The SAT Question Everyone Got Wrong

- In 1982, there was an SAT

question

that all the students answered

wrong

. Here it is. In the figure above, the radius of circle A is 1/3 of 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 be the center of the circle reaches its starting point first. Is it A, 3/2, B, three, C, six, D, 9/2 or E, nine? SAT

question

s are designed to be quick. This exam gave students 30 minutes to solve 25 problems, about one minute each. So feel free to pause the video here and try to fix it.

What is your answer? I'll tell you right now that option B, or option three, is not correct. When I first saw this problem, my intuitive answer was B, because the circumference of a circle is only 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 complete 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 answered question 17 correctly is that the test writers themselves made a mistake. They also thought the answer was three. Therefore, the actual correct answer was not listed as an option on the test. Errors like this are not supposed to happen on the SAT. For decades, it was the only exam that every student had to take to go to college in the United States. He had a reputation for determining people's entire future. As one newspaper of the time said: "If you make a mistake on your SATs, you can forget about it. Your life as a productive citizen is over.

Hang up, son." Of 300,000 test takers, only three students wrote about the error to the College Board, the company that administers the SAT: Shivan Kartha, Bruce Taub and Doug Jungreis. - When I was young, I did a lot of math problems for competitions. I probably did thousands of math problems, read it, and was shocked at how poorly it was written. I just put three. I figured that's what they wanted. - The three students were sure that none of the answers listed were correct and their letters proved it. As one director of the testing service recalled, they did not say that they had found possible alternative answers or that perhaps we were wrong.

They flatly said, "You're wrong," and they proved it. - I discussed it with other people and said, "I think there was a mistake," and most of them said, "No one cares." I wrote a letter to the Educational Testing Service. Shortly after they called us and said I was right. - Here is your argument. The simplest version of this problem is with two identical coins. These have exactly the same circumference. So by our initial logic, this coin should spin exactly once while spinning around the other. So let's try it. Well. But wait, we can see that he's already face up halfway.

So if we finish rolling it around the other coin, it will have rotated not once, but twice. Although the coins are exactly the same size. There are no tricks here, you can try it yourself and I will do it again slowly. That's one, two. This is known as the coin rotation paradox. This paradox also applies to question 17. I have made a scale model of the problem. A helpful tip for standardized tests, even if 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 one, two, three, and four times in total.

So the correct answer to this question is four. Once again, the circle turns once more than we expected. To understand this, let's wrap this larger circle in ribbon. I'll make it the same length as the circumference and then glue it to the table as a straight line. I'm adding some paper here so there's something I can roll on. And now it rolls once, twice, three times. What happens when we convert this straight path into a circular one is that circle A now rolls along the circumference and rotates around a circle. The shape of the circular path itself causes circle A to make an additional rotation to return to its starting point.

So this is the general solution to the problem. Find the relationship between the circumferences of circle B and circle A and then add a rotation to account for the circular path traveled. But there is a way to get three correctly. Let's count the rotations of circle A from the perspective of circle B looking at A. We can see that circle A rotates one, two, three times. And no matter which circle you look from, towards circle A, it also rotates three times to return to its initial position around circle B. Similarly, from the perspective of the coins, we can see that the outer coin only rotates once while rolls around the inner coin.

Using the perspective of a circle is like turning the circumference of the circle into a straight line. Only as outside observers can we see the outer circle travel in a circular path back to its starting point, giving us an additional rotation. But there is 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 revolution is precise. It is a complete orbit around another body. The Earth rotates around the Sun, which is different from the Sun rotating around its axis.

So by the astronomical definition of revolution, circle A only rotates around circle B once. Turn around once. Now, other definitions of revolution include the movement of an object that rotates around its own axis. So there is no definitive answer, but the wording of this question is extremely ambiguous if at least three different solutions can be justified. After reviewing the students' letters, the College Board publicly admitted its error a few weeks later and rescinded the question for all test takers. - They said they were ruling out the problem and were calling us because they were going to break the news and they thought they had to let us know that the news could contact us.

I did a lot of phone interviews and NBC News came to my school. They said he was right and discounted it. That was great. - But there is more to the explanation. - It is easy to find an intuitive reason, but it is very difficult to formally prove that the answer is four. I could give you some proof if you want. - Well, that would be wonderful. I think that would be the case, I'm sure we would appreciate it. - I have a blackboard because I'm a mathematician, so I happen to have a blackboard here. Wait.

Can you see that? - Yes. - It turns out that the amount of rotation of the small circle is always the same as the distance the center travels. Okay, so why is this true? Suppose you have a camera and the camera always points at the center. So in your movie, it seems like the center isn't moving. In the real world, the center revolves around the circle. Let's say it's going at a certain speed V. What is the speed of this point? It is zero and that is because it rolls without slipping. If it had any component in that direction, that would be a slip.

I mean, this is something I think they should have explained in detail in the problem, but when you change your frame of reference, the relative velocities don't change. In motion, the center always has zero velocity. So this point would have to have a negative velocity V. That means the speed at which it rotates is the same as the speed at which the center moves. So if they always have the same speed, they have to travel the same total distance. The total distance it rotates has to be equal to the total distance the center travels. In this problem, the center of the small circle surrounds a circle of radius four.

So the total distance the center travels is eight pi. What is the total amount that the small circle rotates? It rotates four times and its circumference is two pi. It is the same number. If it rolls without slipping, the total distance the center travels is equal to the total amount it rotates. - And this is always true. Take a circle that rolls without sliding on any surface from a polygon to a mass, outside or inside, the distance traveled by the center of the circle is equal to the amount the circle has rotated. So you simply find this distance and divide it by the circumference of the circle to get how many rotations it has made.

