The SAT Question Everyone Got Wrong

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1982
# sat
# mathematics

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Overview

This video explores a flawed SAT math question from 1982 that everyone got wrong. The question involved calculating the number of rotations a smaller circle makes while rolling around a larger circle. The video explains the coin rotation paradox, where a coin rotated around an identical coin makes two rotations, not one as intuitively expected. It then applies this principle to the SAT question, demonstrating that the correct answer wasn't listed. The video delves into the perspectives of the circles themselves and external observers, highlighting the difference between rotations and revolutions. It further connects this concept to astronomy, explaining the difference between solar and sidereal time and their implications for timekeeping on Earth and tracking objects in space.

The Flawed SAT Question
- ❓
  In 1982, an SAT math question stumped every single test-taker.
- 🧮
  The question presented a scenario with a smaller circle rolling around a larger circle, asking for the number of rotations the smaller circle completes.
- ❌
  The intended answer was three, but the test-writers themselves made a mistake.
- ✍️
  Three students wrote to the College Board, pointing out the error and providing their reasoning.
The Coin Rotation Paradox
- 🪙
  The video illustrates the problem using a simpler example: two identical coins.
- 🔄
  Contrary to intuition, a coin rolling around an identical coin makes two full rotations, not one.
- 🤔
  This counterintuitive phenomenon is known as the coin rotation paradox.
Applying the Paradox to the SAT Question
- 📏
  A to-scale model demonstrates that the smaller circle in the SAT question actually makes four rotations.
- 💡
  The video explains that the circular path itself contributes to an extra rotation.
- 🧮
  A general solution is presented: find the ratio of the circles' circumferences and add one rotation.
Perspective and the Number of Rotations
- 👁️
  The video explores how the perceived number of rotations changes depending on the observer's perspective.
- 🔄
  From the perspective of the larger circle, the smaller circle rotates three times.
- 🌎
  This perspective shift is analogous to turning the larger circle's circumference into a straight line.
Beyond the Paradox: Astronomy and Timekeeping
- 🌌
  The video connects the coin rotation paradox to astronomy and the concept of revolutions.
- 🌎
  It explains the difference between a solar day (time for the Sun to return to the same position in the sky) and a sidereal day (time for a star to return to the same position).
- ⏱️
  The difference arises from Earth's simultaneous rotation and orbit around the Sun.

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Key moments

Introduction: The SAT Question Everyone Got Wrong
Introduction of the infamous 1982 SAT math question.
Invitation for viewers to pause and solve the problem.
Reveal that the intended answer is incorrect.
The Flaw and Its Impact
Explanation that the test makers made a mistake in the question.
The significance of the SAT and the pressure it put on students.
Three students identified the error and contacted the College Board.
Understanding the Coin Rotation Paradox
Introduction of the coin rotation paradox using a simpler example.
Demonstration of the paradox with two coins.
Applying the paradox to the SAT question with a to-scale model.
Unveiling the Geometry
Visual explanation of the paradox using a ribbon and a straight line.
How the circular path contributes to an extra rotation.
General solution: Ratio of circumferences plus one rotation.
Perspective Matters
Illustrating how the answer changes when viewed from the perspective of the larger circle.
Both perspectives are valid, highlighting the ambiguity of the question.
Introducing the astronomical definition of "revolution."
Astronomical Implications
Distinction between "rotation" and "revolution" in astronomy.
Earth's rotation and revolution around the sun.
Ambiguity of the question allows for multiple interpretations.
Aftermath and Further Errors
The College Board acknowledges the error and rescores the exam.
Doug Jungreis, one of the students who spotted the error, shares his experience.
Doug provides a mathematical proof for the correct answer.
Beyond a Mathematical Curiosity
The principle applies to a circle rolling on any surface.
Calculating rotations based on distance traveled and circumference.
General formula for rotations: N + 1, N - 1, or N depending on the path.
Timekeeping and the Sidereal Day
The paradox's relevance to astronomy and timekeeping.
Difference between a solar year and a sidereal year.
Explanation of solar and sidereal days.
The discrepancy between solar and sidereal time accumulates to a full day over a year.
Practical Applications and Conclusion
Why we use solar time on Earth and sidereal time in astronomy.
Geostationary satellites rely on sidereal time.
The coin rotation paradox highlights the importance of perspective in understanding the universe.
The Legacy of the Error
Rescoring the SAT impacted students' scores and potential opportunities.
The financial cost of the mistake.
The 1982 error was not an isolated incident.
The Changing Landscape of Standardized Testing
The declining importance of the SAT in college admissions.
Doug Jungreis's perfect score on the math SAT.
Conclusion and Sponsor Message
Encouragement to engage in hands-on learning.
Promotion of Brilliant, a sponsor that offers interactive learning experiences.