Answers

The main problem was to describe the track that allows a square wheel to roll along it so that its center travels a straight horizontal line. The track is a series of catenary arcs. This applies to all wheels that are regular polygons. (IF a wheel is an irregular convex polygon, the track must have arcs that are differently shaped catenaries, one for each side of the wheel.) If the wheel turns with a constant speed, its horizontal speed will vary. For details of the proof I must refer readers to "Rockers and Rollers," by Gerson B. Robison, in Mathematics Magazine for January, 1960, pages 139-144, and the solution to Problem £1668 in The American Mathematical Monthly for January, 1965, pages 82-83.

The riddle's answer is a pair of roller skates.

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