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A pack of 2n cards

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September 5, 2010
modular arithmeticStanfordcollegecombinatorics proposedcombinatorics

Problem Statement

Suppose we have a pack of 2n2n cards, in the order 1,2,...,2n1, 2, . . . , 2n. A perfect shuffle of these cards changes the order to n+1,1,n+2,2,...,n−1,2n,nn+1, 1, n+2, 2, . . ., n- 1, 2n, n ; i.e., the cards originally in the first nn positions have been moved to the places 2,4,...,2n2, 4, . . . , 2n, while the remaining nn cards, in their original order, fill the odd positions 1,3,...,2n−1.1, 3, . . . , 2n - 1. Suppose we start with the cards in the above order 1,2,...,2n1, 2, . . . , 2n and then successively apply perfect shuffles. What conditions on the number nn 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.
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