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A perfect shuffle of a deck of cards can be represented by the following permutation $f\in S_{52}$: $$f(x) =\left\{ \begin{array}{ll} 2x-1 & \text{if }x\in\{1,\ldots,26\}\\ 2(x-26) &\text{if }x\in \{27,\ldots,52\} \end{array}\right.$$

I'm trying to show that if one performs 8 perfect shuffles of a deck of cards, then this returns the cards to their original position.

What I did was get some cycle decomposition:

(2 3 5 9 17 33 14 27) and (4 7 13 25 49 46 40 28) and

when I tried out starting with 6, I am getting a mess since f(31) = 10, but f(6) = 10

How many other cycle decompositions are there?

  • 1
    $f(6) = 2(6)-1 = 11$; what do you mean $f(6)=10$? You are doing well: the next cycle will be (6 11 21 41 30 8 15 29); the next card not yet dealt with is 10.2012-02-13
  • 0
    OK, I will do 10, but how many Total cycle decompositions should I have?2012-02-13
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    the only fixed points are 1 and 52; each cycle will, hopefully, have 2, 4, or 8 elements. So anywhere between 7 and 16. Just go ahead and do it carefully.2012-02-13

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