2 Turns, Not 1: The Coin Paradox That Broke the SAT

2 Turns, Not 1: The Coin Paradox That Broke the SAT Take two identical coins. Roll one all the way around the other and count the turns. You'll bet on one full turn. The real answer is two — and the missing turn hides in plain sight. This is the coin rotation paradox. Since the two coins have the same circumference, intuition screams that one lap equals one rotation. But watch the little arrow on the moving coin: it comes back around having spun twice. One turn comes from rolling along the edge. The second, invisible turn comes from the curved path itself — the center of the moving coin traces a circle of twice the radius, so it travels two full circumferences without slipping. We rebuild the intuition from scratch: flatten the path into a straight line, roll a coin around a square and count four ninety-degree pivots, then smooth the corners into a circle. The pattern snaps into place — a coin rolling around a shape n times its size turns n plus one times. Then the twist with real stakes: in May 1982, this exact problem appeared on the SAT. The test writers expected the answer 3. The true answer was 4. It wasn't even one of the choices. Three students spotted it, and roughly 300,000 exams had to be rescored. Sources: Scientific American, "The SAT Problem That Everybody Got Wrong"; MIT 8.01 rigid-body rotation notes; Wolfram MathWorld, "Coin Paradox"; arXiv 2512.00123 on roulette curves and Aristotle's wheel paradox; GraphicMaths, "Coin rotation paradox." ⏱ Chapters 0:00 The Rolling Coin 0:54 The Paradox Emerges 2:20 Geometry's Extra Turn 2:47 The SAT Problem 3:16 Four Turns Solution ▶ Visualize more mathematics here:    • Origin Math — Visual Proofs, Probability &...   🔔 Subscribe to watch math come alive, one proof at a time. © These animations and narration are original to Origin Math — re-upload or copying is not allowed. #coinrotationparadox #mathparadox #rollingcircle #visualmath #1982SATproblem #SATcirclequestion #epicycloid #rotationsvsrevolutions