Math, Magic, or Luck of the Draw?

Violet Gilbert

Professor Donovan

EDM 390/490

April 11, 2011

Recreational mathematics defined

A look at beliefs

Perfect or faro shuffle defined

• Explore the mathematics behind the

shuffle

Importance of recreational games

• Real world applications

• To educators

• To students

So What is Recreational Math?

Image: crazyengineers.com

“Mathematics with no obvious “use,” particularly if it has a strong aesthetic or brain-teaser component, is likely to be consigned to recreational mathematics” (Deimel 2003).

Recreational Games and Ancient Beliefs

- No one can say exactly who invented

prayer, music, farming, medicine, or

money. The same can be said for

gambling: It is simply older than history.

- But early “gamblers” weren’t simply

playing for amusement: The first ventures

into chance were usually more religious

than recreational.

~ David G. Schwartz

Roll the Bones, 2006

Magic, Luck,, or Mathematics?

Inquiries into a connection between

supernatural magic (luck/fate), games,

chance, and mathematics are recorded in

texts that came after gambling included

card games in the first half of the second

millennium A.D. For example...

(Schwartz, 2006, p. 41)

Magic Explained

But in shewing feats, and juggling with cards, the principall point consisteth in shuffling them nimblie.... Hereby you shall seeme to work wonders....

~ Reginald Scot, 1584 The Discoverie of Witchcraft (Morris, 1998)

Image: Skullsinthestars.com

Image: bsu.edu

Although no one knows for certain

who was the first to investigate

recreational games and mathematics,

Gerolamo Cardono is credited to be

the first one to scientifically explore

gambling in his book, Book on Games

and Chance, in 1526. (Ore, 1965, p.143)

His book on

probability and

chance was not

published until

eighty-seven

years after his

death in 1663. (Schwartz, 2006, p.76)

Image: Greydragon.org

Cardono’s main focus was on

probability and dice games, but he

also made the following

observation about card games after

analyzing primero, which was the

predecessor of basset:

Cardono on Playing Cards:

Cards have this in common with dice, that

what is desired may be got with fraud…

[one] has to do with recognition of the

cards—in its worst form it consists of using

marked cards, and in another form it is more

excusable, namely, when the cards are put in

a special order and it is necessary to

remember this order.

~ Gerolamo Cardono

The Book of Games and Chance (1663)

One game in particular, evolved from

basset in the late 17th century in France,

which exploited the latter form of card

recognition: FARO. (Morris, 1998, p.2)

Faro table: Faro Counter:

Image: petticoatsandpistols.com Image: prices4antiques.com

This is the game that gave rise to the perfect shuffle, faro shuffle (U.S.), or weave shuffle (Great Britain).

What is a faro shuffle and how is it done? Let’s watch a video of Adam West to find out: http://www.youtube.com/watch?v=7lNk7bfkFq8

Now, let’s try our own version…

What Will Happen to Card Position if We Follow These Steps…

1) Evenly cut a 16 card deck in half

2) Put the top card from the left pile in card position one and the top card from the right pile in card position two by slightly overlapping the edges

3) Continue this pattern for cards 3-16

4) Repeat steps 1-3

Abracadabra…

It’s time to find out!

Suggestion for Organizing Data: Shuffle 1 Shuffle 2 Shuffle 3 Shuffle 4 Shuffle 5 Shuffle 6 Etc.

Position 1

Position 2

Position 3

Position 4

Position 5

Position 6

Etc.

The table needs to include all 16 card positions. What patterns do you see after the 1st shuffle? 2nd? Can you predict the card position after the 3rd shuffle?

Using a 16-card Deck, What Happened to Card

Position With Each Perfect Shuffle?

