and Much MoreMaster of Recreational MathematicsAn Interview with Martin Gardner

By Don Albers

On October 21, Martin Gardner cel-ebrated his ninetieth birthday. For 25 ofhis 90 years, Gardner wrote the monthly"Mathematical Games " column for Sci-

entific American. His columns have in-spired thousands of readers to learnmore about the mathematics that heloved to explore and explain. Among hiscolumn correspondents were several dis-tinguished mathematicians and scien-tists, including John Horton Conway,

Persi Diaconis, Ron Graham, Douglas

Hofstadter, Richard Guy, Don Knuth, Sol

Golomb, and Roger Penrose.

he has produced more than 60 books,most still in print; many have been

bestsellers. His Annotated Alice has soldover a million copies, and the 15 volumescollecting his "Mathematical Games"columns have gone through severalprintings. All 15 volumes have been digi-tized and will soon be published by theMAA on a single CD entitled MartinGardner's Mathematical Games.

a few local magicians in Tulsa, LoganWaite and Wabash Hughes, who workedfor the Wabash Railroad.

DA: At what age did this occur?

MG: I was a high school student at thetime. I've never performed magic; it's just

been a hobby. The only time I got paid

for doing magic was when I was a stu-

dent at The University of Chicago; I usedto work at the Marshall Field's depart-ment store during the Christmas seasondemonstrating Gilbert magic sets. Ilearned a lot from the experience. That

was the first time I realized that you're

really not doing a magic trick well until

you've done it in front of an audience

In his ninetieth year, he has returned toOklahoma, where he was born. He is in

good health and full of energy. We look

forward to more from him as he begins

his second 90 years. What follows is asmall portion of an interview done atGardner's home in Hendersonville, NCin the fall of 1990 and spring of 1991.

Gardner's columns have earned him a

place of honor in the mathematical com-munity, which has given him many

awards. But he has always declined invi-

Martin reading on his front porch at age

15,1929Martin at age 10, 1925.Martin and his younger brother

Jim, 1920.

about a hundred times. Then it becomessecond nature, and you know what to say.tations to accept awards in person, on the

grounds that he is not a mathematician."I'm strictly a journalist:' he insists. "Ijust write about what other people aredoing in the field." His modesty is admi-rable, but we insist that he is far morethan a journalist.

Don Albers: As a high school student youwere already writing articles for The

Sphinx, a magazine devoted to magic.Does your interest in magic go back toyour father?

DA: What are the elements of a success.ful magic trick?

MG: The most important thing is tostartle people, and have them wonderhow it's done. Close-up magic that youdo on a table right in front of people isvery different from the stage illusions

that David Copperfield does. It's close-up magic that most intrigues me, espe-

Martin Gardner: Magic wasn't a specialhobby of his, but he did show me somemagic tricks when I was a little boy. Ilearned my first tricks from him, in par-ticular one with a knife and little piecesof paper on it. I then got acquainted with

In addition to his massive contributionsto mathematics, Gardner has writtenabout magic, philosophy, literature, andpseudoscience. Over his first ninety years,

4

cially when it has a mathematical flavor.I did a book on mathematical tricks thathas, for example, a chapter on topologi-cal tricks. I did two massive books for themagic profession: The Encyclopedia ofImpromptu Magic and Martin Gardner

Presents. The first book covers tricks thatdon't require any special equipment. Alot of them are just jokes and gags of the

type 'bet you can't do this:

to read that way. It was very embarrass-ing when I was in first grade, because theteacher would hold up cards that said 'cat'and 'dog' and I was always the first to callout the word. She had to tell me to shut

MG: I was very good at math in highschool. In fact, it and physics were theonly subjects in which I got good grades.I was bored to death by the other classes.I flunked a class in Latin and had to takeit over. I just don't have a good ear for

languages.

DA: You got your B.A. in 1936, thenworked briefly for the Tulsa Tribune as a

reporter, and then came back to The

University of Chicago to the PR office

writing news releases (primarily science

releases), and took a graduate coursefrom Carnap. What else did you do untilthe outbreak of World War II?

DA: Your book Mathematics, Magic, andMystery has been a bestseller for many

years.

