KöMaL 2005 B3841

The problem below appears in the article ``Szalonpóker'' (Hungarian title) of the current issue. Suppose that a deck of 52 cards is suffled in the same way. How many times does it need to be shuffled in order to get back the initial order. Solve the problem for that case, too, when the shuffling starts with the bottom card of the deck on the right, that is, when the card in the 26th place originally is put at the bottom of the new deck. (Little John and Old Firehand drop in the famous casino of Black Jacky to play cards. They play with a deck of 32 cards numbered 1 to 32. Before they agree on the rules, Black Jacky shuffles the cards as follows: He places the deck on the table, removes the top 16 cards and puts them on the table, to the right of the remaining deck without turning over. Then he forms one deck of them, with the bottom card of the deck on the left lying at the bottom, followed by alternating cards from the two decks. Then he repeats the procedure several times with the deck obtained in this way. Little John is sure that this way of shuffling the cards is not fair. Show that repeating the procedure several times will lead to a surprising result.)