Eight perfect shuffles restore the deck

Question

A perfect out-shuffle splits a 52-card deck into two halves of 26 and interleaves them exactly, with the original top card staying on top. (An in-shuffle interleaves so that the original top card becomes second.) 1. How many perfect out-shuffles return a 52-card deck to its original order? 2. How many in-shuffles are needed? 3. Explain the calculation.

Accepted Answer

Eight out-shuffles. Fifty-two in-shuffles. Where each card goes. Number the positions 0 to 51 from the top. In an out-shuffle the top card (position 0) stays at 0 and the bottom card (51) stays at 51; every other card at position k moves to position 2k modulo 51. For example, the card at position 1 moves to 2, the card at 26 moves to 52 minus 51 = 1, and so on. After n shuffles the card from position k is at position 2 to the n times k, modulo 51. Back to the start. The deck is restored when 2 to the n leaves remainder 1 when divided by 51, so that every position maps to itself. Compute the r…

Eight perfect shuffles restore the deck

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