This is a collection of mathemagical card effects with an underlying mathematical explanation. Colm Mulcahy realizes with this book a suggestion that Martin Gardner made to him many years ago. Many of the effects have been described in his blog Card Colm sponsored by the MAA. The mathematics are based on counting, combinatorics and sometimes a bit of probability that applies to the particular constellation of a (sometimes carefully chosen subset of a) card deck. All aspects are enlightened: how to perform the trick, why and how does it work, what mathematics are involved, what are the variations and generalizations, what mnemotechnical tricks can be used, what is the origin, etc.
Colm Mulcahy has Irish roots, with a MSc from the University College Dublin, but he got a PhD from Cornell, and he is teaching at the Math Department of Spelman College, Atlanta (GA) since 1988. He regularly contributes a math column to The Huffington Post, but he is foremost an enthusiastic user of internet media. He has a Math Colm blog, and on his Card Colm website he has many interesting links, for example to material related to his much admired hero Martin Gardner, but most of all to the items of his bimonthly Card Colm blog, that is sponsored by the MAA. In the latter he discusses mathematically based effects (he avoids using the word 'tricks' for some reason) you may obtain by manipulating a card deck. His website has extra videos illustrating the manipulation of the cards, and of course a link to the book under review. This book is the eventual realization of a suggestion made to him by Martin Gardner, many years ago.
So here it is: a book presenting 52 effects arranged in 13 chapters, with 4 sections each, a numerological predestination for a book about card tricks. The actual contents is however a bit liberal with these numbers. There is for example an unnumbered appetizing chapter that also introduces some terminology and card shuffling techniques, and there is a coda that gives some additional information. Also each chapter may have the four `suit'able variations, but they often consist of several introductory sections as well. The effects get some inspiring names like `Full House Blues', `Easy as Pi', or `Twisting the Knight Away' etc. The effects are usually introduced in several steps: 'How it looks' is just describing what an uninformed observer will experience, then `How it works' is explaining what the mathemagician actually is doing, later a section `Why it works' is explaining in more detail what is going on. This is often followed with different options on how to present the magic to the public. Also the origin of the method is referred to, which, no wonder, is often one of his Card Colms blogs. The chapters end with some `Parting Thoughts' which are further elaborations of the previously introduced principles, sometimes in the form of exercises: `prove that...', `what if...', `how to...'.
Mulcahy also gives his effects some Michelin-like ratings in the margins. It can have one ♣ (easy) up to four ♣♣♣♣ (difficult) clubs for mathematical sophistication, similarly hearts grade entertaining value, spades are used to rate the preparatory work needed, and diamonds refers to the concentration and counting that is needed during the performance.
The effects are based on underlying mathematical principles. There are many of them, and most are believed to be original in their application to card magic. Since these principles are introduced but also re-used at several instances, they get some mnemotechnic names of which some recurrent ones are the COAT (Count Out And Transfer) or a generalized overCOAT or underCOAT, and TACO is some kind of inverse, not to be confused with a CATO (Cut And Turn Over). These give rise to neologisms like `minimal underCOATing' of `Fibbing' when it concerns Fibonacci numbers. TOFUH stands for Turn Over And Flip Under Half, sounds healthier than the alternative FAT (Flip And Transfer). In many cases, the order of the suits is important. So the deck can be in cyclic CHaSeD (Clubs, Hearts, Spades, Diamonds) order. And there are many other mnemotechnical tricks to recall certain orders of cards or operations. These witty namings and word plays make the text fun to read.
Since Mulcahy stresses at many places that the mathemagician should not reveal the mathematics underlying the magic, so neither will I uncover them here in this review. As he writes: The best answer to the question `How did you do that?' is to say `Reasonably well, I think'. Unless the audience is really interested in the mathematics, it is unwise to explain what is going on. Otherwise comments like `So is this all you did? It is just mathematics', will kill all the magic. It is absolutely rewarding however to convince young people that mathematics is everywhere and can be fun to play with. So it is perfectly all right if a teacher explains the mathematics to his pupils.
In most cases, the mathematics are not that advanced. Just counting will do, but it is not just plain combinatorics or modulo calculus. The fact that there are 13 different faces and 4 different suits make the counting special. Fibonacci numbers sometimes play a role, occasionally there is some probability involved. The effect may be for example to bring by some seemingly random shuffling a card from the bottom of the deck to the top. Or if the mathemagician is given the sum of the values of 2 or 3 cards that a spectator has randomly selected from a prepared deck, he will be able to name the faces of these cards. As one reads along, the effects become more involved, hence more complicated to perform, but they will have a higher magical alloy and thus be more rewarding. Some involve an accomplice to assist the magician.
One word about the typography. The numbering of the chapters is with cards (A for the first, 2 for the second,..., K for the 13th) and the effects within the chapters are numbered like 6♣ for the first one in chapter 6, or K♠ for the third one in chapter 13 (CHaSeD order). Printed on glossy paper with many colour illustrations, it is not only fun, but also a pleasure to read. The apprentice magician will have a lot to practice on but even the professional magician will find many things to think about while mastering this wonderful calculus of the card deck.