Mathematics, Magic and Mystery

by Martin Gardner

Why do card tricks work? How can magicians do astonishing feats of mathematics mentally? Why do stage "mind-reading" tricks work? As a rule, we simply accept these tricks and "magic" without recognizing that they are really demonstrations of strict laws based on probability, sets, number theory, topology, and other branches of mathematics.

This is the first book-length study of this fascinating branch of recreational mathematics. Written by one of the foremost experts on mathematical magic, it employs considerable historical data to summarize all previous work in this field. It is also a creative examination of laws and their exemplification, with scores of new tricks, insights, and demonstrations.

Dozens of topological tricks are explained, and dozens of manipulation tricks are aligned with mathematical law.

Nontechnical, detailed, and clear, this volume contains 115 sections discussing tricks with cards, dice, coins, etc.; topological tricks with handkerchiefs, cards, etc.; geometrical vanishing effects; demonstrations with pure numbers; and dozens of other topics. You will learn how a Moebius strip works and how a Curry square can "prove" that the whole is not equal to the sum of its parts.

No skill at sleight of hand is needed to perform the more than 500 tricks described because mathematics guarantees their success. Detailed examination of laws and their application permits you to create your own problems and effects.

Genres: Mathematics, Nonfiction, Magic, Science, Puzzles, Games, Psychology

208 pages, Paperback

First published January 1, 1956

About the author

Martin Gardner was an American mathematics and science writer specializing in recreational mathematics, but with interests encompassing micromagic, stage magic, literature (especially the writings of Lewis Carroll), philosophy, scientific skepticism, and religion. He wrote the Mathematical Games column in Scientific American from 1956 to 1981, and published over 70 books.

Reviews

[Yuvaraj kothandaraman · Rating 4 out of 5]

"Mathematics, Magic and Mystery" by Martin Gardner (published in 1956) is a treasury of over 115 mathematical tricks and puzzles that blend conjuring with mathematical principles. Rather than requiring sleight of hand or misdirection, these tricks operate through pure mathematical logic, making them reproducible by anyone willing to understand the underlying principles, regardless of magical skill.

Gardner's genius lies in demonstrating that mathematics itself is the magic.
A card trick doesn't need secret manipulation if the structure of numbers does the work for you. For example, "Findley's Four-Card Trick" uses four cards arranged in a doubling series (1, 2, 4, 8), which can combine to create any total from 1 to 15. Any card the spectator names, you can remove these four specific cards from your pocket and combine them mathematically to represent that card's suit and value.

The book opens with an important observation: mathematical magic is often dismissed by both mathematicians (who consider it trivial play) and magicians (who consider it dull magic). Gardner acknowledges this fairly, these tricks won't hold a non-mathematical audience spellbound. But for people who enjoy both conjuring and mathematical recreations, they're endlessly fascinating.

The first two chapters focus on card tricks, organized by principle rather than difficulty. Some tricks use cards as counting units (like "The Piano Trick," where pairs of cards between fingers reveal a hidden single card through pure arithmetic). Others exploit numerical values, division of colors and suits, or the front/back properties of cards.

One of the most elegant is "The Five Poker Hands." Five hands of five cards are dealt. Each player selects a card. The hands are gathered and redealt. The magician can now instantly identify each selected card from any pile based on which spectator owns which position. The magic is pure mathematics: if spectator number two's card is selected, it will always be in the second position of the fan when that hand is shown. The intersection of two perpendicular organizing principles (which hand, which position) determines everything.

Another standout: "The Missing Card" trick. While your back is turned, someone removes a card from the deck and pockets it. They shuffle the remaining 51 cards. You deal them face-up one at a time, keeping a running total of card values (Jacks=11, Queens=12, Kings=0, others face value). Since all 52 cards total 312 points, you simply subtract your running total of the 51 dealt cards from 312 to find the missing card's value. You can determine the suit with a second pass. This is shocking because it seems impossible to know anything without seeing the cards first.

Chapters on geometric vanishes explore mathematical paradoxes, how rearranging parts of a figure can make objects appear or disappear. The principle is "concealed distribution." When you cut a checkerboard along a diagonal and shift one piece, squares appear to vanish or multiply. The reason: tiny deviations in area from each piece's redistribution add up to exactly one missing square. Gardner explains Sam Loyd's famous "Get Off the Earth" puzzle where a Chinese warrior vanishes when a disk is rotated.

These aren't just visual tricks, they're mathematical proofs in disguise. The Curry Paradox shows how two puzzles can have different areas while appearing identical. Part of the brilliance is that Gardner provides the mathematical formula for creating such paradoxes. If two parts have areas with no common denominator (prime to each other), you can construct a paradox where one unit vanishes.

Chapters on topological effects use rope, string, handkerchiefs, and rubber bands. The Afghan Bands demonstrate that a single twisted band cut down the middle doesn't produce two separate bands,it produces one longer twisted band. Cut it again and you get two interlocked loops. This tricks people's spatial intuition because we expect symmetry where mathematics provides something counterintuitive.

The chapters on pure numbers are particularly satisfying for mathematics lovers.
Gardner explores the "mysteries of nine", how any three-digit number reversed and subtracted from itself produces a number with 9 in the middle position, and where the end digits add to 9. More remarkably, if you scramble a number's digits, reverse the new number, subtract, and add that result to the original, you always get 1089. This constant outcome creates countless prediction tricks.

The concept of "digital roots", repeatedly adding a number's digits until one digit remains, reveals hidden patterns. All multiples of 9 have digital roots of 9. This means if a spectator performs a series of seemingly random operations designed to produce a multiple of 9, you can have them circle any digit in the result and call out the others. Since the total should be 9, you subtract their called digits from 9 to find the circled digit.

