Rising Sequences in Card Shuffling -- Mudd Math Fun Facts
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created, authored and ©1999-2007 by Francis Su
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From the Fun Fact files, here is a Fun Fact at the Medium level:
Rising Sequences in Card Shuffling
In Seven Shuffles we saw that it takes about 7 random riffle shuffles to randomize a deck of
52 cards. This means that after 7 shuffles, every configuration is nearly equally likely.
An amazing fact is that five random riffle shuffles are not enough to randomize a deck of
cards, because not only is every configuration not nearly equally likely, there are in fact
some configurations which are not reachable in 5 shuffles!
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To see this, suppose (before shuffling) the cards in a deck are arranged in order from 1 to
52, top to bottom. After doing one shuffle, what kind of sequences are possible? A
moment's reflection reveals that only configurations with 2 or fewer rising sequences are
possible. A rising sequence is a maximal increasing sequential ordering of cards that appear
in the deck (with other cards possibly interspersed) as you run through the cards from top
to bottom. For instance, in an 8 card deck, 12345678 is the ordered deck and it has 1 rising
sequence. After one shuffle, 16237845 is a possible configuration; note that it has 2 rising
sequences (the black numerals form one, the red numerals form the other). Clearly the
rising sequences are formed when the deck is cut before they are interleaved in the shuffle.
So, after doing 2 shuffles, how many rising sequences can we expect? At most 4, since
each of the 2 rising sequences from the first shuffle have a chance of being cut in the
second shuffle. So the number of rising sequences can at most double during each shuffle.
After doing 5 shuffles, there at most 32 rising sequences.
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But the reversed deck, numbered 52 down to 1, has 52 rising sequences! Therefore the
reversed deck cannot be obtained in 5 random riffle shuffles!
Presentation Suggestions:
You can illustrate examples of rising sequences on the blackboard.
The Math Behind the Fact:
The analysis of shuffling involves both combinatorics, probability, and even some group
theory. Though we've been discussing "random" riffle shuffles in this Fun Fact, you can
also study the mathematics of Perfect Shuffles.
How to Cite this Page:
Su, Francis E., et al. "Rising Sequences in Card Shuffling." Mudd Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
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Rising Sequences in Card Shuffling -- Mudd Math Fun Facts
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References:
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"How
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shuffle a
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UMAP J.
15 (1994),
no. 4,
303--332.
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Keywords:
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Subjects:
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Level: Medium
Fun Fact suggested
by: Francis Su
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form.
Want another Math Fun Fact?
For more fun, tour the Mathematics Department at Harvey Mudd College!
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7/8/09 8:52 AM
Perfect Shuffles -- Mudd Math Fun Facts
http://www.math.hmc.edu/funfacts/ffiles/20001.1-6.shtml
hosted by the Harvey Mudd College Math Department
created, authored and ©1999-2007 by Francis Su
You can now subscribe to our
RSS feed to get the latest Fun Facts!
From the Fun Fact files, here is a Fun Fact at the Medium level:
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.
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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?
Figure 1
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.
How to Cite this Page:
Su, Francis E., et al. "Perfect Shuffles." Mudd Math Fun Facts. <http://www.math.hmc.edu
/funfacts>.
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Perfect Shuffles -- Mudd Math Fun Facts
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Keywords: probability,
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Subjects: algebra,
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Level: Medium
Fun Fact suggested by:
Francis Su
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7/8/09 8:49 AM
Seven Shuffles -- Mudd Math Fun Facts
http://www.math.hmc.edu/funfacts/ffiles/20002.4-6.shtml
hosted by the Harvey Mudd College Math Department
created, authored and ©1999-2007 by Francis Su
You can now subscribe to our
RSS feed to get the latest Fun Facts!
From the Fun Fact files, here is a Fun Fact at the Medium level:
Seven Shuffles
How many shuffles does it take to randomize a deck of cards?
The answer, of course, depends on what kind of shuffle you consider. Two popular kinds of
shuffles are the random riffle shuffle and the overhand shuffle. The random riffle shuffle is
modeled by cutting the deck binomially and dropping cards one-by-one from either half of
the deck with probability proportional to the current sizes of the deck halves.
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© 1999-2007 by Francis Edward Su
All rights reserved.
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In 1992, Bayer and Diaconis showed that after seven random riffle shuffles of a deck of 52
cards, every configuration is nearly equally likely. Shuffling more than this does not
significantly increase the "randomness"; shuffle less than this and the deck is "far" from
random.
In fact, it is possible to show that five shuffles are not enough to bring about the reversal of
a deck (see Rising Sequences in Card Shuffling. So it is somewhat surprising that just two
shuffles later, every configuration is possible and nearly equally likely.
By the way, the overhand shuffle is a really bad way to mix cards: it takes about 2500
overhand shuffles to randomize a deck of 52 cards!
Presentation Suggestions:
Bring a deck of cards in and demonstrate how non-random just 2 or 3 shuffles are by
ordering the deck and then letting someone shuffle. There will still be discernible patterns
after a small number of shuffles!
The Math Behind the Fact:
A well-written account of Bayer and Diaconis' result may be found in the Mann reference.
There are many ideas in this result. Analysis of the "distance from randomness" requires
the choice of a metric betwen probabilities. Combinatorics and probability intertwine in the
analysis of rising sequences generated after a certain number of shuffles, which is an
important part of proving this result.
There are, of course, non-random shuffles: see Perfect Shuffles.
How to Cite this Page:
Su, Francis E., et al. "Seven Shuffles." Mudd Math Fun Facts. <http://www.math.hmc.edu
/funfacts>.
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Stumbleupon |
1 of 2
7/8/09 8:51 AM
Seven Shuffles -- Mudd Math Fun Facts
The Champ
Shuffleboard
#1 Dealer of Shuffle
Board Tables All BrandsGuaranteed Lowest
Prices
http://www.math.hmc.edu/funfacts/ffiles/20002.4-6.shtml
References:
Brad
Mann,
"How
many
times
should
you
shuffle a
deck of
cards?"
UMAP J.
15 (1994),
no. 4,
303--332.
Dave
Bayer and
Persi
Diaconis,
"Trailing
the
dovetail
shuffle to
its lair",
Ann. Appl.
Probab.
2(1992),
no. 2,
294--313.
www.GameTables4Less.com
Play Star Shuffle Now
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Decks
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4.16
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rating
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Suffleboard Tables
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Keywords: card
shuffling, probability
Subjects:
combinatorics,
probability
Level: Medium
Fun Fact suggested
by: Brad Mann
Suggestions? Use this
form.
Want another Math Fun Fact?
For more fun, tour the Mathematics Department at Harvey Mudd College!
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