Circular Puzzle
Puzzles of various sorts have been popular on the math blogosphere lately, so I thought I'd make my own modest contribution. As puzzles go, there is not very much to this one -- no calculations are necessary, you can just look at it and visualize the answer. But the backstory that goes with it is mildly amusing. This appeared as a question on the SAT exam, back many years ago when I was an undergrad.
Imagine a circle of radius 1 unit, sitting on top of a larger circle of radius 3 units, like so:
Now suppose that the smaller circle rolls around the perimeter of the larger circle until it returns to its starting position. In the course of doing this, how many rotations about its axis will the smaller circle make?Before going on, you might want to take a moment to think about what the answer should be... Okay, ready? This particular problem was featured in a story in the news because the answer key used to score the SAT had the wrong value for this question. Let's just call it x for the time being. A high school student who sat for the exam realized that the correct answer should be y, and the math scores had to be retabulated for everyone who took the SAT that session.
The story appeared in our local paper over the weekend. The following Monday, the professor for my differential equations class, who also happened to be serving as the department chairperson that semester, mentioned that he had received a phone call from a newspaper reporter asking about the question. The professor sketched out the problem for us, and said, "I told him that, of course the answer was x." Naturally as soon as class was over, several of us students who had read about the problem over the weekend ran up to the professor and explained to him why he was wrong.
For the next week, the problem continued to stir up debate in our math lab. About half the students who looked at it initially believed the answer was x. Most people quickly changed their minds once it was explained to them, but we did have one hold-out who refused to budge. If I remember correctly, he was a Ph.D. candidate in differential geometry, who insisted that since he had advanced training in a geometrical field, we should defer to his judgment on the problem.
(Now, as to what x and y are... You have almost certainly figured out the correct values for those by now. If you are still unsure, though, I'll put a hint in the comments for you.)
Labels: math
posted by Kurt @ 1:13 AM
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11 Comments:
If you are still unsure, here are a couple of things to think about...
Suppose the smaller circle was rolling around the inside perimeter of the larger circle (imagine one of those Spirograph toys). Would the answer change?
How many rotations about its axis does the Moon make each time it orbits the Earth?
is the answer 0 or 3, Kurt? (Or something else?!) It's not clear to me how to define a rotation "about its axis."
Yeah, my guess was 3, with 1 as my alternate...
Hmmm, I'm not sure where those 0 and 1 values are coming from...
Now, there is a little bit of ambiguity about what constitutes a rotation here, but the most obvious interpretation what we want. The frame of reference is the background of the picture, and the big circle is fixed in position with respect to that background. The little circle, as it rolls around the big circle, is rotating about its own center like a wheel. How many times does it need to rotate before it returns to its starting position? We're assuming it is traveling only in one direction (say, clockwise), and by definition it is going to complete one 'orbit' around the big circle - that's not the number we want.
You could figure out the answer empirically by cutting circles out of cardboard and rolling the little circle around the big one. Now that I think about it, this problem would be perfect for a little Flash or Java animation... I'm not going to be able to do anything like that in the immediate future, but if anyone else would like to volunteer that would be great.
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?
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?
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?
Yep, that's it!
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.
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.
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.
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