Colm's Cards -- Mathematical Card Tricks

Card Colm debuted in Oct 2004, and appears every two months at MAA Online. The Oct 2006 one includes a recent interview with Martin Gardner.

Here's a guide to topics we've explored. "Levels U/P" refers to two things, how much mathematics is Used and how hard the tricks are to Perform. Sometimes the underlying mathematics can be at quite a different level from the chops needed to do the trick for an audience! Also, some of the ideas (e.g., the First Gilbreath Principle) have much simpler, less mathematically rococo, applications than those indulged in here.

Title	Date	Contents	Level U/P
Additional Certainties	Feb 2008	Fibonacci Numbers and Zeckendorf Representations	Some/Moderate
Plurality Events, Standard Deviations and Skewed Perspectives	Dec 2007	Mean, Standard Deviation & Skewness	Some/Moderate
A Magic Timepiece Influenced By Martin Gardner (Celebrating His 93rd Birthday Incidentally)	Oct 2007	Cycling quadruple sum forces	Some/Easy
Sixy Alpha Omegas	Aug 2007	Alternating triple sum forces	Some/Easy
Gibonacci Braclets	Jun 2007	Fibonacci sequences mod m	Little/Easy
Magic Circles of Eight	Apr 2007	Magic circles	Little/Easy
Quasi-Masked Forcing Kind of Magic Squares	Feb 2007	Magic squares	Little/Easy
Quantitative Reasoning in Small Groups	Dec 2006	Z₂, Z₃, Klein-4 group	Some/Easy
Martin Gardner's Magic Spells	Oct 2006	Interview & 10-card pyramid speller	None/Easy
The Second Norman Invasion	Aug 2006	The Second Gilbreath principle	Little/Moderate
Better Poker Hands Guaranteed	Jun 2006	Birthday card matches applications	Little/Easy
Bill Simon's Sixty-Four Principle	Apr 2006	Two ways to force 4 of 8 cards	Some/Moderate
Many Fold Synergies	Feb 2006	Parity principles	Little/Easy
Luckation Is Everything	Dec 2005	Completing to 13 and advantages of knowing the 26th card	Little/Easy
Subtraction Is Addictive	Oct 2005	Subtraction principles	Little/Moderate
The First Norman Invasion	Aug 2005	The First Gilbreath principle	Some/Easy
A Little Erdos/Szekeres Magic	Jun 2005	Any 5 has 3 in order (two person magic)	Some/Moderate
The Down Under Deal (aka the Australian Shuffle)	Apr 2005	Down under deals and powers of 2	Little/Easy
Fitch Four Glory	Feb 2005	3 cards predict the 4th (two person magic)	Some/Hard
Sheer Luck	Dec 2004	Elimination deals	Some/Moderate
Low Down Triple Dealing	Oct 2004	Triple deal principle/ quad run false shuffle	Little/Easy

(NEW) A curious spelling/numerical duality is explored with cards in "One of Six and Another of Half a Dozen" in the May 2007 issue of Word Ways (The Journal of Recreational Linguistics),

(NEW) An article called "An ESPeriment with Cards," combining the Erdos/Szekeres and Gilbreath principles, was published in MAA Horizons in February 2007.

An article on the ever popular "Fitch Cheney Five-Card Trick and Generalizations" was published in MAA Horizons in February 2003. It also appears in the recent book The Edge of the Universe--Celebrating Ten Years of Math Horizons book (MAA, August 2006), edited by Deanna Haunnsperger & Stephen Kennedy.

On 21 October 2004, Martin Gardner turned 90. Here is a birthday present written for him: Low Down Triple Dealing. (There is a great website called g4g4 dedicated to all sorts of stuff inspired by Martin.) That article served as the inaugural Card Colm at MAA Online. It also appeared in print in the November 2004 issue of MAA Focus. To get some idea of the extent of Martin's influence, look at the dedication to the book Winning Ways (Acedemic Press, 1982), by Elwyn R. Berlekamp, John H. Conway & Richard K. Guy: "To Martin Gardner, who has brought more mathematics to more millions than anyone else."

The effects in Low Down Triple Dealing use an alternative to the simple card cycling which forms the basis of a family of card tricks first explored in print 35 years ago by George Sands, while avoiding the prime (or more correctly, coprime) constraints of those. (Peter Duffie and Robin Robertson have recently published several tricks extending Sands to the coprime context.) Variations of the triple (actually quadruple!) deal, in which some cards are turned face down, have been developed as well, and will appear in a forthcoming book.

This can also be used for a "quad run" false shuffle: take a packet of n cards in one hand and run off k (reversing their order), where k is at least half of n, before dropping the rest on top. Repeating this three times (for a total of four runs) restores the packet to its original order, despite the appearance of adequate shuffling. (Repeating it just twice brings the bottom k cards to the top, suitably reversed.)

The triple deal version was used to produce money from a stack of seemingly empty envelopes at the November 2004 meeting of the Georgia Magic Club (International Brotherhood of Magicians Ring 9) as follows: world-renowned magician Dan Garrett first checked that all envelopes were devoid of cash, and then selected a magician's name at random (he picked "David Copperfield"). Upon spelling out that name three times over (dealing envelopes into a pile, and then dropping the remainder on top), he was surprised to find a $5 bill in the top envelope. Of course it helped that some cad had ensured that the orginal bottom envelope was pre-loaded...

Highlights from The Seventh Gathering for Gardner (g4g7) (March 2006), by Ivars Peterson.

Slides from The Sixth Gathering for Gardner (g4g6) (March 2004), courtesy of the NYT.

An article on five simple mathematical card tricks was published in MAA Horizons in February 2004.

