1. Card Shuffling How many perfect shuffles will return a full deck
of cards to their original order?

2. Perfect Shuffle Divide 52 cards into 2 equal piles
Shuffle by interlacing cards
Keep top card fixed (Out Shuffle)
8 shuffles => original order

3. For example

4. A Model for Card Shuffling
Label the positions 0-51
Then
0->0 and 26 ->1
1->2 and 27 ->3
2->4 and 28 ->5
… in general?
f(x) = 2x mod 51

5. The Order of a Shuffle
Minimum integer k such that 2 k x = x mod 51 for all x in {0,1,…,51}
In particular, this has to hold when x = 1
Minimum integer k such that 2 k - 1= 0 mod 51
Thus, 51 divides 2 k - 1
k= 6, 2 k - 1 = 63 = (3)(3)(7)
k= 7, 2 k - 1 = 127 (prime)
k= 8, 2 k - 1 = 255 = (5)(51) = 0 mod 51

6. The Out Shuffle

7. The In Shuffle

8. Model for shuffling n Cards
p = position
In Shuffles
Out Shuffles

9. Order of a Shuffle 8 Out Shuffles for 52 Cards In General?
o (O,2n-1) = o (O,2n)
o (I,2n-1) = o (O,2n)
=> o (O,2n-1) = o (I,2n-1)
o (I,2n-2) = o (O,2n)
Therefore, only need o (O,2n)

10. o (O,2n) = Order for 2n Cards
1 shuffle: O(p) = 2p mod (2n-1), 0<p<N-1
2 shuffles: O2(p) = 2 O(p) mod (2n-1) = 22 p mod (2n-1)
k shuffles: Ok(p) = 2kp mod (2n-1)
Order: o (O,2n) = smallest k such that
Ok(p) = p mod (2n-1) for all p between 0 and 2n
Which means 2k = 1 mod (2n-1) => (2n – 1) | (2k – 1)

11. The Orders of Perfect Shuffles
n o(O,n) o(I,n) n o(O,n) o(I,n)