The SAT Question Everyone Got Wrong

Summary

This video recounts the story of a 1982 SAT question that baffled students and even the test creators. The problem involved a rolling circle, and all provided answers were incorrect. Three students challenged the College Board, proving the question was flawed. The video explains the mathematical concept behind the 'coin rotation paradox' and its connection to real-world phenomena like timekeeping in astronomy. The College Board eventually nullified the question, though the mistake impacted student scores. The video also highlights hands-on learning through real-world exploration and promotes learning with Brilliant.org.

Takeaways

• 🔄 In 1982, an SAT math question stumped every student because the correct answer wasn’t listed among the options.
• 🔍 The question involved two circles: circle A with 1/3 the radius of circle B, asking how many revolutions it would take for A to roll around B and return to its starting point.
• ❌ The intended correct answer was three revolutions, but this was wrong. None of the provided options were correct due to a mistake by the test writers.
• 🌀 The correct solution is four revolutions due to the coin rotation paradox, where a circle rotates one extra time when rolling around another circle.
• 🎓 Three students wrote to the College Board pointing out the mistake, leading to the question being nullified for all test takers.
• ⚖️ Rolling a circle without slipping means the distance traveled by its center is equal to the number of rotations, which explains why it took four full rotations in the problem.
• 📏 A general rule is that the number of rotations a circle makes when rolling around another is the ratio of their circumferences plus one.
• 🔭 The concept also applies to astronomy, where Earth’s rotations differ based on whether time is measured relative to the sun or the stars (sidereal time).
• 📉 Nullifying the question affected students' scores by 10 points, which could have impacted college admissions for some.
• 🎥 The script concludes by highlighting the importance of hands-on exploration in learning, with Brilliant offering interactive lessons in science and math.

Q & A

What was the key error in the SAT question from 1982?

The SAT question involved a circle rolling around another circle, and the test writers believed the answer was three full revolutions. However, the correct answer was four revolutions, which wasn't listed as an option. The mistake stemmed from not accounting for the extra rotation caused by the circular path.

Why did all students get the SAT question wrong?

All students got the question wrong because the SAT writers made a mistake in the answer choices. They didn't account for the extra rotation that occurs when a circle rolls around another circle, leading to no correct option being listed.

What concept does the 'coin rotation paradox' demonstrate?

The 'coin rotation paradox' shows that when one circle (or coin) rolls around another circle of the same size, it rotates twice instead of once, even though their circumferences are identical. This paradox highlights the additional rotation caused by the circular path.

How did the three students prove the SAT question was wrong?

The students demonstrated that none of the listed answers were correct by explaining the additional rotation that occurs when a circle rolls around another. They confidently pointed out the test writers' error and provided proof that the answer should have been four.

What general principle explains the number of rotations of a rolling circle?

The general principle is that the distance traveled by the center of a rolling circle is equal to the total number of rotations it makes. This principle holds whether the circle rolls around another shape or along a flat line.

What impact did the rescoring of the SAT question have on test takers?

The rescoring of the SAT question moved students' scores up or down by 10 points out of 800. While this doesn't seem significant, it could have impacted university admissions or scholarships for some students who were on the edge of cutoff scores.

Why is the wording of the SAT question considered ambiguous?

The wording is ambiguous because the question asks how many 'revolutions' the circle makes. In astronomy, a revolution refers to an orbit, which would result in one revolution. However, other definitions include rotations about the circle's own axis, leading to different interpretations.

How does the coin paradox relate to astronomy and timekeeping?

The coin paradox helps explain the difference between solar and sidereal time. While a solar day accounts for Earth's rotation and orbit around the Sun, a sidereal day only accounts for Earth's rotation. This distinction is crucial in astronomy, where accurate timekeeping is needed for observing celestial objects.

What is the difference between a solar day and a sidereal day?

A solar day is the time it takes for the Sun to be directly overhead again, which is about 24 hours. A sidereal day, used in astronomy, measures the time it takes for a distant star to be overhead again, which is slightly shorter at 23 hours, 56 minutes, and 4 seconds.

What is the importance of sidereal time in astronomy?

Sidereal time is important in astronomy because it ensures that telescopes are aligned with the same region of space each night. This timekeeping method accounts for Earth's rotation without factoring in its orbit around the Sun, providing more consistent observations of stars and celestial objects.