MG: I waste a lot of time on magic. DaiVernon was one of the great inventors of

magic. He was a great influence on Persi

Diaconis. Persi traveled with Dai for a

long time. I knew Vernon very well. I

knew Persi when he was a student at

NYU. You probably heard the story how

he got into Harvard.

MG: I had various jobs. I worked as a caseworker for the Chicago Relief Adminis-

tration, I had to visit 140 families regu-larly in what was called the Black Belt. Ialso had several odd jobs: waiter, sodajerk, etc. Remember, this was at theheight of the Great Depression.

DA: As I recall, he gave you some creditfor writing a letter of recommendationto Fred Mosteller, the Harvard statisti-cian.

DA: In December of 1941, the U.S. en-tered World War II and you enlisted inthe Navy.

Gardner as a navy sailor, 1941

MG: Mosteller is a magic buff. WhenPersi said he wanted to get into Harvard,I wrote to Fred and said that Persi cando the best bottom deal and second dealof anybody I know, and that got him intoHarvard. I talked to Fred on the phoneabout it and he said, "Is he willing tomajor in statistics?" And Persi said surehe'd major in statistics if that would gethim into Harvard. So he went up toHarvard, and they had a session together,maybe doing card tricks. Mosteller gothim into Harvard.

MG: I ended up serving on DE 134, a

destroyer escort, in the Atlantic. I was

miserably seasick for about three days,and then I was never seasick again. Icouldn't wait for the war to end, but laterI looked back at it as a rather pleasurable

time of my life. You're on a ship, you

make friends with your shipmates, you

got liberties now and then, and you didn'thave to worry about anything.

I've had migraine headaches all my life

that were fairly severe when I was in high

school. When I enlisted in the Navy, I didnot list my migraines because I was afraidthey wouldn't take me. I feared that Imight develop migraine headaches dur-ing battle situations. We were part of aso-called "killer group" of six destroyerslooking for German submarines. Duringmy four years in the Navy, I never had amigraine headache. I'm convinced that

they're associated with periods of anxi-

ety. When you're in the Navy, you don't

worry about what you're going to do to-morrow, what tie to put on, etc. You justfollow orders. In a way, you have a bigsense of freedom. Otherwise, I have noother explanation.

DA: What did your mother do?

MG: She was a kindergarten teacher be-fore marriage, but then became a house-

wife, caring for three children. Her hobbywas painting, and I have a number of herpaintings hanging in the house. Both ofmy parents lived into their nineties. I hada brother and sister, both younger, whoare deceased.

Martin Gardner with the Mad Hatter inCentral Park, New York City.

up, to give the other children a chance tolearn how to read.

I learned to read before I went to school.My mother read The Wizard of Oz to mewhen I was a little boy, and I looked overher shoulder as she read it. I learned how

DA: As a kid, do you remember otherstrong interests in addition to magic?

5

Martin and grandson Martin.Gardner with his wife Charlotte, andtheir two sons Jim, left, and Tom.

Martin Gardner with his brother Jim and

sister Judith.

ematician who has to teach a course in

mathematics, and then write. To me, it's

hard to imagine how a professional

mathematician would have time to evenwrite a book. I had nothing else to do,except research for those columns, andwrite them up.

York City, because for writers that'swhere all the action is. I had a friend whoworked for Parents' Institute, and whowas in charge of their periodicals for chil-dren. They were starting a new magazinecalled Humpty Dumpty, and were look-

ing for activity features, where you foldthe page or stick something through thepage, or cut; where you destroy the page.So he hired me to do the activity featuresfor Humpty Dumpty, as well as a short

story for every issue and a poem of moraladvice.

DA: At the end of the war, you promptlywent back to Chicago.

MG: Yes, I went back, and I could havehad myoid job back in the public rela-tions office at The University of Chicagobecause there was an understanding thatif you enlisted in the service you couldget your old job back. But the one rea-

son I didn't go back to the PR office was

that I sold a story, my first sale, to Es-

quire. The title of the story, "The Horse

on the Escalator;' came from a joke go-

ing around at the time about a man who

entered Marshall Fields department store

on a horse, and the elevator operator toldhim he couldn't take the horse on the el-evator.And he said,"Butlady, he gets sickon the escalator!" It was a shaggy dog jokeabout a horse. The story is about a man

who collected horse jokes, and his wife

didn't think any of them were funny, but

she laughed heartily every time he told

one to conceal the fact. So that was myfirst story. I decided that maybe I couldmake a living as a freelance writer, and Ivery quickly sold Esquire a second story,and that was the "No-Sided Professor;'about topology.