1st Shuffle Original Position

2nd Shuffle 3rd Shuffle 4th Shuffle

1 A 2 B 3 C 4 D 5 A 6 B 7 C 8 D 9 A 10 B 11 C 12 D 13 A 14 B 15 C 16 D

1 A 2 A 3 B 4 B 5 C 6 C 7 D 8 D 9 A 10 A 11 B 12 B 13 C 14 C 15 D 16 D

1 A 2 A 3 A 4 A 5 B 6 B 7 B 8 B 9 C 10 C 11 C 12 C 13 D 14 D 15 D 16 D

1 A 2 C 3 A 4 C 5 A 6 C 7 A 8 C 9 B 10 D 11 B 12 D 13 B 14 D 15 B 16 D

1 A 2 B 3 C 4 D 5 A 6 B 7 C 8 D 9 A 10 B 11 C 12 D 13 A 14 B 15 C 16 D

Types of Magic Card Tricks Based on

Variations of Faro Shuffling…

Image: Graphicsfuel.com

• Stay-Stack Principle: a partial faro

shuffle allows a spectacular double-

fanning of cards

• “The Seekers” uses an incomplete

faro shuffle and the magician “seeks”

a card chosen by an audience member

• Many tricks that require magicians

to find a certain card in the deck rely

on partial faro shuffles; some tricks

use faro shuffling in tandem with

card counting (Morris, 1998)

Connecting Recreational Math

to “Real” Mathematics:

Image: capearagorubberstamps.com

British physicist Roger Penrose’s discovery of two simple polygons that tile the plane only in a nonperiodic fashion like this:

Led to the constructions of 3-D analogs of quasicrystals, which is a hot field of research by crystallographers. (Morris, 1998, p. xviii)

Crystal & Tile Images: euler.phys.cmu

Another Connection:

Image: Thinkgeek Work done on the faro theory for the fun of it proved to be of great importance in computer science with the development of dynamic memory. (Morris, 1998, p. xviii)

The out- and in-shuffles can be used as interconnections to dramatically improve the performance of shift-register memory. (Morris, 1998, p. 82)

Why is Recreational Mathematics Important to Educators?

Puzzles Magic Square

Dice &

Card

Games

Card Image: photos-public-domain School Image: wordpress.com

Image: games.4you4free.com

From an Educator’s Perspective:

Image: mason.gmu.edu

Example: Which of the five strands of mathematical proficiency did we use today exploring the faro/perfect shuffle?

The Five Strands of Mathematical Proficiency

Five Strands Defined:

(1) Conceptual understanding refers to the “integrated and functional grasp of mathematical ideas”, which “enables them [students] to learn new ideas by connecting those ideas to what they already know.” A few of the benefits of building conceptual understanding are that it supports retention, and prevents common errors.

(2) Procedural fluency is defined as the skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.

(3) Strategic competence is the ability to formulate, represent, and solve mathematical problems.

(4) Adaptive reasoning is the capacity for logical thought, reflection, explanation, and justification.

(5) Productive disposition is the inclination to see mathematics as

sensible, useful, and worthwhile, coupled with a belief in diligence

and one’s own efficacy. (NRC, 2001, p. 116)

(mason.gmu.edu)

From a Student’s Perspective:

Calvin and Hobbes by Bill Watterson Image: wordpress.com

Do not train a child to learn by force

or harshness; but direct them to it by

what amuses their minds, so that you

may be better able to discover with

accuracy the peculiar bent of the

genius of each.

~ Plato

(thinkexist.com)

References

Cardono, G. (1965). The book on games of chance (S. H. Gould, Trans.).

New York, NY: Dover Publications, Inc. (Original work published 1663)

Deimel, L. (2007, May 24). Biography. Retrieved from

http://www.deimel.org/biography/biography.htm

Mason.gmu.edu. (n.d.). The five strands of mathematical proficiency. Retrieved from

http://mason.gmu.edu/~jsuh4/teaching/strands.htm

Morris, S. B. (1998). Magic tricks, card shuffling and dynamic computer

memories. U. S. A.: The Mathematical Association of America.

Ore, O. (1965). Cardono the gambling scholar. New York, NY: Dover

Publications. (Original work published 1953)

Plato. (n.d.). Retrieved from:

http://thinkexist.com/quotation/do_not_train_a_child_to_learn_by_force_or/259396.html

Schwartz, D. G. (2006). Roll the bones. New York, NY: Gotham Books.

West, A. (2007, February 4). 8 perfect faro shuffles [Video file]. Video

posted to http://www.youtube.com/watch?v=7lNk7bfkFq8

PowerPoint Template: www.presentationmagazine.com