CALCULATED TO AMUSE -- I love mathematical card tricks. There are those in which math is the underlying secret and/or there are those in which math is a key element of the plot itself. Some I perform for my friends who are engineers and scientists, some I perform for high school math classes, some I perform for anyone. I favor them because, unlike card tricks involving sleight of hand or gaffed decks, what happens in a mathematical card trick often really happens. I find it cool that the seemingly impossible (or at least highly improbable) result is, in fact, inevitable.

Even this propensity for such niche card entertainment didn't prepare me for how much I am enjoying Magical Mathematics, by Persi Diaconis and Ron Graham. Yes, there are some mathematical card tricks in the book, and good ones at that. Richard Hatch immediately inserted Baby Hummer, an in-audience-hands trick, into his Hatch Academy theater show. Steve Freeman's Royal Hummer is even stronger. Ronald Wohl creates a miracle with a cyclic-stacked deck. Gilbreath's Principle allows a spectator to shuffle the deck and then deal a winning hand to her partner along with a strong hand to the mark. Hummmer's three-card monte spawns multiple variations. Or you mail a deck to your victim, instruct him to cut and shuffle the deck multiple times, remember and bury the top card, shuffle again, why not, and return the deck. You identify the remembered card.

But it's not the tricks themselves that make this book so enjoyable. It's the symbiosis between math and magic, between mathematicians and magicians, and the similar creative and intellectual challenges that beset them.

Some of the math is dizzying. Like many of my friends in magic, I was a physics major, and so was exposed to a fair amount of calculus, differential equations, and statistics. Alas, this was of little help, for, as the authors point out, "Curiously, we know of essentially no magic tricks that lean on calculus." Instead, the reader enters the world of combinatorics. We are introduced to de Bruijn sequences (not Gray codes, as they have been mistakenly called in card trick literature), Eulerian circuits, Hamiltonian cycles, Steiner points, and other esoteric constructs. At places there are so many nested square roots and power functions that it would tax my word processor as well as my intellect just to place this stuff on the page. A highlight in the world of combinatorics was a Banff meeting in 2004 titled Generalizations of de Bruijn sequences and Gray codes, bringing together 25 or so top researchers. Its makeup, of experts convening on a focused topic, reminded me of Tamariz's Escorials or the recent Erdnaseum get together in Montana. It is simply fun to share cutting edge information with like-minded folks. Along the way, the authors leave many problems and proofs up to the student. Some of these are for your own amusement and edification; one of the others, if you solve it, will earn you a million dollars.

Title of a new Caleb Wiles trick?

Although there are two authors -- Ron Graham is a juggler and trampolinist in addition to being a mathematician -- it is Persi Diaconis's voice that I detect most often, and in many ways the book is a biography of Persi. Here is the thirteen-year-old future math professor encountering the binary number system via Elmsley's faro shuffle work. Here is the thirteen-year-old future magic legend receiving the gift of the five-volume Greater Magic from Lou Tannen. Here is the fourteen-year-old hitting the road with Dai Vernon. Here is the young math aspirant getting into Harvard with a testimonial from Martin Gardner.

Three guys you'd want on your team in Math Bowl.

The authors clearly enjoy their vocations and avocations, and they humanize the arcane worlds they walk in. A few highlights that caught my attention:

• The math behind faro shuffles, stay-stack shuffles, Monge shuffles, and Australian shuffles.
• A statistical estimate of how many magic tricks existed in 1584.
• A chilling classroom presentation on evil by students of Robert E. Neale.
• A surprising Stewart James comment on how many decks of cards he possessed.
• A big "So there!" to Martin Kruskal.
• Al Baker taking the wind out of a boring "math wizard."
• One hundred Henry Christ card tricks that Persi hopes to publish some day, one being a very fair ace assembly described in this book.
• A bittersweet comment by Martin Gardner on his hopes for a novel, leading to a pact between the authors that is as fine a philosophy of life as I have ever read.

