Wikipedia. I judge the article as poorly written, especially when it says the paradox is about two wheels. A comment on the Talk page agrees. The quote from Mechanica, written more than 2,000 years ago, describes the paradox as about two circles. Two circles can depict one wheel, e.g. like on a car or truck, with the smaller circle depicting a metal rim. Or the two circles can depict one tire -- not mounted on a rim -- with the smaller circle depicting its smallest circumference, the bead or lip. They can depict a roll of tape.
If the rigidly coupled circles are rolled a full revolution, then all points on both circles have the same position relative to their common center at start and end. Every point's translation vector has the same direction -- parallel to the horizontal surface -- and length as the center's translation vector. Such length is 2*pi*R, where R is the radius of the larger circle. This necessary fact about translation elegantly solves the paradox. Every point on the smaller circle must move 2*pi*R. This shows that the smaller circle's circumference 2*pi*k*R, where k is its circumference divided by R, is irrelevant for one rotation and the given setup. How far the smaller circle moves horizontally is dictated by its center.
The Wikipedia article is better written now. I edited it to remove any mention of two wheels and added my solutions in a new Analysis and Solutions section. Since others can edit Wikipedia articles at any time, I cross my fingers that somebody won't butcher it. I added the same text to the Talk Page. I think others can alter that too, but perhaps they are less likely to do so.
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