A Miscalculation on an SAT Question In 1982, a SAT problem involving a small circle rolling around a larger one prompted the expected answer of three rotations, based on a simple ratio of circumferences. However, the problem concealed an extra twist—an additional rotation due to the circular path—which none of the provided options captured. A few students recognized this discrepancy and challenged the test makers. Their insistence revealed that the exam’s error, overlooked even by its creators, invalidated the standard answer.
The Coin Rotation Paradox Unveiled A fundamental demonstration involves a coin rolling around an identical coin, where intuition might expect a single revolution but in fact yields two complete rotations. The extra turn arises because the coin’s journey along a curved path adds an additional spin. This intriguing effect, known as the coin rotation paradox, lays the foundation for understanding the unexpected result. It highlights how circular motion can defy straightforward reasoning.
Extra Rotation Through Path Length Calculation The underlying principle is that a rolling circle’s total rotations equal the distance its center travels divided by its own circumference. When the circle rolls externally around another, the circular path contributes one extra revolution beyond the simple ratio computation. This result emerges because the path’s curvature effectively adds the circle’s own circumference to the journey. The phenomenon is a striking reminder that geometric motion can yield counterintuitive outcomes.
Extending the Principle to Celestial Mechanics The same extra rotation concept applies to planetary motions and timekeeping in astronomy. Although Earth appears to rotate 365.24 times in a solar day count, external observation reveals an extra rotation over the course of an orbit, explaining the 366.24 sidereal days. This subtle difference is crucial for accurately aligning telescopes and synchronizing satellite orbits. The coin paradox thus provides a tangible insight into how rotation and revolution are interrelated in the cosmos.
Impact on Testing and Lasting Educational Legacy The SAT error forced a rescoring that altered thousands of students’ scores, highlighting how a miscalculation in a single question can have far-reaching academic consequences. The controversy underscored the importance of precise wording and deep mathematical insight in high-stakes assessments. It also marked a turning point in discussions about standardized testing reliability and fairness. The incident remains a powerful example of how a mathematical curiosity can influence educational practices and policies.