DA: Most people that I've ever talked toabout your Scientific American columns

roo,' that it was your job, but they're stillawe! I 'by the fact that you turned out

something really sparkling every month.It's one thing to write something everymonth, but that doesn't mean that it'sgoing to be inspirational or great fun toread each time.

DA: Your work with children's magazineswent up to about 1956. By 1957 you wereat Scientific American. So there was notmuch of a hiatus between HumptyDumpty and Scientific American. MG: I miss doing those columns, they

were a lot of fun, and I met many fasci-

nating people while doing them. Once

the column got started I began hearingfrom people like Sol Golomb and John

Conway, who were really doing creativework that had a recreational flavor. Thatkept the column going. It became much

more interesting after I began gettingfeedback from people like Conway, Ron

Graham, Don Knuth, and many others.

MG: No, I stopped working for HumptyDumpty to start "Mathematical Games"

at Scientific American. I couldn't do both.It started with the sale in December 1956,

of an article on Hexatlexagons. That was

not a column, but that led to the column.

When Gerry Piel, the publisher of Scien-

tific American, called me and suggested

the column. That was when I resigned

from Parents.

Probably my most famous column wasthe one in which I introduced Conway'sgame of Life. Conway had no idea when

he showed it to me that it was going totake off the way it did. He came out on avisit, and he asked me if I had a Go board.I did have one, and we played Life on theGo board. He had about 50 other thingsto talk about besides that. I thought that

DA: That had to give you a lot of confi-dence, helping to convince you that youcould earn a living as a writer.

DA: A lot of people are astonished thatanybody could turn out one of those col-umns on mathematical games and rec-reations every single month for ScientificAmerican.

MG: That's right, but Esquire changededitors after I had sold them several sto-

ries. The new editor had a differentpolicy, and he didn't care for the kind ofstories I was writing. So I moved to New

MG: Perhaps they don't realize I had noother job. I'm not a professional math-

6

Life was wonderful - a fasci-nating computer game. When I

did the first column on Life, itreally took off. There was evenan article in Time magazine

about it.

DA: You've read a lot of contem-porary material, and you've reada lot by those who have beengone a long time. Are there anyof those departed people that

you'd like to sit down with over

dinner, or sit down here in your

library and chat with them?VA: Can you tell me a little bit

more about how you actually

approach writing? You previ-

ously said something about how

you did your monthly columns

over a long period of time. Youwrite about many other things

as well. Do you have a differentstyle or a different mode whenyou write about pseudoscience?

MG: I'd love to chat with Godelfor example. He had some

strange cosmological views, and

I'd like to talk to him about that,about time travel into the past. Inever could quite understandthat. And of course he was a

dedicated Platonist. He thought

all of mathematics was out

there, including the transfinitenumbers. I'd enjoy talking tohim about that. Of course I'dlove to talk with Einstein andNeils Bohr. Among puzzle mak-ers, I'd most want to talk withHenry Dudeney and Sam Loyd.I also would enjoy talking toBertrand Russell. He's one ofmy heroes.

~

ii~

Gardner doing some table magic

MG: I don't think so. I've neverworried about style. I just write asclearly as I can, and I suppose it'simproved over the years. I get in-terested in a topic, and I do asmuch research as I can on it. I havemy library of working tools, so Ican do a lot of research right here.I usually rough out the topic first,just list all the things that I have tosay, and then I sit down and try toput it together on the typewriter.It's all kind of a sequence that ishard to explain. It comes easy forme, I enjoy writing and I don't suf-fer from writer's block, where I sitand wonder for an hour how I'mgoing to phrase the opening sen-tence.

DA: Here's an equally easy ques-tion for you. Once you've de-parted this life, let's suppose youhad an opportunity to comeback in a hundred years. Whatquestions would you most wantto know the answers to that

might have been developedduring that time?

