# The SAT Question Everyone Got Wrong

How an SAT question became a mathematical paradox. Head to https://brilliant.org/veritasium to start your free 30-day trial, and the first 200 people get 20% off an annual premium subscription.

I invented Snatoms, a molecule modeling kit where the atoms snap together magnetically. Try it at https://ve42.co/SnatomsV

Huge thanks to Dr. Doug Jungreis for taking the time to speak with us about this SAT question.

Thanks to Stellarium, a wonderful free astronomy simulator – https://ve42.co/Stellarium

Thanks to Newspapers.com, a database of historical newspapers – https://ve42.co/Newspapers

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References:

Summary of this problem by MindYourDecisions – https://www.youtube.com/watch?v=kN3AOMrnEUs

More cool math about this problem by Kyle Hill – https://www.youtube.com/watch?v=_zCDbB9wvrc

Murtagh, J. (2023). The SAT Problem That Everybody Got Wrong. Scientific American – https://ve42.co/SATSciAm

United Press International (1982). Error Found in S.A.T. Question. New York Times – https://ve42.co/SAT-NYT

Yang (2020). What’s the hardest SAT math problem that you’ve seen? Quora – https://ve42.co/SATQuora

Coin rotation paradox via Wikipedia – https://ve42.co/CoinParadox

Simmons, B. (2015). Circle revolutions rolling around another circle. MathStackExchange. – https://ve42.co/CircleRoll

Sidereal time via Wikipedia – https://ve42.co/SiderealWiki

Solar Time vs. Sidereal Time via Las Cumbres Observatory – https://ve42.co/SiderealLCO

Images & Video:

Zotti, G., et al. (2021). The Simulated Sky: Stellarium for Cultural Astronomy Research – https://ve42.co/Stellarium

Newspapers from 1980s – 1990s via Newspapers.com – https://ve42.co/Newspapers

SAT Practice Test via the College Board – https://ve42.co/PracticeSAT

Revolution Definition via NASA – https://ve42.co/RevolutionNASA

Revolution Definition via Merriam-Webster – https://ve42.co/RevolutionWebster

Earth motion animation via NASA – https://ve42.co/OrbitNASA

Satellite animation via NASA – https://ve42.co/SatNASA

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Directed by Emily Zhang

Written by Emily Zhang and Gregor Čavlović

Edited by Peter Nelson

Animated by Ivy Tello and Fabio Albertelli

Filmed by Derek Muller

Produced by Emily Zhang, Gregor Čavlović, and Derek Muller

Thumbnail by Ren Hurley

Additional video/photos supplied by Getty Images and Pond5

Music from Epidemic Sound

This was a mentally challenging video to watch first thing in the morning. I’m awake now

Wait, it is night

It is 10 PM where I live and now I can’t sleep😂

Bruh it’s 16:46 where I am

Got back from school and just did some homework now I’m eating snacks then I will play games

@@zayansaifullah2008 same

That part about the circle rotating around the triangle was mind-blowing. You instantly understand why it’s not the same if the circle rolls on a flat line or rolls on a curved line

That was the “aha” moment for me too.

This

There were 3 aha moments for me

if you divide the straight line in half and start to roll along it at the “top” to the end you then can make a 180, roll around to the “bottom” and then go in the other direction, make another 180 and keep going until you reach your starting point. These two 180 needed for the direction change add the 4th rotation 🤯

The earth around the sun was a fantastic example for why the frame of reference matters, especially with the graphic

It’s so impressive how you made this seemingly basic math question into a really interesting and well thought out video. I hadn’t even considered the idea of a Siderial day, it’s so cool!

Thou ne maketh a full point, anything of mathematics must be really interesting.

Agreed

No such thing as a sidereal day! A day is and has always been noon to noon! We made the clock based on a day, NOT the rotation of the Earth to the stars!

Having the small circle rotating 3 times with the camera rotating is the best intuitive explanation of what’s going on I’ve ever seen for something like this

Great video. I have been working for months on my own updated video with the topic of sidereal days, so those interested may find a few surprises not covered here. It’s a wonderful topic and makes me realize how little astronomy we learn.

I really felt the absence of “Hey, this is Presh Talwalkar here!” at the start of this video

Ic thanke thee.

Doing some math with sidereal days and orbits of moons are what made me figure all of this out as well. Was figuring out phases of the moon with sine waves and stuff and noticed that indeed, if you tell that the moon does 12 orbits in 1 year, you actually end up with 11 full moons per year because it needs to orbit a bit more than a full orbit to end up at full moon again. This is what made me realize that the number of full rotations in an orbit (which the SAT question effectively is asking about) is the amount of “days” + 1. So ‘lo an behold how confused I was when the SAT question didn’t have a 4 in the answers lol

For me this is was quite a great watch. Funnily enough I pointed out this same problem in the solution of one of my Professors when we did Parametrization of curves, I now instantly recognized the problem, but when I tried figuring out what was wrong with their solution it took me a while to come up with a reason of why that would be the case. As always incredible video, I really appreciate the fact that you introduce more people to math and science problems with great and easy to understand explanations!

I loved the “I hope so” answer from Doug at the end. It highlights the most important lesson I learned during my education: “I might be wrong.”

I feel like I already had that lesson before education. I feel like the most important lesson for me – that helped me grapple with how to be effectively wrong – is how to think in terms of probability than binaries.

A harder lesson still is, “I might be wrong and I’ll never know it.” This is why people who fear the Scientific Method really shouldn’t. It’s also a primer in the Scientific Method, perfectly demonstrating why the goal isn’t to prove a hypothesis is correct. Rather, the goal is to prove a hypothesis is NOT correct. Similarly, it demonstrates why the strongest theories are those derived from inductive reasoning (multiple specific cases lead to a generalized conclusion), rather than deductive reasoning (a generalized case leads to multiple specific conclusions).

Agreed! The most important thing I learned when learning math or physics or any objective knowledge is that by admitting the probability your are wrong is the best you can do to advance in those fields. I love to think that the physics, as we human know and define it, is always more correct than before but never (at least in the foreseeable future) completely right.

I always thought this way, but I learned in the working world that if you acknowledge that you could be wrong other people will assume you’re wrong.

This was a lot more interesting than I initially expected. Great explanation and visuals that made it easy to understand all of the facets of the paradox. Kudos!

Another way to see it is: imagine (or try at home) a circle turning around an infinitesimal (aka really small) point. It will need a full rotation + the perimeter of the infintisemal point, roughly, one roation in total. So, just the fact of completing a close circuit, requires one full rotation, and then add the distance of the perimeter of the object that has been circled around.

I sent an email to you about prime numbers. Check the spam bin, just in case…

Since I took Astronomy classes after my math classes, the first thing I thought was “the answer is technically one, but that seems too easy”. It reminds me of the sentiment of how we define words. We use words and concepts that seem to have no way to define things themselves. Rather that perspective and experience are a part of every definition and inform our ability to draw conclusions.

You are technically correct, which is the best type of correct. I do no envy those that have to create word problems as even with multiple editors, it can be easy to miss out on an ‘obvious’ alternative answer. One of my first jobs out of high school was working for a paint retailer, and we had to do Paint Pro University as part of our training. The company had been asking the same questions for 50+ years and I pointed out that one of them was wrong. It was a True/False question that stated “The correct way to store a paint brush is to hang it up.” I chose ‘False’ as the correct way to store a brush is to hang it down. It was the only question I got wrong, and I challenged my boss, and ultimately the HR department on it. The question was changed a couple months later to “The correct way to store a paint brush is to hang it.” That I would have said ‘True’ to.