 Perfect Shuffle

A perfect shuffle deck of cards with an even number of cards is accomplished by splitting the deck of cards into an upper half and a lower half and then interlacing the cards alternately, one at a time from each half of the deck. For example with 4 cards

1 2 3 4

the split gives half deck 1 2 and 3 4. The perfect shuffle where a card is taken first from the top half and then from the bottom half gives

1 3 2 4

Repeating gives a split of 1 3 and 2 4

1 2 3 4

So two shuffles returns the cards to the original order. This sequence where a card is taken first from the top half is called and Out-Shuffle. On the other hand if the first card is taken from the bottom half, the following sequence is obtained:

1 2 3 4

3 1 4 2

4 3 2 1

2 4 1 3

1 2 3 4

So, 4 shuffles are required to return the deck to the original order. This is called an In-Shuffle.

For a deck of 8 cards, we might show the format for two shuffles as: The perfect shuffle is a non-random process. A shuffle is a permutation of n elements (8 in the case above). Clearly, each shuffle produces a new permutation or returns to a previous on. Therefore at some point the process would return to the original order. However, there are n! permutations of a set of n elements.

QUESTIONS

1. How many out-shuffles for 8 cards is needed to return to the original order? How many in-shuffles?

2. Repeat the process for

6 cards

10 cards

12 cards

14 cards

3. Is there a pattern?

4. How many out-shuffles are needed to return to the original order for a standard deck of 52 cards? How many in-shuffles.

Comment:

The analysis might be made easier by following the 'movement' of one card through the deck. For example on the out-shuffle for 8 cards, given above, the first and last cards are always in the same place. Follow the movement of the 2. In the first shuffle, the 2 moves to the 3rd position.

In the next shuffle, the 2 moves from the 3rd position to 5th position.

Then the 2 moves from the 5th position to the 2nd position.

Thus there are 3 shuffles to move return to the original order: For the in-shuffles of 8 cards, the movement of the 2 is

First shuffle -- 4th position

Second shuffle -- 8th position

Third shuffle -- 7th positon

Fourth shuffle -- 5th position

Fifth shuffle -- 1st position

Sixth shuffle -- 2nd position

Comment:

Spreadsheets can easily be set up to generate the mapping patterns of the shuffles. Click HERE for a Spreadsheet to show mapping for 8, 12, or 52.

Comment:

APPLICATION: see http://www.mathscarves.org/ for a discussion of using Perfect Shuffle patterns in knitting.

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