Figure 1

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 outshuffle is one in which the top card stays on top.
The inshuffle is one in which the top card moves
to the second position of the deck. Figure 1 shows an
outshuffle.
Surprise: 8 perfect outshuffles will restore the deck
to its original order!
And, in fact, there is a nice magic trick that uses
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 an outshuffle for a 0 and
and inshuffle for a 1. Voila! You will now have the top
card at position N. (See the reference.)
Example. Since 6 is 110 in binary notation, then the sequence ININOUT will move the top card (position 0) to position 6 (the seventh card).
Presentation Suggestions:
Have students go home and determine
how many inshuffles 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! outshuffles, we must hit
some configuration at least twice (and then cycle from there).
But 8 is so much smaller than (52!). See Making History By Card Shuffling.
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.
How to Cite this Page:
Su, Francis E., et al. "Perfect Shuffles."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
