Chaos, an introduction to dynamical systems by Kathleen T. Alligood & all - Exercises3.15

3.15. (Party trick.) (a) A perfect shuffle is performed by dividing a 52-card deck in half, and interleaving the halves, so that the cards from the top half alternate with the cards from the bottom half. The top card stays on top, and so it and the bottom card are fixed by this operation. Show that 8 perfect shuffles return the deck to its original order. [Hint: Number the original card order from 0 to 51. Then a perfect shuffle can be expressed as the map f (n) = 1 2n if 0