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.)