DA: Which of your more than sixtybooks is in some sense a favorite? MG: I guess I'd be interested to

know if various famous un-

solved problems had beenI solved, such as the Goldbach

~ Conjecture. But I don't have any

Martin Gardner's form letter, often sent as a response to re- great desire to come back andQuests he received from readers. learn what modern mathemat-

ics is up to. You're giving me

credit for being more of a mathemati-

cian than I really am. I'm strictly a jour-

nalist. I just write about what other

people are doing in the field.

MG: I think my Whys of a Philo-sophical Scrivener is my favorite be-cause it is a detailed account of ev-erything I believe.

DA: Let's move back to math for a . - --

minute. You've lived long enough now tosee a lot of really interesting mathemati-cal ideas hit the scene, and there are alsosome really beautiful ideas that were herelong before you were on the scene. First,during your own lifetime, what ideas,what discoveries just kind of knockedyour socks oft?

physics, and in particular the develop-ment of superstring theory. That cameas a complete surprise to me. It's a beau-tiful theory of particles, and it mayormay not be true, but it's the hottest thingin town now in particle physics. It opensup the possibility that higher dimensionsare not just artifacts but actually real.

MG: Well, I think the most interesting de-velopments are mainly in mathematical

Thanks to Jim Gardner for supplying thephotos that accompany this interview.

Low Down Triple Dealing

By Calm Mulcahy

Dedicated to Martin Gardner on the occasion of his 90th birthday

times) on a packet of thirteen cards,which was then split into two piles. Drawattention to the two cards originally se-lected. Say, "Wouldn't it be surprising if,after all that triple dealing based on thevalues of two randomly selected cardsfrom a shuffled deck, there were cards

intimately related to the two you selectedat the bottoms of the two piles now onthe table?" Have the piles on the tableturned over: one of the cards exposed is9+ and the other is 4.. "A curious align-ment with the selected cards."

Consider the following three demonstrations of mathemagic:

word being spelled out and the size ofthe "quarter deck" which the spectatorstarts with, and the fact that you mustsomehow know the identity of one cardin the spectator's hand from the begin-

ning! It should come as no surprise thatthe card in question is the bottom card:asking the spectator to hold the cards inher hand in preparation for the spellingis just to give you an added opportunityto peak at this card, if you haven't alreadydone that as she completed her shuffling.You must do whatever it takes to discoverthat card's identity!

1. A deck of cards is handed to a specta-tor, who is invited to shuffle freely. She isasked to callout her favourite ice-creamflavour; let's suppose she says, "Choco-late." Next, she is asked to cut off about aquarter of the deck and hold it ready fordealing. You take another quarter of thedeck and demonstrate a spelling deal,dealing cards into a pile, one for each let-ter in the word "chocolate;' before drop-ping the rest of your quarter deck on top.Set those cards aside and have the spec-tator perform this spelling routine threetimes with the cards in her hands. You

correctly name the top card in her pile at

the conclusion of her triple dealing.

3. Have each of three volunteers in turnpick a card at random, and then have thecards returned to anywhere in the deck.

Shuffle with abandon. Ask a fourth per-

son to name their favourite magician.

Assume they say, "Harry Houdini." Hold

the deck in the right hand, and peel cardsoff the bottom into a pile in the left hand,without altering their order, one for eachletter, as you spell out the whole name.Hand the stack of twelve cards to the firstvolunteer and ask him to spell out

HOUDINI while dealing out seven cards,

then dropping the other five on top. Now

give the cards to the second volunteerand give the same directions, and finallyto the third volunteer for one last deal ofthe same type. Take the cards behindyour back and immediately producethree cards, handing one to each volun-teer face down. Have the chosen cardsnamed, as they are turned over, to revealthat vou have correctly located each one.

This is the scoop on the ice cream trick:

Claim 1: Start with n cards, the bottomone of which is known. If k cards are dealtout into a pile, thus reversing their or-der, and the remaining n - k cards are

dropped on top as a unit, and this typeof deal is repeated twice more, then theknown card rises like cream to the top- provided that n ~ 2k.

