A memorably phrased puzzle from The Graham Dial: “Consider a vertical girl whose waist is circular, not smooth, and temporarily at rest. Around the waist rotates a hula hoop of twice its diameter. Show that after one revolution of the hoop, the point originally in contact with the girl has traveled a distance equal to the perimeter of a square circumscribing the girl’s waist.”
Since the ratio of the diameters is 2:1, so is the ratio of the circumferences. This means that a point on the girl oscillates back and forth between two opposite points on the hoop, passing through the hoop’s center on the way and producing a straight line (the “Tusi couple”).
“The perimeter of circumscribing square equals four girl diameters or two hula hoop diameters which is the total displacement of initial point of contact between hula hoop and the aforementioned vertical girl.”
From L.A. Graham, The Surprise Attack in Mathematical Problems, 1968.
