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Brain teaser: rolling one quarter around another

The other of my two favorite brain teasers involves rolling one quarter (a U.S. 25-cent piece) around another. What makes this such a goodie, of course, is that the correct answer is very counter-intuitive; plus it seems that the puzzlers who pose it are themselves unaware of all the interesting ramifications.

The problem is this: Lay 2 quarters on a flat surface so that their edges are in contact. Hold one still and roll the other one around it. How many rotations does the moving quarter make?

Just to make sure we're on the same wave-length and there are no quibbles, understand that there is no slippage as the rolling quarter moves; no sliding or skidding, like a well-behaved tire on a dry road. By "rotations", we mean, how many time does George Washington's head spin around by the time the quarter returns to its starting point?

Clear enough? You still have time to think about it; I'm not ready to spill the beans quite yet.

First, here's a newspaper article from 1982 that discusses the same problem in a slightly different guise. I got such a charge out of this. From the Washington Post, May 25, 1982:

College Board's Math Proved Wrong

New York, May 24 (UPI) - Math scores are being recalculated for 300,000 students who took Scholastic Aptitude Tests across the nation May 1 because three students proved that the correct answer to one of the questions was not among the possible choices on student's answer sheets, the College Board said today.

It was not the first time students discovered a flawed answer on the tests taken by more than 1 million high school juniors and seniors each year and scored on a scale of 200 to 800. Three others have been discovered during the past two years. In this instance, three students wrote to the College Board after taking the test, complaining that none of the five possible answers offered for a math question was correct.

The board said it will not identify the students involved in the expose until permission is granted.

"The problem came to light Friday and today we sent Mailgrams to 3,000 colleges advising them recalculated scores would go out to them within the next 10 days," said Barrie Kelly, the board's executive director of communication.

Daniel B. Taylor, executive vice president for operations, said as a result of the flawed question, he anticipates adjustments 10 points up or down on the tests.

The disputed math question shows a large circle, B, and to the left of it, a small circle, A - touching B.

"In the figure above," the problem states, "the radius of circle A is one third the radius of circle B. Starting from position shown in figure, circle A rolls around circle B. At the end of how many revolutions of circle A will the center of circle A first reach its starting point?"

The choices given:
(a) 3/2
(b) 3
(c) 6
(d) 9/2
(e) 9

"The answer to this question should have been 4, not 3," Kelly said.

This explanation, proving the students right, was given:

"The circumference of the large circle is three times the circumference of the small circle. If the small circle were to rotate among [along] a straight line segment equal in length to the circumference of the large circle, it would make three revolutions.

"So, the intended answer to this problem was choice (b) 3. However, the motion of the small circle is not in a straight line, but rather around a large circle. This revolving action around the large circle contributes an extra revolution as circle A rolls around circle B. Thus, the answer to this question should have been 4, not 3."

There, do you have the answer to our "2 quarters" problem now? The moving quarter actually makes two rotations in its single trip around the stationary quarter.

Perhaps the above explanation from the College Board didn't make this clear. For a start, they confuse matters by their sloppy use of the word "revolve". They use it to refer to both revolution and rotation. We will not be so careless. Just remember that the earth revolves about the sun, and the earth rotates about its axis. In the college board problem, circle A rotates 4 times about its axis while it revolves 1 time around circle B.

The College Board's great communicator may also have given the impression that this unexpected extra rotation comes about because the small circle moves "around a large circle". "Largeness" has nothing to do with it. Whether the stationary circle has the diameter of a pinhead, or a button, or a Love Me Do 45, or the equator, or the entire Galaxy, the analysis is exactly the same: the rolling circle always picks up one extra rotation for each complete revolution. You calculate how many rotations it would make if the path were a straight line - and then add 1.

Thus it is with our "2 quarters" version of the problem. The intuitive answer (that is, the "straight line" answer) is 1. Add 1 to that to get the "circular motion" answer, 2. Pull out two quarters and give it a whirl - it can be quite startling the first time you see how George's head spins.

It seems a waste that the newspaper article, for as long as it was, didn't make more of a point of this general solution. Moreover, it could have mentioned examples of this phenomenon in our everyday lives.

