A perfect shuffle of a deck of 2n cards is given by the permutation: 1 2 3 ... n

Question

A perfect shuffle of a deck of 2n cards is given by the permutation:

1 2 3 ... n n+1 n+2 n+3 ... 2n

2 4 6 ... 2n 1 3 5 ... 2n-1

Find the least number of perfect shuffles that need to be performed on a deck of 52 cards before the cards are back in their original order.

Explanation / Answer

In reverse order, the cycles of permutations are: (length 1) 0 -> 0 (length 1) 51 -> 51 (length 8) 50 -> 25 -> 38 -> 19 -> 35 -> 43 -> 47 -> 49 -> 50 (length 8) 48 -> 24 -> 12 -> 6 -> 3 -> 27 -> 39 -> 45 -> 48 (length 8) 46 -> 23 -> 37 -> 44 -> 22 -> 11 -> 31 -> 41 -> 46 (length 8) 42 -> 21 -> 36 -> 18 -> 9 -> 30 -> 15 -> 33 -> 42 (length 8) 40 -> 20 -> 10 -> 5 -> 28 -> 14 -> 7 -> 29 -> 40 (length 2) 34 -> 17 -> 34 (length 8) 32 -> 16 -> 8 -> 4 -> 2 -> 1 -> 26 -> 13 -> 32 So the answer is LCM of {1, 2, 8}. Therefore, it takes 8 shuffles.

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