact to rotate back around into contact, the circle has to rotate 360+120 degrees the first time, 360+90 the second time, and 360+150 the third time. The additional rotation adds up to 360 degrees, so 4. Or you could just note that the center of circle a is 4u from the center of circle b. If it rotated inside it would be 2u.

By: arvindn

arvindn — Tue, 10 Jun 2008 23:19:07 +0000

when i was a kid i read a book called “mathematics can be fun” by yakov perelman, also author of the more famous “physics can be fun.” the book was lifechanging in that looking back, it significantly affected my career choice. anyway, it had a detailed discussion of this question. 3 and 4 are both valid answers (0 is ruled out by the wording of the question — “rolling instead of moving”). to see why 3 might be valid, imagine you’re an observer sitting inside the smaller circle. to all other observers, the answer is 4.

By: Kurt

Kurt — Mon, 28 Apr 2008 16:38:02 +0000

Yep, that’s it!

By: Anders

Anders — Mon, 28 Apr 2008 16:16:16 +0000

The guy who walks around the outside of the big circle has travelled a distance of 6pi, and rotated one revolution in the same direction as the smaller circle would do if rolling. So I’m guessing y=4.If we put the little man inside the circle he will again walk 6pi but rotate in the opposite direction compared to a rolling circle, so for the case where the small circle is inside the big circle I would guess the answer is 2?

By: Kurt

Kurt — Mon, 28 Apr 2008 14:31:44 +0000

Hi Mathmom, you are absolutely right. (By the way, feel free to discuss the answer in the comments if you want.) The way the value ‘3′ is justified typically goes like this: The perimeter of the big circle is 6pi, and you can even imagine “unwrapping” the big circle to make a straight line 6pi long. The small circle has a perimeter of 2pi, and if we assume there is no slippage as it rolls along, it’s going to have to rotate 3 times to cover that distance.Now, the question is, can you stop there, or is there more to be considered to come up with the final answer? Here is one more “visualization aid” for anyone else still working through the problem: Instead of a small circle, imagine there is a little person standing at the “north pole” of the big circle. If the person walks around the circle back to their starting point, have they done a flip in the process? If so, was the flip in the same direction or the opposite direction of the rotating circle?

By: mathmom

mathmom — Mon, 28 Apr 2008 05:00:21 +0000

I’m going to assume that x is 3. That was the original answer I came up with by comparing circumferences, but all the while, I was wondering if I was missing something about “rotating around its axis”. Then I did the visualization thing. I thought about what the orientation of the little circle would be when it had moved 1/3 of the way around the big circle. And I thought about where the little circle would be on the big circle when it was first re-oriented the way it is at the top of the page. And thus came up with a different answer (larger than 3) that I think is y. But I gather that Kurt would rather the numbers not be put in the comments here, so I won’t post mine (yet anyhow).I’m having a much harder time visualizing the case where the little circle rolls around the inside perimeter! But I think the answer is smaller than 3 in that case. Am I on the right track?