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00:00In 1982, there was one SAT question that every single student got wrong.
00:07Here it is.
00:08In the figure above, the radius of circle A is one-third the radius of circle B. Starting
00:14from the position shown in the figure, circle A rolls around circle B. At the end of how
00:20many revolutions of circle A will the center of the circle first reach its starting point?
00:27Is it A, 3 halves, B, 3, C, 6, D, 9 halves, or E, 9?
00:38SAT questions are designed to be quick.
00:40This exam gave students 30 minutes to solve 25 problems, so about a minute each.
00:46So feel free to pause the video here and try to solve it.
00:51What is your answer?
00:53I'll tell you right now that option B, or 3, is not correct.
00:58When I first saw this problem, my intuitive answer was B, because the circumference of
01:03a circle is just 2 pi r.
01:06And since the radius of circle B is three times the radius of circle A, the circumference
01:12of circle B must also be three times the circumference of circle A.
01:16So logically it should take three full rotations of circle A to roll around circle B, so my
01:23answer was three.
01:26This is wrong.
01:28But so are answers A, C, D, and E.
01:34The reason no one got question 17 correct is that the test writers themselves got it wrong.
01:40They also thought the answer was three.
01:43So the actual correct answer was not listed as an option on the test.
01:50Mistakes like this aren't supposed to happen on the SAT.
01:53For decades, it was the one exam every student had to take to go to college in the US.
01:58It had a reputation for determining people's entire futures.
02:02As a newspaper from the Times stated, if you mess up on your SAT tests, you can forget it.
02:07Your life as a productive citizen is over.
02:10Hang it up, son.
02:13Of 300,000 test takers, just three students wrote about the error to the College Board,
02:19the company that administers the SAT.
02:22Shivin Kartha, Bruce Taub, and Doug Youngrice.
02:27I did a lot of mouth problems when I was young.
02:30For the competitions, I probably did thousands of mouth problems.
02:33I read it, and I was amazed how badly it's worded.
02:37I just put three down.
02:38I figured that's what they wanted.
02:39The three students were confident none of the listed answers were correct, and their
02:43letters showed it.
02:44As a director at the testing service recalled, they didn't say they had come up with possible
02:49alternative answers or that maybe we were wrong.
02:51They said flat out, you're wrong, and they proved it.
02:55I discussed it with some other people and said I think there was a mistake, and they
03:00mostly said no one cares.
03:03I wrote a letter to the educational testing service.
03:07It was a little while later.
03:08They called us and said I was correct.
03:13Here is their argument.
03:15The simplest version of this problem is with two identical coins.
03:20These have the exact same circumference.
03:22So by our initial logic, this coin should rotate exactly once as it rolls around the other.
03:28So let's try it.
03:30Okay.
03:33But wait, we can see it's already right side up at the halfway point.
03:38So if we finish rolling it around the other coin, it'll have rotated not once, but twice.
03:45Even though the coins are the exact same size.
03:48There are no tricks here.
03:50You can try it for yourself, and I'll do it again slowly.
03:54So one, two.
04:05This is known as the coin rotation paradox.
04:08This paradox also applies to question 17.
04:12I've made a two scale model of the problem.
04:16One useful tip for standardized tests.
04:19Even though they say their images are not to scale, they almost always are.
04:24So when we roll circle A around circle B, we can see that it rotates once, twice, three
04:38times, and four times in total.
04:44So the correct answer to this question is actually four.
04:48Once again, the circle rotates one more time than we expected.
04:52To understand this, let's wrap this larger circle in some ribbon.
04:57And I'll make it the same length as the circumference.
05:04And then I will stick it down to the table as a straight line.
05:08I'm adding some paper here.
05:12So there's something for this to roll on.
05:17And now rolls one, two, three times.
05:23What's happening when we turn this straight path into a circular one is that circle A is
05:27now rolling the length of the circumference and it's going around a circle.
05:32The shape of the circular path itself makes circle A do an additional rotation to return
05:37to its starting point.
05:39So this is the general solution to the problem.
05:42Find the ratio between the circumferences of circle B and circle A and then add one rotation
05:47to account for the circular path traveled.
05:52But there is a way to correctly get three.
