Out shuffle

An out shuffle is a type of perfect shuffle done in two steps:

If this shuffle keeps the top card on top and the bottom card on bottom, then it is an out shuffle, otherwise it is known as an in shuffle.

For simplicity, we will use a deck of six cards.

The following shows the order of the deck after each out shuffle. Notice that a deck of this size returns to its original order after 4 out shuffles.

The number of out shuffles required to return a deck of cards of even size, N, to original order is given by the multiplicative order of 2 modulo (N − 1).

For example, for a deck size of N = 2, 4, 6, 8, 10, 12 …, the number of out shuffles needed are: 1, 2, 4, 3, 6, 10, 12, 4, 8, 18, 6, … (sequence in the OEIS).

For a standard deck of 52 playing cards, the number of out shuffles required to return the deck to its original order is 8.

The analogous operation to an out shuffle for an infinite sequence is the interleave sequence.

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Wikipedia