For example, the moon always shows the same face to us. One might hastily conclude that this is because "the moon doesn't rotate!" In fact, though, it must rotate once per revolution about the earth in order to show the same face to the earth all the time. Our analysis would be, take 0 rotations (the "straight line" answer in which the Man in the Moon scrapes his nose along his orbit) and add 1, to get the correct answer: 1 rotation.

How many times does the earth rotate during the 365 day year? The hasty answer would be 365. The correct answer is 366; again, you have to add 1 because of the revolution around the sun. Sketch this out on a piece of paper. Notice that every day the earth has moved a little further along in its orbit, so it has to rotate a hair more than 360 degrees to get the sun directly overhead the same spot again. Those little extra bits, accumulated over one whole revolution add up to an extra 360 degrees.

The above astronomical examples point up the fact that the stated requirement of "no skidding" is not necessary at all. The revolving (orbiting) body may spin as wildly or as lazily as it wants; it does not have to roll along its orbit like a tire on a road. Our solution of "add 1 rotation per revolution" is even more general, more powerful! And note that the orbit doesn't have to be a circle, or an ellipse, even - or anything so well-behaved. It could be amoeba-shaped, or a square, or an irregular heptagon, even.

Tying this all together we could reformulate our brain teaser solution as follows: A body that spins a given number (N) of times as it moves along some path back to its starting point will appear to spin one less (N-1) times to an observer located anywhere within the area bounded by that path. Or, put the other way around, if an observer sees an orbiting body spin N times for each revolution, it really has to spin N+1 times to give that appearance. (The stationary George only saw the orbiting George spin 1 time, which was our "straight line" answer.) Or, put mathmatically, actual_rotations = apparent_rotations + 1.

Note that spin direction is significant. Spinning counter to the orbital motion is viewed as negative spins. For example, suppose your moon looks like it's spinning backwards once per revolution. To accomplish that, it must not rotate at all ( 0 = -1 + 1 ). Act this out with your two quarters - it's easier than the rolling experiment.

Sorry to get so mathematical-sounding there - back to the college board goof-up...

Why I got such enjoyment out of it is that, for me, all this was 7th-grade stuff. (In the U.S., 7th grade is for 12-year-olds. See note at the bottom.) Anybody can make a mistake, but I presume those SAT questions are checked over by whole battalions of test-makers.

And while I tip my hat (with some envy) to the 3 students who caught the error, I must admit that I am amazed that it was only 3 out of 300,000. It should have been more like the other way around: 3 students who didn't catch it.

And we're not through yet. There is a bizarre follow-up to all of this. The same problem popped up 10 years later (March, 1992) in a Marilyn vos Savant column. When I say the same, I mean the same. She has two circles, called circle A and circle B. The smaller circle rolls around the larger one. The larger circle has a circumference 3 times as big as the smaller one. Brainiac (vos Savant) even uses the word "revolution" where she means rotation.

Like the college board people, she made no attempt to step beyond this special case to the more general case of anything rolling around anything. No, she had the whole world running around desperately trying to find 2 disks, one exactly 3 times as wide as the other.

She certainly had plenty of opportunity to discuss the general case. There was an extensive follow-up column, prior to which I had written to her about the simpler - and more effective - case of the 2 quarters. I suggested, "Rather than a special case illustration, you might try to explain that the rolling circle always picks up one extra rotation for each revolution." I clued her in on how this relates to our friendly neighborhood astronomical bodies.

Instead, she strokes herself with a fawning letter from an MIT professor with a Ph.D. (Now why would the world's smartest person need praise from such peabrains?) She exclaimed, "Wow. Out of all that mail, I found only one letter that agreed with me." That means she didn't read mine. Or couldn't understand it. (Sorry. As John Lennon once said, "Silly to be so bitter...")

THANKS: to my 7th-grade math teacher, Mr. Lieske at Johnnycake Junior High School, for all of the above, plus so many more things (e.g. different-sized infinities, irrational numbers, the usefulness of memorized squares of integers up to 25, special relativity...) which, years later, I was shocked to find that my college-mates didn't know. The quote on your classroom wall, "Understanding is not memorization", has frequent reason to come to mind. I try to work "right rectangular parallelopiped" into every conversation I can. Thanks for the introduction to calculus in 9th grade. And thanks for making it all so much fun. By the way, how's your friend (colleague? mentor?) Gushalotta H2O?


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