05:55Let's count the rotations of circle A from the perspective of circle B looking out at A.
06:00We can see circle A rotates one, two, three times.
06:14And it doesn't matter which circle you are looking from.
06:17To circle A, it also rotates three times to come back to its starting position around circle
06:21B.
06:22Similarly, from the perspective of the coins, we can see that the outer coin only rotates
06:28once as it rolls around the inner coin.
06:33Using the perspective of a circle is just like turning the circle's circumference into
06:37a straight line.
06:39It's only as external observers that we actually see the outer circle travel a circular path back
06:44to its starting point, giving us the one extra rotation.
06:50But there's even another answer.
06:56If you look closely at question 17, it asks how many revolutions circle A makes as it rolls
07:02around circle B back to its starting point.
07:05Now in astronomy, the definition of a revolution is precise.
07:08It's a complete orbit around another body.
07:12The earth revolves around the sun, which is different from it rotating about its axis.
07:16So by the astronomical definition of a revolution, circle A only revolves around circle B once.
07:23It goes around one time.
07:26Now other definitions of a revolution do include the motion of an object rotating about its
07:31own axis.
07:32So one isn't a definitive answer, but the wording of this question is extremely ambiguous
07:37if you can justify at least three different solutions.
07:44After reviewing the letters from the students, the College Board publicly admitted their mistake
07:48a few weeks later and nullified the question for all test takers.
07:51They said they were discounting the problem and they were calling us because they were going
07:57to tell the news and they thought that we should be warned that the news might contact us.
08:03I did a bunch of phone interviews and NBC news.
08:06They came to my school.
08:08They said they were, they said I was right and they were discounting it.
08:10So that was great.
08:11But there's more to the explanation.
08:15It's easy to get an intuitive reason, but it's really hard to formally prove that the answer is four.
08:23I could give you some proofs if you want.
08:25Well, that would be wonderful.
08:26I think that would be, we'd appreciate that for sure.
08:29I have a whiteboard because I'm a mathematician.
08:31So I just happen to have a whiteboard here.
08:33Hold on.
08:36Can you see that?
08:37Yep.
08:38It turns out that the amount the small circle rotates is always the same as the distance
08:44the center travels.
08:46All right.
08:46So why is this true?
08:49Suppose you had a camera and the camera was always pointed at the center.
08:53So in your movie, it looks like the center doesn't move.
08:56In the real world, the center is going around the circle.
08:59Let's say it's going at some speed V.
09:02What's the velocity of this point?
09:04It's zero.
09:04Right.
09:05And that's because it's rolling without slipping.
09:08If it had any component in that direction, that's what slipping would be.
09:13I mean, this is something I think they should have spelled out in the problem.
09:15But when you change your frame of reference, the relative velocities don't change.
09:20In the movie, the center always has velocity zero.
09:24So this point would have to have velocity negative V.
09:27So that means the speed that this is turning is the same as the speed the center is moving.
09:34So if they always have the same speed, they have to go the same total distance.
09:38The total distance this turns has to be the same as the total distance the center moves.
09:43In this problem, the center of the small circle goes around a circle of radius four.
09:50So the total distance that the center moves is eight pi.
09:55What's the total amount that the small circle rotates?
09:58It rotates four times and its circumference is two pi.
10:01It's the same number.
10:02If it rolls without slipping, the total distance the center travels is the same as the total amount it turns.
10:11And this is always true.
10:14Take a circle rolling without slipping on any surface, from a polygon to a blob,
10:19on the outside or the inside.
10:22The distance traveled by the center of the circle is equal to the amount the circle has rotated.
10:27So just find this distance and divide it by the circle's circumference to get how many rotations it's made.
10:33This is an even more general solution than our answer to the coin paradox,
10:36where we just took our expected answer, which we'll call n, and added one.
10:41And it reveals where this shortcut comes from.
10:44If a circle is rolling continuously around a shape, the circle's center goes around the outside,
10:50increasing its distance traveled by exactly one circumference of the circle.
10:54So the distance traveled by the circle's center is just the perimeter of the shape
10:58plus the circle's circumference.
11:00When we ultimately divide this by the circle's circumference to get the total number of rotations,
11:04we get n plus one.
