In shuffle

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

  1. Split the cards exactly in half (a bottom half and a top half) and then
  2. Interweave each half of the deck such that every-other card came from the same half of the deck.

If this shuffle moves the top card to be 2nd from the top then it is an in shuffle, otherwise it is known as an out shuffle.

Contents

Example

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

The following shows the order of the deck after each in shuffle. Notice that a deck of this size returns to its original order after 3 in shuffles.

Step Top
Card		 2 3 4 5 Bottom
Card
Start
1
2
3

Mathematics

The number of in 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 in shuffles needed are: 2, 4, 3, 6, 10, 12, 4, 8, 18, 6, 11, ... (sequence A002326 in OEIS).

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

Computer Science

In computer science the problem is stated as an in shuffle of sequences A {a .. an} and B {b1 .. bn} such that the resulting in shuffled sequence

In-Shuffle(A,B) → { bi, ai } | i: 1→n

A linear time O(n) algorithm that performs this in constant space O(1) complexity is presented.

References