Shuffles to restore the deck

A 5252-card deck is shuffled as follows: split into positions 112626 and 27275252, then interleave so the bottom half (27275252) takes the odd positions 1,3,,511, 3, \dots, 51 and the top half (112626) takes the even positions 2,4,,522, 4, \dots, 52. What is the minimum number of such shuffles needed to return the deck to its original order?

Show hints (2)+
  1. The shuffle is a fixed permutation; the answer is its order. This is an in-shuffle of 5252 cards.
  2. Its order is ord53(2)\operatorname{ord}_{53}(2) (smallest kk with 2k1mod532^k\equiv1\bmod 53), which is 5252.

Answer

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52

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Asked at: Jane Street, Two Sigma

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