The coin rotation paradox is the

counter-intuitive A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictor ...

mathematical fact that, when one coin is rolled without slipping around the rim of another coin of equal size, the moving coin completes not one but two

full rotation The turn (symbol tr or pla) is a unit of plane angle measurement that is the measure of a complete angle—the angle subtended by a complete circle at its center. One turn is equal to radians, 360 degrees or 400 gradians. As ...

s after going all the way around the stationary coin, when viewed from an external reference frame. The problem can be further generalized to coins of different radii.

Description

Start with two identical

coin A coin is a small object, usually round and flat, used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in large quantities at a mint in order to facilitate trade. They are most often issued by ...

s touching each other on a table, with their "head" sides displayed and parallel. Keeping coin A stationary, rotate coin B around A, keeping a point of contact with no slippage. As coin B reaches the opposite side, the two heads will again be parallel; B has made one revolution. Continuing to move B brings it back to the starting position and completes a second revolution. Paradoxically, coin B appears to have rolled a distance equal to twice its circumference. In reality, as the circumferences of both coins are equal, by definition coin B has only rolled a distance equal to its own circumference. The second rotation arises from the fact that the path along which it has rolled is a circle. This is analogous to simply rotating coin B "in situ". One way to visualize the effect is to imagine the circumference of coin A "flattened out" into a straight line, by which means it can be observed that coin B has rotated only once as it travels along its, now flat, path. This is the "first rotation". Equally, sliding coin B around the circumference of coin A, instead of rolling it, whilst maintaining its current specific point of contact, will impart a rotation representative of the "second rotation" in the original scenario. As coin B rotates, each point on its perimeter describes (moves through) a

cardioid In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal ...

curve.

Analysis and solution

From start to end, the center of the moving coin travels a circular path. The circumference of the stationary coin and the path of the centre form two concentric circles. The radius of the outer circle is the sum of the coins' radii; hence, the circumference of the path of the moving centre is twice either coin's circumference. The center of the moving coin travels twice the coin's circumference without slipping; therefore, the moving coin makes two complete revolutions. How much the moving coin rotates around its own center en route, if any, or in what direction – clockwise, counterclockwise, or some of both – has no effect on the length of the path. That the coin rotates twice as described above and focusing on the edge of the moving coin as it touches the stationary coin are distractions.

Unequal radii and other shapes

A coin of radius ''r'' rolling around one of radius ''R'' makes ' + 1 rotations. That is because the center of the rolling coin travels a circular path with a radius (or circumference) of ' = ' + 1 times its own radius (or circumference). In the limiting case when ''R'' = 0, the coin with radius ''r'' makes + 1 = 1 simple rotation around its bottom point. The shape around which the coin is rolled need not be a circle: one extra rotation is added to the ratio of their perimeters when it is any

simple polygon In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a Piecewise linear curve, piecewise-linear Jordan curve consisting of finitely many line segments. The ...

or closed curve which does not intersect itself. If the shape is

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

, the number of rotations added (or subtracted, if the coin rolls inside the curve) is the absolute value of its

turning number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that the curve travels counterclockwise around the point, i.e., the curve's number o ...

Applications

Sidereal time

The paradox is related to

sidereal time Sidereal time ("sidereal" pronounced ) is a system of timekeeping used especially by astronomers. Using sidereal time and the celestial coordinate system, it is easy to locate the positions of celestial objects in the night sky. Sidereal t ...

: a

sidereal day Sidereal time ("sidereal" pronounced ) is a system of timekeeping used especially by astronomers. Using sidereal time and the celestial coordinate system, it is easy to locate the positions of celestial objects in the night sky. Sidereal t ...

is the time Earth takes to rotate for a distant star to return to the same position in the sky, whereas a

solar day A synodic day (or synodic rotation period or solar day) is the period for a celestial object to rotate once in relation to the star it is orbiting, and is the basis of solar time. The synodic day is distinguished from the sidereal day, which is ...

is the time for the sun to return to the same position. A year has around 365.25 solar days, but 366.25 sidereal days to account for one revolution around the sun. As a solar day has 24 hours, a sidereal day has around × 24 hours = 23 hours, 56 minutes and 4.1 seconds.

Group theory

A version of the puzzle arises in

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, specifically the study of the

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

known as the split real form of G₂. One construction of this group uses the fact that a ball rolling around another ball with three times its radius will make four full turns, rather than three.

1982 SAT scoring error

In May 1, 1982, one of the US college admissions tests, the

SAT The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and Test score, scoring have changed several times. For much of its history, it was called the Scholastic Aptitude Test ...

, had a multiple choice question concerning the coin rotation problem. The test had to be regraded after three students proved there was no correct answer among the choices provided.

References

External links

* ::This upvoted answer includes animations and intuitive explanations about the original question where r of the "outer coin" was 1/3 of the inner coin's radius. * {{cite AV media , last=Muller , first=Derek , date=2023-11-30 , title=The SAT Question Everyone Got Wrong , url=https://www.youtube.com/watch?v=FUHkTs-Ipfg , via=

YouTube YouTube is an American social media and online video sharing platform owned by Google. YouTube was founded on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim who were three former employees of PayPal. Headquartered in ...

, publisher=

Veritasium Derek Alexander Muller (born 9 November 1982) is a Science communication, science communicator and media personality, best known for his YouTube channel Veritasium, which has over 17.8 million subscribers and 3.3 billion views as of April 2025. ...

Mathematical paradoxes Rotation Circles