This is an even more general solution than our answer to the coin paradox where we simply take our expected answer, which we'll call N, and add one, and it reveals where this shortcut comes from. If a circle rolls continuously around a shape, the center of the circle circles the outside, increasing the distance traveled by exactly one circumference of the circle, so that the distance traveled by the center of the circle is just the perimeter of the shape plus the circumference of the circle. When we finally divide this by the circumference of the circle to get the total number of rotations, we get N plus one.

If a circle rolls continuously within a shape, the distance traveled by the center of the circle decreases by one circumference of the circle, making the total number of rotations N minus one. If the circle rolls along a plane line, the distance traveled through the center of the circle is equal to the length of the line, which, divided by the circumference of the circle, is exactly N. This general principle extends far beyond a fun math fact. In fact, it is essential in astronomy to measure time accurately. When we count 365 days that pass in a year, 365.24, to be precise, we say that we are simply counting how many rotations the Earth makes in an orbit around the Sun.

But it is not that simple. All this counting is done from the perspective of you on earth. To an outside observer, they will see that the Earth makes an additional 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, meaning sidereal with respect to the stars where an external observer would be. But what about that extra day? A normal solar day is the time it takes for the sun to return to being directly overhead you on Earth.

But the Earth not only rotates, it also orbits the Sun at the same time. So in a 24-hour solar day, the Earth actually has to rotate more than 360 degrees for the Sun to be directly overhead again. But the Earth's orbit is insignificant for distant stars. To see a star directly above us again, the Earth only needs to rotate exactly 360 degrees. So while it takes the sun exactly 24 hours to be directly overhead again, a star at night takes just 23 hours, 56 minutes and four seconds to be directly overhead you again. That is 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 will see them slowly diverge throughout the year. After six months, the sidereal day would be 12 hours ahead of the solar day, meaning noon would be midnight, and it would continue to advance until it was finally a full day ahead of the solar day, at which point a new year would begin. and orbit. starts. 365.24 days that have 24 hours each are equivalent to 366.24 days that have 23 hours, 56 minutes and four seconds each. Therefore, it makes no sense to use sidereal time on Earth, because six months later, day and night would be completely interchanged.

But equally, it's useless to use solar time while tracking objects in space, because the region you're observing would change between, say, 10:00 p.m. one night and 10:00 p.m. of the following night. So instead, astronomers use sidereal time in their telescopes to make sure they're looking at the same region of space each night. And all geostationary satellites, such as those used for communications or navigation, use sidereal time to keep their orbits synchronized 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 new 1982 SAT score wasn't all good news. By removing question 17, students' scores were scaled without it, moving their final result up or down by 10 points out of 800. Now, while that doesn't sound like much, some universities and scholarships use strict minimum score limits in the tests. And as one admissions expert put it: "There are cases, even if we don't consider them justified, where 10 points can have an impact on a person's educational opportunities. It may not keep someone out of law school, but it may affect which one I might go to." This mistake not only cost him points on the exam.

According to the testing service, "Rescoring would cost more than $100,000, money that came out of test takers' pockets. The question 17 circle problem was far from the last mistake in theSAT. But mistakes are probably the least of his worries these days. "I mean, the SAT is slowly becoming a thing of the past. After COVID-19, almost 80% of undergraduate colleges in the US no longer require any standardized tests. And that exam 1982, well, it didn't turn out that way either. bad for some. How did you do on your math SAT, if I may ask? - I got an 800. Even before that, it was clear that I was going to go into math.

I did math competitions. math. I really liked math. - Do you end up writing any math questions these days? - A while ago I wrote problems for a math competition. - And were you careful with the way you wrote them and the wording? - That's it. I hope. I tried - Today's deep dive into an SAT The question demonstrates that there is no substitute for hands-on exploration to understand and appreciate the everyday phenomena of our world. But there is no need to observe the Earth from space or make cardboard cutouts to practice mathematics, science and cutting-edge technology.

In fact, you can do it from anywhere right now with this video sponsor, Brilliant, and you can get started for free. Just go to shiny.org/veritasium. Brilliant will make you a better thinker and problem solver in everything from science to math, data programming, technology, you name it. Simply set your goal and Brilliant will design a path to get you there. If you liked today's video, I recommend you check out one of my favorite courses at Brilliant, which is Scientific Thinking. Scientific Thinking takes you on an interactive tour of our physical world. You'll engage with key scientific principles and theories, from simple machines like gears and pulleys to Einstein's special theory of relativity.

Whether you're comparing circuits to understand voltage and current, playing pool to learn the rules of collisions, or even planning your itinerary for an intergalactic music festival on a space-time diagram, you'll learn by doing and foster deep understanding without stagnate. in heavy mathematical formulas. They even have a version of the coin paradox. And in the end, this course will change the way you think about the world around you. Beyond science, Brilliant has a huge library of content in math, data science, programming, and technology, all with the same hands-on approach that makes concepts incredibly easy to understand. And the best part is that you can learn with Brilliant while you travel.

It's like having interactive versions of our videos in your pocket. To try everything Brilliant has to offer free for 30 days, visit shiny.org/veritasium. I'll put that link in the description. And the first 200 of you will get 20% off Brilliant's annual premium subscription. So I want to thank Bright for sponsoring this video and I want to thank you for watching it.