Coin Rotation Paradox

The coin rotation paradox is a famous math problem with an unintuitive solution:

If you roll a coin around the edge of another coin of the same size, from an external perspective, how many rotations does the coin make by the time it returns to its original position?

Spoiler warning: solution is below, don’t scroll down if you’re not ready to view the solution

Solution to the Coin Rotation Paradox

Two full rotations.

Mathematical Explanation

The intuitive answer you might have guessed is one rotation. After all, the circumferences of the two coins are the same, so if you roll one coin around the other without slipping, it should have only travelled the distance of one of the circumferences.

However, that considers the point of contact between the two coins. If we want to analyze how many times the outer coin rotates around its own center, we must instead consider the center of that coin. The distance from the center of the inner coin to the center of the outer coin is twice the radius of either coin, meaning the circle that the outer coin’s center traces has double the circumference of either coin. Since the center of this outer coins travels double the distance, it make two full rotations.

Coin rotation paradox visual demonstration
Source: cmglee, Mark Morgan, 636Buster, CC BY-SA 4.0, via Wikimedia Commons

Intuitive Explanation

Intuitively, you can think of it this way: if the outer coin rolled along a straight line the length of its circumference, it would indeed make one rotation. But since it’s rolling around another coin, the revolution around the inner coin “contributes” one extra rotation.

Coin Rotation Paradox on the SATs

In fact, the 1982 SAT featured a similar question on the test, and all provided answers were incorrect!

The SAT question showed two circles: a small circle (A) that was to be rotated around a large circle (B).

The question asked: “In the figure above, the radius of circle A is one-third the radius of circle B. Starting from position shown in figure, circle A rolls around circle B. At the end of how many revolutions of circle A will the center of circle A first reach its starting point?” (Source: New York Times, May 25, 1982)

There were five answer choices: (a) 3 over 2. (b) 3. (c) 6. (d) 9 over 2. (e) 9.

But none of these choices were correct. The test-makers intended (b) 3 to be the correct answer, but in reality, the correct answer is 4. This is because, as explained above, the rotation around an inner circle contributes one extra rotation.

Leave a Reply

Your email address will not be published. Required fields are marked *