The SAT Question Everyone Got Wrong

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Published 2023-11-30
How an SAT question became a mathematical paradox. Head to 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 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 – ve42.co/Stellarium
Thanks to Newspapers.com, a database of historical newspapers – ve42.co/Newspapers

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References:
Summary of this problem by MindYourDecisions –    • Why did everyone miss this SAT Math q...  
More cool math about this problem by Kyle Hill –    • The SAT Question NO ONE Got Right  
Discussion of a solar day by MinutePhysics –    • Why December Has The Longest Days  
Murtagh, J. (2023). The SAT Problem That Everybody Got Wrong. Scientific American – ve42.co/SATSciAm
United Press International (1982). Error Found in S.A.T. Question. New York Times – ve42.co/SAT-NYT
Yang (2020). What's the hardest SAT math problem that you've seen? Quora – ve42.co/SATQuora
Coin rotation paradox via Wikipedia – ve42.co/CoinParadox
Simmons, B. (2015). Circle revolutions rolling around another circle. MathStackExchange. – ve42.co/CircleRoll
Sidereal time via Wikipedia – ve42.co/SiderealWiki
Solar Time vs. Sidereal Time via Las Cumbres Observatory – ve42.co/SiderealLCO

Images & Video:
Zotti, G., et al. (2021). The Simulated Sky: Stellarium for Cultural Astronomy Research - ve42.co/Stellarium
Newspapers from 1980s - 1990s via Newspapers.com – ve42.co/Newspapers
SAT Practice Test via the College Board – ve42.co/PracticeSAT
Revolution Definition via NASA – ve42.co/RevolutionNASA
Revolution Definition via Merriam-Webster – ve42.co/RevolutionWebster
Earth motion animation via NASA – ve42.co/OrbitNASA
Satellite animation via NASA – 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, Han Evans, Gregor Čavlović, and Derek Muller

Thumbnail by Ren Hurley
Additional video/photos supplied by Getty Images and Pond5
Music from Epidemic Sound

All Comments (21)
  • @5MadMovieMakers
    This was a mentally challenging video to watch first thing in the morning. I'm awake now
  • @peter9477
    My brain didn't fully accept this until I pictured a circle going "around" a straight line segment in the same manner. Picture a horizontal line segment, circle positioned above it at the left end, bottom (not right or left side) of circle touching the end of the line segment. The circle travels to the right along the length of the line. Then to flip itself around the right tip of the line to the bottom side it has to undergo a 180 degree turn, but while doing so it travels no additional distance along the line. (Its centre travels a distance along a semicircle, but the part touching the tip of the line does not.) Then back along the bottom of the line to the left, then another 180 degree rotation back around the left tip, to the top again. Total distance traveled is just twice the length of the line. Number of rotations is some amount to accomplish that traveling, PLUS one additional complete rotation. Same thing for any convex shape that it travels completely around.
  • @user-kb6mj7zq8t
    What is so interesting about your videos is that almost 100% of the I couldn't care less about the topic. Yet, I'm still enthralled through the whole thing. That is most definitely a compliment just to be clear. I love that you love to teach. That's all that matters.
  • @tc6818
    10:44 The circle traveling on the outside of the triangle helped me visualize the solution best.
  • This is great! I want to point out that the more general solution doesn't seem to apply when the shape is not convex. I need to explore this more deeply and check the literature for details and edge cases.

    To see why convexity matters, take the example of a small circle (circumference c) rolling around a large circle (circumference C), and cut out a segment of the larger circle that the smaller one can fit through, leaving a circular arc of length of C-c so that it rolls around the outside and then the inside before returning to its starting place. The smaller center then travels (C-c+c)=C for going around the outside + (C-c-c)=C-2c for going around the inside and c/2+c/2=c for moving between inside and outside 2 times, so a total of 2C+2c-c= 2C + c. The perimeter of the shape is 2(C-c)= 2C-2c, so the difference isn't c, but 3c, meaning the smaller circle revolved N+3 times if C=N•c.

    N+3 isn't a general rule for a circle rolling along nonconvex shapes, just for this example.

    This will characterize nonconvexity in an interesting way that seems like it must appear in the literature but I don't remember a name for it. Reminds me of the Euler characteristic, but this is not exactly a topological property.
  • @ElectroBOOM
    This was a great video! Blew my mind when I realized how I was wrong!! Good to know question wordings can be so important, eh?! 😁😉
  • @WankiTank
    I really love your channel. I'm recently "healing" my relationship with math and logic problems, as I realized I always had issues with remembering numbers for calculations but some recent tests showed that I am not actually that bad with (number-based) logic problems itself, making me enjoy this type of content without panicking I might be stupid because I don't get something right away. the fact that you actually show it haptically and not just by throwing numbers on a board, is just soooo wonderful and non-threatening as compared to my experiences in school :')
  • I figured it had to do with the discrepancy between the center of the circle (what the question was asking about) and the edge of the circle (on which all the solving strategies were relying). It really clicked for me when the guy showed his first proof which formally described exactly that effect far more clearly than I could have. The graphics right after that proof were also great for understanding the generalization.
  • Your way to develop content for your video from a question on a quiz is impressive. Thank u so much XD
  • What a beautiful problem, knowing in advance the question is vague but the explanation made it so interesting to watch & uncover alienated topics too (physics too).
  • @darrylpioch2055
    My brain just melted in a shower of understanding.

    If anyone besides veritasium explained this I’d have been lost lol. Veritasium is a legend
  • @KevinJDildonik
    To all the 1st posters: YouTube takes up to 15 minutes to gather data on a video before showing stats. Everyone in the first 15 minutes all think they're first.
  • @Radon-220
    Interesting yet simple, as a person who uses logic rather than actually doing the question to find the answer.
    This video was fun to watch .
  • Love this channel!!
    Considering the appalling quality of education in general, and it's quite obvious when reading comments in various social media platforms, I think that "Brilliant" should be completely free and not just for a thirty day trial. This world needs all the help it can get to raise the level of intelligence and not just knowledge.
  • @skindude9251
    Fantastic! Only just caught this channel and love the fascinating subjects it brings up and the beautifully clear explanation.
  • @kiwiyss9969
    The centre point of the turning outer circle travels the perimeter of an unseen circle that has the circumfrence of r1 + r2. Thus in the first qurstion the circle should revolve 4 times around itself.
  • 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 think this is the most interesting video to date on this channel... well done!