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The coin paradox considers a circle of radius 
that rolls without slipping around the
outside of a fixed circle of radius 
. During one circuit, the center
of the rolling circle traverses a circumference
of length 
. Therefore, an observer in the plane
sees
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complete rotations. Rolling along the fixed circle accounts for 
turns relative to the radius joining the two centers.
The direction of this radius itself makes one additional
turn.
In the equal-radius case 
, the rolling circle
consequently makes two complete rotations. A marked
point on it traces the cardioid that is the equal-radius
case of an epicycloid.
Question 17 on the May 1982 SAT used 
and expected the answer 3. The correct count is instead
, which was absent from the five choices. Three of
the approximately 300,000 students who took the test reported the error, and the
examinations were rescored (Murtagh 2023, Veritasium 2023).