Because Persi Diaconis has a reputation for closely guarding secrets, the publication of this book met with a certain degree of surprise, indeed prompting a Genii forum thread titled "I Never Would Have Believed This." Although there are secrets in the book, ferreting them out will be a far harder task than watching YouTube to find out how Liu Chien sticks his hand through a table. (Regarding secrets, Persi takes to task "a nameless cad" who forwarded a Marlo secret to Karl Fulves, who allegedly then published it without permission. About this sort of thing the author is also philosophical: "Of course, this also means his books have some good magic in them.") Magical Mathematics, far from being a casual exposé, is a book for readers who are already seriously in the grip of magic and/or mathematics or, even better, for intelligent prospects who are willing to be seduced. The authors do a fine job a conveying what life is like for a magician or mathematician, conveying their love of minutiae, of problem solving, and of the joy that solutions can lead to even more problems to be conquered. I was reminded of the early books of Robert Parrish, favorites that seduced me back in grade school.

All of this is wrapped up in an unusually well-written text and handsomely laid out, with color photo trick instructions; graphs; I Ching excerpts; letters from Stewart James, Henry Christ, and Charles Jordan; an image re the three-object divination trick from Bachet's 1612 French magic book; a program featuring a Henry Christ performance; publicity photos of Robert Hummer; excerpts from Marlo's notebooks, in his own hand; and lots of fascinating equations.

As I began above, I love math-based card tricks, and I'll close by describing a favorite, along with the math behind it. This has nothing directly to do with the review of Magical Mathematics, other than to indicate that, if you like this, you are going to love the book.

I first encountered this trick as Dai Vernon's Affinities, in More Lost Inner Secrets/The Vernon Chronicles, Volume Two, by Stephen Minch. The basic trick is much older.

Method and Effect: Holding the cards of a fairly shuffled deck facing you, in your left hand, begin thumbing off  cards into your right.  Suppose the first face card is a 6.  Then  thumb off some cards, silently counting 6-7-8-9-10-J-Q-K.  Slap this pile (8 cards) onto the table;  the 6 is  the top card of the face-down pile.  You don't count aloud, as you  don't want the spectators to realize just what you are doing.  You are just thumbing off a  "random" bunch of cards. Do this again, beginning with the next face card, always beginning with the face card value and counting through King.  For this trick,  the court cards are their actual values, not just 10.   Repeat this, placing various piles onto the table.  (If a King is the initial face card, count it off as a one-card pile.) Stop when you  finally don't have enough cards in your hand to complete an x-to-K run. Ask the spectator to roll a pair of invisible dice, then boldly announce that he has rolled a 10. The spectators then designate any three piles;  these are the piles you  will use.  All the other piles go back into the deck in your hand. Of the three piles on the table, ask  the spectator to select two, and you  turn the top card up of each of these two.  Let's say a 6 and an 8  appear.  Sum these getting 14, and then add the "10" from the dice.    That yields 24,  and you discard 24 cards off the top of the deck. (Again, a Jack  is an 11, a Queen is a 12, etc. when forming the sum.) Okay, you have a few cards left in your hand.  Count them.  Their  quantity will equal the value of the top card of the remaining pile.   If you have 7 cards, the top card will be a 7.  I find that  cool, counterintuitive. Remember, the deck is shuffled, any piles can be used, any two cards turned up.

Math: All you need for this is a smattering of algebra. The solution was provided in a letter from my friend Jim Held, a Caltech grad who lives in San Diego. I've altered his letter only a little to make certain values more clear.

Steve,
I kinda like the algebra that's involved.
Let N = the total cards left on the table at the end (in the three piles).
ND = the cards in the deck (in your hand) at the end.
A = the value of top card #1
B = the value of top card #2
x = the value of the unknown card
Then, ND = 52 - N.
But N = (14-A) + (14-B) + (14-x)
          = 42 - (A + B) - x
Therefore ND = 10 + (A + B) + x
or x = ND - (10 + A + B)
QED
Jim

Isn't math fun?

In Affinities, Vernon uses some amusing bluffing to also name the suit, with some mumbo jumbo to indicate why it has to be that suit. In fact, you needn't bluff to name the suit, as you looked at the piles as you formed them. You know what the suits are (if your memory is better than mine). If you wish, you could place the piles at a slightly different angle for each suit or in slightly different quadrants of the table to assist your memory. (The invisible dice bit was my own idea. You can of course concoct other excuses for why you need to add 10.)

Getting back to the book, I thoroughly enjoyed it, and it is unlike virtually any other magic book that I own. It certainly meets the authors' goals as stated at the end of the Neat Shuffles chapter: "We hope the reader finds some good tricks and enjoys the mathematical mortar as well." Hardback from Princeton University Press, foreword by Martin Gardner, 244 pages, $29.95.