The 1982 SAT Circle Problem: Uncovering the Coin Rotation Paradox

- In 1982, there was one SAT question that every single student got wrong. Here it is.

In the figure above, the radius of circle A is 1/3 the radius of circle B. Starting from the position shown in the figure,

circle A rolls around circle B. At the end of how many revolutions of circle A will the center

of the circle first reach its starting point. Is it A, 3/2, B, three, C, six, D, 9/2, or E, nine?

SAT Questions are designed to be quick. This exam gave students 30 minutes to solve 25 problems, so about a minute each.

So feel free to pause the video here and try to solve it. What is your answer? I'll tell you right now that option B,

or three, is not correct. When I first saw this problem, my intuitive answer was B, because the circumference of a circle is just two pi r,

and since the radius of circle B is three times the radius of circle A, the circumference of circle B must also be three times

the circumference of circle A. So logically, it should take three full rotations of circle A to roll around circle B.

So my answer was three. This is wrong, but so are answers A, C, D, and E. The reason no one got question 17 correct

is that the test writers themselves got it wrong. They also thought the answer was three. So the actual correct answer was not listed

as an option on the test. Mistakes like this aren't supposed to happen on the SAT. For decades, it was the one exam every student had to take

to go to college in the US. It had a reputation for determining people's entire futures.

As a newspaper from the time stated, "If you mess up on your SAT tests, you can forget it. Your life as a productive citizen is over.

Hang it up, son." Of 300,000 test takers, just three students wrote about the error

to the College Board, the company that administers the SAT, Shivan Kartha, Bruce Taub, and Doug Jungreis. - I did a lot of math problems when I was young

for the competitions. I probably did thousands of math problems and I read it and I was amazed how badly it's worded.

I just put three down. I figured that's what they wanted. - The three students were confident none of the listed answers were correct,

and their letters showed it. As a director at the testing service recalled, they didn't say they had come up

with possible alternative answers or that maybe we were wrong. They said flat out, "You're wrong," and they proved it.

- I discussed it with some other people and said, I think there was a mistake, and they mostly said, "No one cares."

I wrote a letter to the Educational Testing Service. It was a little while later they called us and said I was correct.

- Here is their argument. The simplest version of this problem is with two identical coins.

These have the exact same circumference. So by our initial logic, this coin should rotate exactly once

as it rolls around the other. So let's try it. Okay.

But wait, we can see it's already right side up at the halfway point. So if we finish rolling it around the other coin,

it'll have rotated not once, but twice. Even though the coins are the exact same size There are no tricks here,

you can try it for yourself, and I'll do it again slowly. That's one,

two. This is known as the coin rotation paradox. This paradox also applies to question 17.

I've made a to scale model of the problem. One useful tip for standardized tests, even though they say their images are not to scale,

they almost always are. So when we roll circle A around circle B, we can see that it rotates once, twice,

three times, and four times in total. So the correct answer to this question is actually four. Once again, the circle rotates one more time

than we expected. To understand this, let's wrap this larger circle in some ribbon

And I'll make it the same length as the circumference, and then I will stick it down to the table as a straight line.

I'm adding some paper here so there's something for this to roll on. And now it rolls one, two, three times.

What's happening when we turn this straight path into a circular one is that circle A is now rolling the length of the circumference

and it's going around a circle. The shape of the circular path itself makes circle A do an additional rotation

to return to its starting point. So this is the general solution to the problem. Find the ratio between the circumferences of circle B

and circle A and then add one rotation to account for the circular path traveled. But there is a way to correctly get three.

Let's count the rotations of circle A from the perspective of circle B looking out at A. We can see circle A rotates one, two,

three times. And it doesn't matter which circle you are looking from, to circle A, it also rotates three times

to come back to its starting position around circle B. Similarly, from the perspective of the coins, we can see that the outer coin only rotates once

as it rolls around the inner coin. Using the perspective of a circle is just like turning the circle's circumference

into a straight line. It's only as external observers that we actually see the outer circle travel

a circular path back to its starting point, giving us the one extra rotation. But there's even another answer.

If you look closely at question 17, it asks how many revolutions circle A makes as it rolls around circle B back to its starting point.

Now, in astronomy, the definition of a revolution is precise. It's a complete orbit around another body.

The earth revolves around the sun, which is different from it rotating about its axis. So by the astronomical definition of a revolution,

circle A only revolves around circle B once. It goes around one time. Now, other definitions of a revolution do include the motion

of an object rotating about its own axis. So one isn't a definitive answer, but the wording of this question is extremely ambiguous

if you can justify at least three different solutions. After reviewing the letters from the students, the College Board publicly admitted their mistake

a few weeks later and nullified the question for all test takers. - They said they were discounting the problem

and they were calling us because they were gonna tell the news and they thought that we should be warned

that the news might contact us. I did a bunch of phone interviews and NBC News, they came to my school.

They said I was right and they were discounting it. So that was great. - But there's more to the explanation.

- It's easy to get an intuitive reason, but it's really hard to formally prove that the answer is four.

I could give you some proofs if you want. - Well, that would be wonderful. I think that would be, we'd appreciate that for sure.

- I have a whiteboard because I'm a mathematician, so I just happen to have a whiteboard here. Hold on.

