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