Earlier, several mathematical card tricks were discussed in an article at the AMS's October 2000 What's New in Mathematics site.

A four-hour MAA minicourse on "An Introduction to Mathematical Card Tricks" was given (jointly with Jeffrey Ehme) at the joint winter meetings, January 2004, in Phoenix. Similar minicourses were given at the joint winter meetings , January, 2002, in San Diego, and at MAA Mathfest in August 2001, in Madison.

A two day minicourse on "Mathematical Card Tricks" was given (jointly with Derek Smith) at the Math Jubilee at the University of Georgia, Athens, Georgia, April 12 & 13, 2002.

English faro shuffle and false count expert Alex Elmsley with Colm Mulcahy, London, November 2002.

Alex Elmsley, one of the most delightful magicians I ever had the pleasure of meeting, died on Sunday 8th January 2006 in London. His groundbreaking work on faro shuffles (with their fascinating tie in with binary arithmetic) goes back to the 1950s, and forms the basis for some of the finest example of mathematical magic ever. There is probably more elegant mathematics ingeniously applied to magic in the superb Collected Works of Alex Elmsley, Volumes I & II (edited by Stephen Minch), L & L Publishing, 1991, than in any other book(s) in print. He also left behind some excellent videos.

He was equally famous for his false count. He called it the ghost count, everyone else knew is as the Elmsley count. (An aside: everytime I withdraw money from a cash machine, I think of him as I purposely count it in front of the security camera, just in case I'm short changed. I pause to wonder at the fun he could have in similar circumstances. Would banks recognize videocamera evidence if they knew what he and all his magic disciples could do if they chose to be less than honest?!)

Two talks exploring a new family of mathematical tricks (inspired by Alex Elmsley creations) were given by Jeff Ehme and me at the Joint Winter Meeting in San Antonio, on Friday 13th January, 2006.

("All You Need Is Cards" also got a fresh airing at this meeting. This fab paper was published in January 2002 in Puzzlers' Tribute - A Feast for the Mind (A.K. Peters), a tribute to Martin Gardner.)

BAMA (Bay Area Mathematical Adventures) Lecture: Magic with Math--Eine Kleine Nachtmagie, 19th November, 2005, Santa Clara University, CA.

Banquet speaker: Eine Kleine Nachtmagie, 29th October, 2005, Geneseo, NY.

A Mini-Lecture on "Mathematical Principles in Card Tricks" was given 17th January, 2005, at the Georgia Magic Club (International Brotherhood of Magicians Ring 9).

A Teach-a-Trick Session on "Mathematical and Self-Working Magic" was given 17th December, 2001 at the Georgia Magic Club (International Brotherhood of Magicians Ring 9).

A three-hour MAA short course on "An Introduction to Mathematical Card Tricks" was given (jointly with Jeffrey Ehme) on 10th March 2000, at the 79th Annual Southeastern Section Meeting, in Charlotte, NC.

"Pick a card, any card!"

Rumours that the cards were rigged--the night before, in the hotel room--are entirely without foundation, as this photograph makes abundantly clear. The blue book is Martin Gardner's 1956 classic Mathematics, Magic and Mystery (Dover), which contains some excellent card tricks.

Abstract: Card tricks can be used to liven up many classes for mathematics students, from precalculus and discrete math to abstract algebra and number theory. Perhaps more importantly, card tricks are an invaluable tool in convincing non-mathematics students that math can be fun, and in instilling in them a healthy scepticism for "performance magic". This fits in well with the philosophy of those non-math major courses which aim to equip students with quantitative reasoning and quantitative literacy skills.

Of course, card tricks have proved to be a popular form of diversion outside of the classroom for hundred of years, and armed with a good card trick one and relatives and highly-trained professional mathematicians. Moreover, experience suggests that one's chances of fooling, er, entertaining members of any of these groups is just about equal.

This course will present an elementary, interactive introduction to card tricks that are based on simple mathematical ideas, rather than slight of hand. Working in small cluster groups, participants will learn a variety of tricks---from very simple tricks one can do alone to more sophisticated ones requiring an accomplice---along with the underlying principles. The mathematics can be explained in simple language to any interested parties, regardless of the level of mathematical training possessed.

Martin Gardner's 1956 classic Mathematics, Magic and Mystery (Dover) was the first book to gather together in one place some of the great mathematics-based card tricks, and many of his more recent books are a good source also. Karl Fulves has two Dover books from some time back with many relevant tricks---minus the mathematics. A whole series of recent books by Bob Longe, published by Stirling, is also worthy of attention.

Some mathematicians are also highly-skilled performers of card (and other) magic, such as Perci Diaconis, Ron Graham, Brent Morris, John H. Conway and Art Benjamin, many of whom have published on the subject. Brent Morris has a recent MAA book detailing sophisticated mathematical card tricks which utilize the faro (perfect) riffle-shuffle, a technique which is unfortunately far beyond the level of card handling expertise of the casual amateur.

Will Jeff find her card successfully? Yes, thanks to his amazing ability to compute mentally twisted aperiodic spectral sequences of the second type, over non-associative quasi-algebras with neither elements nor operations.

(Just kidding...the fact is that while that's what she thinks he's up to, on a good day Jeff can count to ten without moving his lips too obviously.)

Magician, inventor, The Magic Show interactive book co-author and generous faro shuffle teacher Mark Setteducati with Colm Mulcahy, Atlanta, March 2002.

Swedish card maestro Lennart Green with Colm Mulcahy, Atlanta, 23rd April 2004.

(This is http://www.spelman.edu/~colm/cards.html, and it was last updated on 27 Mar 2008. Click here to return to main page.)

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