- 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 - Today's deep dive on
one SAT question proves there's no substitute for
hands-on exploration to understand and appreciate the everyday
phenomena of our world. But you don't have to
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