Can you see that? - Yep. - It turns out that the amount the small circle rotates is always the same as the distance the center travels.

All right, so why is this true? Suppose you had a camera and the camera was always pointed at the center.

So in your movie, it looks like the center doesn't move. In the real world, the center is going around the circle.

Let's say it's going at some speed V. What's the velocity of this point? It's zero, and that's because it's rolling without slipping.

If it had any component in that direction, that's what slipping would be. I mean, this is something I think they

should have spelled out in the problem, but when you change your frame of reference, the relative velocities don't change.

In the movee, the center always has velocity zero. So this point would have to have velocity negative V. So that means the speed that this is turning

is the same as the speed the center is moving. So if they always have the same speed, they have to go the same total distance.

The total distance this turns has to be the same as the total distance the center moves. In this problem,

the center of the small circle goes around a circle of radius four. So the total distance that the center moves is eight pi.

What's the total amount that the small circle rotates? It rotates four times, and its circumference is two pi. It's the same number.

If it rolls without slipping, the total distance the center travels is the same as the total amount it turns.

- And this is always true. Take a circle rolling without slipping on any surface from a polygon to a blob, on the outside or the inside,

the distance traveled by the center of the circle is equal to the amount the circle has rotated. So, just find this distance

and divide it by the circle circumference to get how many rotations it's made. This is an even more general solution than our answer

to the coin paradox where we just took our expected answer, which we'll call N, and added one, and it reveals where this shortcut comes from.

If a circle is rolling continuously around a shape, the circle center goes around the outside, increasing its distance traveled

by exactly one circumference of the circle so the distance traveled by the circle center is just the perimeter of the shape

plus the circle's circumference. When we ultimately divide this by the circle circumference to get the total number of rotations, we get N plus one.

If a circle is rolling continuously within a shape, the distance travel by the circle center decreases by one circumference of the circle,

making the total number of rotations N minus one. If the circle is rolling along a flat line, the distance travel by the circle center

is equal to the length of the line which, divided by the circle circumference, is just N. This general principle extends far beyond

a mathematical fun fact. In fact, it's essential in astronomy for accurate timekeeping.

When we count 365 days going by in a year, 365.24, to be precise, we say we're just counting how many rotations

the earth makes in one orbit around the sun. But it's not that simple. All this counting is done from the perspective

of you on earth. To an external observer, they'll see the earth do one extra rotation

to account for its circular path around the sun. So while we count 365.24 days in a year, they count 366.24 days in a year.

This is called a Sidereal year, Sidereal meaning with respect to the stars where an external observer would be.

But what happens to that one extra day? A normal solar day is the time it takes the sun to be directly above you again on earth.

But the earth isn't just rotating, it's orbiting the sun at the same time. So in a 24-hour solar day,

earth actually has to rotate more than 360 degrees in order to bring the sun directly overhead again. But Earth's orbit is negligible to distant stars.

To see a star directly overhead again, Earth just needs to rotate exactly 360 degrees. So while it takes the sun exactly 24 hours

to be directly above you again, a star at night takes only 23 hours, 56 minutes, and four seconds to be above you again.

That's a Sidereal day. This explains where the extra day goes in the Sidereal year. If we start a solar day

and a Sidereal day at the same time, we'd see them slowly diverge throughout the year. After six months,

the Sidereal day would be 12 hours ahead of the solar day, meaning that noon would be midnight, and it would keep moving up

until it's finally one full day ahead of the solar day, at which point a new year and orbit begins. 365.24 days that are each 24 hours long

are equal to 366.24 days that are each 23 hours, 56 minutes, and four seconds long. So it makes no sense to use Sidereal time on earth,

because six months down the line, day and night would be completely swapped. But equally, it's useless to use solar time

while tracking objects in space, because the region you're observing would shift between say, 10:00 PM one night and 10:00 PM the next night.

So instead, astronomers use Sidereal time for their telescopes to ensure that they're looking at the same region of space each night.

And all geostationary satellites, like those used for communication or navigation, they use Sidereal time

to keep their orbits locked with the Earth's rotation. So the coin paradox actually explains the difference between how we track time on earth

and how we track time in the universe. The rescoring of the 1982 SAT wasn't all good news. With question 17 scrapped,

students' scores were scaled without it, moving their final result up or down by 10 points out of 800.

Now, while that doesn't seem like much, some universities and scholarships use strict minimum test score cutoffs.

And as one admissions expert put it, "There are instances, even if we do not consider them justified, in which 10 points can have an impact

on a person's educational opportunities. It might not keep someone out of law school, but it might affect which one he could go to."

This mistake didn't only cost points off the exam. According to the testing service, "Rescoring would cost them over $100,000,

money that came outta the pockets of test takers. The question 17 circle problem was far from the last error on the SAT.

But errors are likely the least of their concerns these days. I mean, the SAT is slowly becoming a thing of the past.

After COVID-19, nearly 80% of undergraduate colleges in the US no longer require any standardized testing. And that 1982 exam, well,

it didn't turn out too badly for some. How did you do on your math SAT, if I can ask? - I got an 800.

Even before that, it was clear I was gonna go into math. I did math competitions. I really liked math. - Do you end up writing any math questions these days?

- A while back I wrote problems for a math competition. - And were you careful with how you wrote them, the wording? - I hope so. I tried

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