Perfect Shuffles

Perfect Shuffles By Christine Mac, Tanya de Nobrega, Paula Hulse, Keith Kennison

WIKI • http://perfectshuffles.wikispaces.com

In Shuffle • Left Hand Shuffle • Disjoint Cycle Representation(1,2,4,8,16,32,11,22,44,35,17,34,15,30,7,14,28,3,6,12,24,48,43,33,13,26,52,51,49,45,37,21,42,31,9,18,36,19,38,23,46,39,25,50,47,41,29,5,10,20,40,27) • There is one cycle of length 52. • It would take 52 shuffles to return the deck to its original order.

In Shuffle • Written as transpositions: • (1,27)(1,40)(1,20)(1,10)(1,5)(1,29)(1,41)(1,47)(1,50)(1,25)(1,39)(1,46)(1,23)(1,38)(1,19)(1,36)(1,18)(1,9)(1,31)(1,42)(1,21)(1,37)(1,45)(1,49)(1,51)(1,52)(1,26)(1,13)(1,33)(1,43)(1,48)(1,24)(1,12)(1,6)(1,3)(1,28)(1,14)(1,7)(1,30)(1,15)(1,34)(1,17)(1,35)(1,44)(1,22)(1,11)(1,32)(1,16)(1,8)(1,4)(1,2) • There are 51 transpositions indicating this is an odd permutation.

Out shuffle • Right Hand shuffle • Disjoint Cycle Representation(2,3,5,9,17,33,14,27)(4,7,13,25,49,46,40,28)(6,11,21,41,30,8,15,29)(10,19,37,22,43,34,16,31)(12,23,45,38,24,47,42,32)(18,35)(20,39,26,51,50,48,44,36) • There are 6 cycles of length 8 and one cycle of length 2. • The LCM (8,2) = 8. • It will take 8 shuffles to return to the deck to its original order. • http://www.natedog.com/cards/faro.html

Out shuffle • Written as transpositions: • (2,27)(2,14)(2,33)(2,17)(2,9)(2,5)(2,3)(4,28)(4,40)(4,46)(4,49)(4,25)(4,13)(4,7)(6,29)(6,15)(6,8)(6,30)(6,41)(6,21)(6,11)(10,31)(10,16)(10,34)(10,43)(10,22)(10,37)(10,19)(12,32)(12,42)(12,47)(12,24)(12,38)(12,45)(12,23)(18,35)(20,36)(20,44)(20,48)(20,50)(20,51)(20,26)(20,39) • There are 43 transpositions indicating this is an odd permutation.

Mixture of In and Out shufflesto move top card to position n • Move top card to position 20. • Write 20 in binary: 10100 • Do an in shuffle for every 1 and an out shuffle for every 0 • In, Out, In, Out, Out • http://www.natedog.com/cards/faro.html

Using shuffles to your advantage • Consider: It is your turn to deal and you are playing cards with a group of six people. As you gather the cards, you place an ace on the top of the deck. How could you shuffle the deck to ensure that you deal the ace to yourself? • To make sure that the dealer deals the ace as the sixth card to himself, the top card would need to move five positions down the deck. • Write five as a binary number: 5 =101 • You need to do an in shuffle (1), an out shuffle(0), and in shuffle(1).

Mixture of In and Out shufflesto move card n to top position • Deck of 2n cards, define r 2r-1<2n<2r, so for 2n=52 then r=6 • Let’s say you want to move card 35 to the top • Let

Mixture of In and Out shufflesto move card n to top position • Correction terms s=2nt-2rp Out, in, in, in, out, outhttp://www.natedog.com/cards/faro.html

5 perfect shuffles • Rising sequence: Increasing sequential ordering of cards in deck • Ordered deck has one rising sequence • Cut of deck splits rising sequence into two, so each shuffle doubles rising sequences • After 5 shuffles we have at most 25 rising sequences • But the reversed deck has 52 rising sequences – so we cannot reach this configuration with 5 shuffles.

7 riffle shuffles • A riffle shuffle is a shuffle where the deck is not cut exactly in half and cards are not perfectly interlaced. • Every shuffle has a small error which makes the result random • We need at least 7 shuffles for this error to propagate through the whole deck and randomize the deck • After 7 shuffles every card is equally likely to be at all positions in the deck

Perfect Shuffles