Colm's Cards -- Mathematical Card Tricks

Follow @cardcolm

Here's a guide to topics we've explored in Card Colm over the years. "Level U/P" refers to two things, firstly (U) how much mathematics is Used and secondly (P) 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
Aug 2013 Some/Moderate
Jun 2013 Some/Moderate
Api 2013 Some/Moderate
Feb 2013 Some/Moderate
Jun 2013 Some/Moderate
Dec 2012 Some/Moderate
Jun 2012 Some/Moderate
Aug 2012 Some/Moderate
Something Old, Something True, Something Borrowed, Something NewJun 2012 Poker hand control via Bill Simom principle, position parity, and monge shuffles Some/Moderate
Splitting the Pot Apr 2012 Hall's marriage theorem Some/Moderate
Amazon Arrays (Large Action) Feb 2012 Graeco Latin squares Some/Easy
Magical Mathematics: Recurring Cycles of Ideas of Cycles Dec 2011 Universal cycles Some/Easy
A Third Selection of Mathemagical Amusements
with Cards in Martin Gardner's Own Words
Oct 2011 Various Some/Easy
A Call For A New Shuffle Principle (Need Rot Sextet?)Aug 2011 New Gilbreath principle? Some/Easy
Out Of This Whirled (Twisted Mess) Jun 2011 Parity principle Some/Easy
Twisting the Knight Way (No Big Deal) Apr 2011 Twisting principle Some/Easy
Low-Down Double Dealing With the Big BoysFeb 2011 Double deal palindrome principle Some/Easy
It's Red or Black and Blue All OverDec 2010 Small linear ternary code
(two person magic)
The Three Piles (This Isn't That Trick)Nov 2010 Small linear binary codes
(two person magic)
Celebration of MindOct 2010 Martin Gardner events) ---/---
Also In His Own Words: More Mathemagical Games (and Tricks) With Cards From Martin GardnerAug 2010 Highly varied Some/Moderate
In His Own Words: Mathemagical Card Tricks From Martin Gardner (1914-2010)Jun 2010 Highly varied Some/Moderate
Mathematical Idol 2010Apr 2010 2 items predict the 3rd
(two person magic)
Tighter Ascertainments: Matching Interest RatesFeb 2010 Summand deduction using balanced ternary representations
The Boldgach ConjectureDec 2009 Matches from shuffled palindromic stacks
Poker-Faced Over the PhoneOct 2009 Small linear binary codes
(two person magic)
The Bligreath PrincipleAug 2009 Gilbreath twistSome/Easy
Two Summer Difference CertaintiesJun 2009 Sum-rich (Sidon) sets Some/Moderate
The Yummie Deal and VariationsApr 2009 Special deals Some/Easy
Esteem SynergismFeb 2009 Spelling miracles Little/Easy
What's Black and Red and Red All Over?Dec 2008 Binary de Bruijn sequencesSome/Moderate
Monge Shuffle CliquesOct 2008 Monge shuffle invarianceSome/Moderate
(A) Pi Evolved Set--Harmonic Split DrillAug 2008 Palindromic stack invariance Some/Moderate
Sum-Rich CirculantsJun 2008 Summand deduction from cycling sums Some/Moderate
Projective GeometryApr 2008 The Fano planeSome/Easy
Additional CertaintiesFeb 2008 Summand deduction via Fibonacci Numbers and Zeckendorf representationsSome/Moderate
Plurality Events, Standard Deviations and Skewed PerspectivesDec 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 OmegasAug 2007 Alternating triple sum forces Some/Easy
Gibonacci Braclets Jun 2007 Fibonacci sequences mod m Little/Easy
Magic Circles of Eight
Apr 2007 Magic circlesLittle/Easy
Quasi-Masked Forcing
Kind of Magic Squares
Feb 2007 Magic squaresLittle/Easy
Quantitative Reasoning in
Small Groups
Dec 2006 Z2, Z3, Klein-4 groupSome/Easy
Martin Gardner's Magic SpellsOct 2006Interview & 10-card pyramid spellerNone/Easy
The Second Norman InvasionAug 2006The Second Gilbreath principleLittle/Moderate
Better Poker Hands GuaranteedJun 2006Birthday card matches applicationsLittle/Easy
Bill Simon's Sixty-Four PrincipleApr 2006Two ways to force 4 of 8 cardsSome/Moderate
Many Fold SynergiesFeb 2006Parity principlesLittle/Easy
Luckation Is EverythingDec 2005Completing to 13 and advantages of knowing the 26th cardLittle/Easy
Subtraction Is AddictiveOct 2005Subtraction principlesLittle/Moderate
The First Norman InvasionAug 2005The First Gilbreath principleSome/Easy
A Little Erdös/Szekeres MagicJun 2005Any 5 has 3 in order
(two person magic)
The Down Under Deal
(aka the Australian Shuffle)
Apr 2005Down under deals and powers of 2Little/Easy
Fitch Four GloryFeb 20053 cards predict the 4th
(two person magic)
Sheer LuckDec 2004Elimination dealsSome/Moderate
Low Down Triple DealingOct 2004Triple deal principle/
quad run false shuffle

