SAT question becomes a mathematical paradox and everyone gets it wrong

TLDR

The video discusses a notorious SAT math problem from 1982, where all test-takers incorrectly answered a question about the revolutions of one circle around another. The test makers themselves had the wrong solution, assuming three revolutions, while the correct answer was not among the provided options. Three students recognized and reported this error, leading the College Board to acknowledge the mistake and nullify the question. The explanation involves the coin rotation paradox, where a circle rotates one more time than expected when rolling around another circle. This principle has broader applications, such as in astronomical timekeeping. The incident had minor implications for students’ scores and highlighted potential issues with standardized testing.

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This article is a summary of a YouTube video “The SAT Question Everyone Got Wrong” by Veritasium

Key insights

Incorrect SAT Question: In 1982, an SAT math question asked about the number of revolutions one circle makes around another. Despite all students choosing one of the provided options, none of these were correct.
Coin Rotation Paradox: The video explains the coin rotation paradox, which reveals that a circle rolling around another completes one more rotation than intuitively expected. This principle was key to understanding why the SAT question’s assumed answer was incorrect.
Correct Answer Not Provided: The actual correct answer to the SAT question was not among the options given. The test creators, like most students, incorrectly assumed the answer was three revolutions.
Student Advocacy: Only three out of 300,000 test-takers recognized the error and wrote to the College Board, leading to the acknowledgment of the mistake.
Impact on Test Scores: The College Board eventually nullified the question, slightly adjusting the test scores of all students. This decision had minor but significant implications for students’ college admissions and scholarship opportunities.
Implications for Standardized Testing: This incident highlighted potential flaws in standardized testing and raised questions about the reliability and accuracy of such exams.
Application in Astronomy: The principle illustrated by the SAT question has broader applications, such as in astronomical timekeeping, differentiating between solar and sidereal days.
Changing Views on SATs: The video concludes by noting the decreasing relevance of SATs in college admissions, especially post-COVID-19, with many universities no longer requiring standardized tests.

Timestamped Summary

0:00-0:22: Introduction to a unique SAT problem from 1982, involving the revolutions of one circle around another.
0:22-0:58: Description of the SAT question, which asked about the number of revolutions circle A makes around circle B.
0:58-1:37: Explanation that the correct answer was not among the given options, with most, including the test makers, incorrectly assuming it was three.
1:37-2:16: Only three students out of 300,000 identified the mistake and reported it to the College Board.
2:16-3:15: Students’ argument based on the coin rotation paradox, demonstrating the actual answer should have been four, not three.
3:15-4:47: Visual demonstration with a scale model, showing circle A rotates four times around circle B.
4:47-6:00: Exploration of different perspectives: from circle B’s viewpoint, circle A appears to rotate only three times.
6:00-7:26: Discussion of the astronomical definition of a revolution, suggesting another possible answer could be one revolution.
7:26-7:46: The College Board’s admission of the error and decision to nullify the question.
7:46-10:33: Discussion of the broader implications of the question, including its relevance to astronomical timekeeping.
10:33-15:38: Reflection on the minimal but significant impact of the SAT question on students’ college opportunities and the overall reliability of standardized testing.
15:38-End: Concluding thoughts on the decreasing relevance of the SAT in college admissions and the broader implications for educational testing.

Q&A

What was the notorious SAT question in 1982 that everyone got wrong?

The question involved calculating the number of revolutions made by a smaller circle (Circle A) rolling around a larger circle (Circle B), where Circle A’s radius was one-third that of Circle B.

Why did everyone, including the test makers, get the SAT question wrong?

Most people, including the test makers, incorrectly assumed that the answer was simply based on the ratio of the circumferences of the two circles. They overlooked the additional rotation caused by the circular path.

What is the correct answer to the 1982 SAT circle problem and why?

The correct answer is that Circle A makes four complete rotations. This accounts for the three rotations based on circumference ratio and one additional rotation due to the circular path it follows around Circle B.

How did the error in the SAT question come to light, and what was the response?

Only three students out of 300,000 test takers identified the error and wrote to the College Board. The College Board later admitted the mistake and nullified the question for all test takers.

What are the broader implications or learnings from this SAT question error?

This incident highlights the importance of precise problem formulation and solution verification in mathematics. It also sheds light on the nature of standardized testing and the potential for human error in test design.

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