11:07If a circle is rolling continuously within a shape,
11:10the distance traveled by the circle's center decreases by one circumference of the circle,
11:14making the total number of rotations n minus one.
11:16If the circle is rolling along a flat line, the distance traveled by the circle's center is
11:22equal to the length of the line, which divided by the circle's circumference is just n.
11:27This general principle extends far beyond a mathematical fun fact.
11:32In fact, it's essential in astronomy for accurate timekeeping.
11:37When we count 365 days going by in a year, 365.24 to be precise,
11:43we say we're just counting how many rotations the earth makes in one orbit around the sun.
11:48But it's not that simple.
11:50All this counting is done from the perspective of you on earth.
11:54To an external observer, they'll see the earth do one extra rotation to account for
11:59its circular path around the sun.
12:01So while we count 365.24 days in a year, they count 366.24 days in a year.
12:10This is called a sidereal year, sidereal meaning with respect to the stars,
12:15where an external observer would be. But what happens to that one extra day?
12:22A normal solar day is the time it takes the sun to be directly above you again on earth.
12:27But the earth isn't just rotating, it's orbiting the sun at the same time.
12:31So in a 24-hour solar day, earth actually has to rotate
12:35more than 360 degrees in order to bring the sun directly overhead again.
12:40But earth's orbit is negligible to distant stars.
12:44To see a star directly overhead again, earth just needs to rotate exactly 360 degrees.
12:53So while it takes the sun exactly 24 hours to be directly above you again,
12:57a star at night takes only 23 hours, 56 minutes, and 4 seconds to be above you again.
13:06That's a sidereal day.
13:09This explains where the extra day goes in the sidereal year.
13:14If we start a solar day and a sidereal day at the same time, we'd see them slowly diverge
13:20throughout the year. After 6 months, the sidereal day would be 12 hours ahead of the solar day,
13:26meaning that noon would be midnight, and it would keep moving up until it's finally one full day
13:32ahead of the solar day, at which point a new year and orbit begins.
13:39365.24 days that are each 24 hours long are equal to 366.24 days that are each 23 hours,
13:4856 minutes, and 4 seconds long.
13:50So it makes no sense to use sidereal time on earth, because six months down the line,
13:58day and night would be completely swapped. But equally, it's useless to use solar time while
14:04tracking objects in space, because the region you're observing would shift between say 10pm
14:10one night and 10pm the next night. So instead, astronomers use sidereal time for their telescopes
14:16to ensure that they're looking at the same region of space each night. And all geostationary satellites,
14:22like those used for communication or navigation, they use sidereal time to keep their orbits locked
14:28with the earth's rotation. So the coin paradox actually explains the difference between
14:34how we track time on earth and how we track time in the universe.
14:43The rescoring of the 1982 SAT wasn't all good news. With question 17 scrapped, students' scores
14:50were scaled without it, moving their final result up or down by 10 points out of 800.
14:55Now while that doesn't seem like much, some universities and scholarships use strict minimum
15:00test score cutoffs. And as one admissions expert put it, there are instances, even if we do not
15:06consider them justified, in which 10 points can have an impact on a person's educational opportunities.
15:11It might not keep someone out of law school, but it might affect which one he could go to.
15:17This mistake didn't only cost points off the exam. According to the testing service,
15:21rescoring would cost them over $100,000, money that came out of the pockets of test takers.
15:27The question 17 circle problem was far from the last error on the SAT.
15:35But errors are likely the least of their concerns these days. I mean, the SAT is slowly becoming a
15:40thing of the past. After COVID-19, nearly 80% of undergraduate colleges in the US no longer require
15:47any standardized testing. And that 1982 exam, well, it didn't turn out too badly for some.
15:53How did you do on your math SAT, if I can ask?
15:57I got an 800. Even before that, it was clear I was going to go into math. I did math competitions.
16:04I really liked math.
16:06Do you end up writing any math questions these days?
16:09A while back, I wrote problems for a math competition.
16:12And were you careful with how you wrote them, the wording?
16:14I hope so. I tried.
16:18Today's deep dive on one SAT question proves there's no substitute for hands-on exploration
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18:30you
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