7 Oct 2010 talk on Mathemagic with Cards at Clayton State University (50 minute video).

Card Colm -- Celebration of Mind.

Since summer of 2020, Colm has given over a dozen public lectures in various cities on "The Mathematics Magic & Mystery of Martin Gardner" and the Gardner inspired Celebration of Mind events.

AJC guest Colm on The best friend mathematics ever had (21 Oct 2010).

Click on Martin Gardner for information and reminiscences of the man being honoured.

Click here for 90 second video clip of Card Colm in action in 2010 (filmed in June, released in September).

Card Colm debuted on 21 Oct 2004, at MAA Online, to mark the 90th birthday of Martin Gardner (1914-2010), and has appeared every two months since then. The Oct 2006 one includes an interview with Martin from seven months earlier.

Several mathematical card tricks -- including a new two-person effect based on (the first non-trivial case of) a well-known combinatorial result by Erdös & Szekeres -- were featured on the AMS's What's New in Mathematics site (October 2000).

See the table below for a guide to topics we've explored in Card Colm over the years.

The Order In the Ranks puzzle in the New York Times (10 May 2010) was a cardless reworking -- wrapped up in a convoluted plot line involved the President and Vice President of the USA as guests on the Daily Show -- of the Erdös/Szekeres trick mentioned above. (That five-card trick had also featured in the June 2005 Card Colm.) Some quick witted NYT readers, over time, figured out the solution (7 June 2010) .

Upon reflection, it seems that the above ranking trick was the second substantive Card Colm creation, which is a curious coincidence. Six months before the NYT ran the politicized version mentioned above, the first Card Colm trick (invented in early 1999) was presented in the NYT, on 7 December 2009, in the form of a puzzle in Magic Phone Calls.

(It's the second effect pulled off by the Wizard there, and unlike the first, it involves no hokey pokey whatsoever; the communication is entirely mathematical. It's the Fitch Four Glory (Feb 2005) four card twist on the classic Fitch Cheney Five-Card Trick, which was first published in 1950 as a telephone effect. We urge interested readers to try to solve it before clicking on the link below.)

A spirited discussion of the Magic Phone Calls effects ensued on the associated NYT blog, and eventually one clever reader figured out precisely how it could all be done. The beans were finally spilled a few weeks later, in the delightfully illustrated Wizard's Clock (24 December 2009).

A big thanks to Pradeep Mutalik for featuring these puzzles in the NYT!

In August 2009, Colm gave the MAA Lecture for Students at MathFest in Portland, Oregon, with a presentation entitled: "Mathemagic with a Deck of Cards on the Interval Between 5.700439718 and 80658175170943878571660636856403766975289505440883277824000000000000."

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),

An article called "An ESPeriment with Cards," combining the Erdös/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...

For a video of a lecture entitled, "Mathemagic with a Deck of Cards" at the University of Oklahoma (Norman) on 31 March 2008, click here (55 minutes, broken up into segments). Accompanying pdf here.

For a video of a lecture entitled, "New Mathemagical Principles Applied to Card Tricks" at the 2008, University of St. Thomas (St. Paul, MN) on 16 April 2008, click here (55 minutes, takes a while to load). Accompanying pdf here.

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.

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, and it was last updated on 21 Oct Aug 2010. Click here to return to main page.)

There were 25,000 visitors between 26 March 2005 and 26 March 2006, another 23,500 visitors between 26 March 2006 and 26 March 2007, and, it appears, another 10,200 visitors between 27 March 2007 and 27 March 2008. (The method of counting ``repeated'' hits from the same person seems to have changed. It was never supposed to do this, but the reported numbers dropped by 50% starting in the spring of 2007.)
Number of visitors since 27 March 2008: