Suppose we have a pack of 2n cards, in the order 1,2,...,2n. A perfect shuffle of these cards changes the order to n+1,1,n+2,2,...,n−1,2n,n ; i.e., the cards originally in the first n positions have been moved to the places 2,4,...,2n, while the remaining n cards, in their original order, fill the odd positions 1,3,...,2n−1.
Suppose we start with the cards in the above order 1,2,...,2n and then successively apply perfect shuffles.
What conditions on the number n are necessary for the cards eventually to return to their original order? Justify your answer.[hide="Remark"]
Remark. This problem is trivial. Alternatively, it may be required to find the least number of shuffles after which the cards will return to the original order. modular arithmeticStanfordcollegecombinatorics proposedcombinatorics