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It will take 8 shuffles.
To see this, write a shuffle as a permutation, using cycle notation:
(1)(2 3 5 9 17 33 14 27)(4 7 13 25 49 46 40 28)(6 11 21 41 30 8 15 29)(10 19 3722 43 34 16 31)(12 23 45 38 24 47 42 32)(18 35)(20 39 26 51 50 48 4436)(52)
where this means card 1 -> position 1, card 2 -> position 3, card 3 -> position5, ..., card 27 -> position 2, etc.
So the shuffle permutation is a produceof 2 1-cycles, 6 8-cycles and a 2-cycle. Shuffling twice corresponds to thecomposition of this permutation with itself, and so on. The number of shufflesneeded to restore the deck to its original order is the least common multiple ofthe lengths of the cycles: 8.
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