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       #Post#: 107--------------------------------------------------
       PERFECT SHUFFLESS
       By: eba95 Date: July 30, 2010, 6:32 am
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       Perfect Shuffles
       We know from the Fun Fact Seven
       Shuffles that 7 random riffle
       shuffles are enough to make almost
       every configuration equally likely
       in a deck of 52 cards.
       But what happens if you always
       use perfect shuffles, in which you
       cut the cards exactly in half and
       perfectly interlace the cards? Of
       course, this kind of shuffle has no
       randomness. What happens if you
       do perfect shuffles over and over
       again?
       There are 2 kinds of perfect shuffles:
       The out-shuffle is one in which the
       top card stays on top. The in-shuffle
       is one in which the top card moves
       to the second position of the deck.
       Figure 1 shows an out-shuffle.
       Surprise: 8 perfect out-shuffles will
       restore the deck to its original order!
       And, in fact, there are some nice card
       tricks that use out and in shuffles to
       move the top card to any position
       you desire! Say you want the top
       card (position 0) to go to position N.
       Write N in base 2, and read the 0's
       and 1's from left to right. Perform
       and out-shuffle for a 0 and and in-
       shuffle for a 1. Voila! You will now
       have the top card at position N. (See
       the reference.)
       Presentation Suggestions:
       Have students go home and
       determine how many in-shuffles it
       takes to restore the deck to its
       original order. (Answer: 52.) You can
       also have students investigate decks
       of smaller sizes. As a project, you
       might even tell them part of the
       binary card trick and see if they can
       figure out the rest: whether 0 or 1
       stands for an in/out shuffle, and
       whether to read the digits from left
       to right or vice versa.
       The Math Behind the Fact:
       This fact may come as somewhat of
       a surprise, because there are 52!
       possible deck configurations, and
       since there is no randomness, after
       52! out-shuffles, we must hit some
       configuration at least twice (and
       then cycle from there). But 8 is so
       much smaller than (52!)!
       Group theory concerns itself with
       understanding sets and properties
       preserved by operations on those
       sets. For instance, the set of all
       configurations of a deck of 52 cards
       forms a group, and a shuffle is an
       operation on that group.
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