The Gergonne Pile Problem is mesmerizing: a spectator thinks of a card in a 27-card packet. You deal into three face-up piles. They tell you which pile has their card. You repeat twice more. After three deals, you can name the exact position of their card (1-27), bring it to any position they specify beforehand, or directly name the card. The trick works through a process of narrowing,each deal reduces 27 possibilities to 9, then 9 to 3, then 3 to 1. The mathematics is pure elimination.

Gardner writes with clarity and genuine delight. He credits original creators, provides historical context, and explains not just how to perform each trick but why it works mathematically. He notes that Charles Sanders Peirce, the American logician and philosopher, created card tricks based on "cyclic arithmetic" in 1860,showing that mathematical tricks have scholarly pedigree.

The book opens doors to a vast field. Gardner writes that mathematical magic is "a field in its infancy" and invites readers to invent new effects. He provides the principles so readers can understand how to create their own tricks rather than just memorizing procedures.

The organization is logical. Tricks build on previous principles, you understand the simpler applications before tackling complex ones. By the end, readers grasp the underlying mathematics deeply enough to improvise variations.

The book requires mathematical comfort. While Gardner explains everything clearly, readers without interest in or aptitude for mathematics will struggle. The section on trigonometry in measuring card positions requires understanding spatial relationships and numerical sequences.

Some tricks genuinely do take a long time to perform. Gardner acknowledges that the Gergonne Pile Problem, while mathematically elegant, requires three dealing rounds. For entertainment value, it's weak spectators aren't wowed by mathematics; they're bored by repetition. Gardner honestly states these tricks won't captivate non-mathematical audiences.

One delight is Gardner's attribution. He credits Paul Curry, Stewart James, Charles T. Jordan, Bob Hummer, Sam Loyd, and many others. These weren't mathematical recreations invented by mathematicians, they were magicians who discovered mathematical principles. Gardner acknowledges that magicians, not mathematicians, drove innovation in mathematical magic during the 20th century.

If you're a recreational mathematician or puzzle enthusiast, this book is invaluable. If you're a magician seeking self-working tricks (tricks that don't require sleight of hand), it's essential. If you want to understand how mathematics creates seemingly impossible outcomes, it's illuminating.

My Rating:4/5 starts

[Nicholas Ball · Rating 5 out of 5]

Mathematics and magic are usually a tedious affair ("spelling" card tricks or the unendurable "21 card trick" are the worst offenders), and after an early chapter acknowledging them, Martin moves on to a lot of richer material. (He briefly lists some of his favourite tricks in "Scarne on Card Tricks" as well, which is an excellent collection of self working card tricks for any magic curious readers)

I won't replicate Martin's impressive bio here but he was a tireless mathematical reader and researcher - and fewer people knew he was also very interested in magic as well - so this is a wonderful book for him to write.

The most interesting (for my money) part of the book is the section on what Gardner calls the Principle of Concealed Distribution where a 2d diagram containing a number of drawn items (or grid-squares) will end up having a different amount after sliding some pieces around. Some good amusement for teachers or parents wanting to baffle children (Martin includes an Easter example of his own where one of some bunnies disappears and is replaced with an egg), but be warned the section includes the most famous example being Sam Loyd's original Get Off The Earth diagram (complete with disappearing "Chinaman").

The last few segments of the book have other curiousities (modular maths, casting out nines etc) which might not be new concepts to readers but give rise to some interesting ideas for tricks.

All in all a mathematics book first, but also a good tour of mathematical magic concepts as well (Martin specifically lists original magicians or individuals who worked on concepts with the diligence and honesty of a true scholar).

[Robert Lewis · Rating 4 out of 5]

I've spent many years studying both magic and mathematics (separately and together). When it comes to joining these fields together, there have been a number of important individuals who've made a number of important contributions, but Martin Gardner is the grandfather of us all. This book represents a treasury of over a hundred mathematical magic tricks, more than enough to keep the reader busy for years.

The wonderful thing about this book is its scope. Despite being a relatively short book, it covers a lot of territory, featuring mathematical magic tricks using a wide range of props ranging from mere numbers themselves to playing cards and ropes and much more. If you want the largest collection of mathematical trickery I've ever seen conveniently presented in a single reference, this is the book for you.

However, if you're looking for a detailed exploration of the mathematics behind those tricks, you may be disappointed. While Gardner's other works clearly demonstrate that he was up to the task, he didn't provide much mathematical depth in this book. It's a rare page that even contains a single equation. The result is a book that, though it covers mathematical ground, is much better suited to the amateur magician than to the amateur mathematician.

[Zebulon Fross · Rating 5 out of 5]

A surprisingly fun and thorough description of easy magic tricks

The book started out being tedious because the tricks are explained from start to finish, so you just have to imagine in your mind the trick being done to know what is going on. But stick with it because this book is incredible. There is such a huge variety of fun tricks in here, card tricks, number tricks, handkerchief tricks, and on and on.

[Angel L. Jubete · Rating 2 out of 5]

De lo peor del autor, no es interesante ni desde las matemáticas ni desee la magia. Me ha defraudado.

[Volodymyr Andrusenko · Rating 5 out of 5]

One of a kind puzzle book. The book is well organized and thought out by the author. It has great illustrations as well. Very entertaining and stimulating, excellent mix of puzzles and magic tricks (I would say, logical tricks as well). Fun read. 5 starts

[Lost Planet Airman · Rating 4 out of 5]

Excellent join of mathematics and magic tricks for the adult crowd.