2. A deck of cards is handed to two spec-

tators, each of whom is invited to shuffle

at will and then choose a card (of not too

Iowa value) and place it face up on the

table. Let's suppose that 4+ and 9. areselected and displayed. You run throughthe deck face up, tossing out all of theAces, 2s and 3s - saying, ('Sorry, I should

have eliminated the low cards earlier."

Then riffle shuffle a few times. Remark,

"Since a 9 was selected, let's count out

nine cards:' dealing into a pile on thetable. Shuffle overhand and continue,"We'll need four more:' as you peel offthat many cards as a single unit, withoutchanging their order. Drop these on topof the other nine. (The rest of the deck isignored from now on.) Pick up this pileof thirteen cards and demonstrate a dealof the nine top cards into a pile, revers-ing their order, and then putting the re-maining four on top. Have the first spec-tator do this deal three more times, andhand the cards to the second spectator.Have the second volunteer deal eitherfour or nine cards into a pile, with theremainder placed beside this to form asecond pile. Recap: the two numbers (4and 9) being used were determined byfreely selected cards, and as a result a dealof nine cards was performed done (three

In the case of the nine-letter wordCHOCOLATE, the trick works providedthat the portion of the deck selected bythe volunteer contains at most eighteen

cards. If MINT CHOCOLATE CHIP

(seventeen letters) is named, you'll ask

for between a third and half of the deck.(If RUM is selected, try to force RUM

RAISIN!)

The triple deal described is actually 75%of a rather interesting quadruple deal.This is the real scoop:The same purely mathematical principle

underlies each of these demonstrations,with a little more magic thrown in for

good effect as we progress to the second

and third tricks. We gradually reveal this

principle below, and discuss how each of

the tricks is done as we go, before finally

explaining why the principle works.

Claim 2: Start with n cards, and assume

that n ~ 2k ~ 2n. If k cards are dealt outinto a pile, thus reversing their order, andthe remaining n - k cards are dropped

on top as a unit, and this deal is repeatedthree more times, the entire packet of ncards is restored to its original order.

Let's start with the first effect. There aretwo secrets working behind the scenes foryou here: an unadvertised but importantrelationship between the length of the

Now consider the second effect above.Two cards (not Aces, 2s or 3s) are chosenand set aside face up. Let's suppose they

8

the cards}. Actually, it's easy to see in allcases: Suppose for the sake of concrete-ness that n = 13 and k = 8. Let's agree to

represent a pile of thirteen cards in a par-ticular order by a sequence of gray-scale

panels in decreasingorder of brightness,from white for thetop card to black forthe bottom card, asdepicted in the im-age.

Have these cards returned, one at a time,to the deck and then control them to thebottom - this means that you appear toallow free choice of where to put the

cards, but you actually use elementary

Then the results ofthe four deals - each

of eight cards into apile with the otherfive dropped on top- is given by the suc-

cessive images in the

pictures.

~

Since the last imageshows a fully restoredpile, the deal in ques-tion has period 4: af-ter four deals we arealways back to where

we started. After

three such deals, theoriginal bottom card (black) has risen tothe top - in preparation for its finaljourney back to the bottom under onemore deal. Moreover, it is clear that theeight bottom cards become the eight top

cards, suitably reversed, after three deals.There are just three portions of the packet- of sizes 5, 3 and 5 here - to keep track

of: and they move around intact, subjectat most to some internal reversals. Theonly relationship between 13 and 8 whichis needed to make this sequence of im-ages totally generalizable is the fact that8;?: 13/2.

are 4. and 9.. As you run through thedeck face up, ostensibly to toss out thelow valued cards, what you really focuson doing is cutting the 9. and 4. to thetop and bottom respectively. They will

stay there if you are

careful how you

riffle shuffle. Con-tinue as describedearlier: reversingnine cards into a pileand then doing

some overhand

shuffling whose

purpose is to bringthe bottom card tothe top. Peel fourmore off the topwithout reversingthem and drop ontop of the othernine. You now have

thirteen cards with

the desired two

cards at the top and

bottom of that

packet. Your subse-quent demonstra- -tion of dealing nineand dropping four isjust the first of a series of four deals: thefirst spectator does the next three deals,thereby restoring the packet to its initialstate. The second spectator deals (eitherfour or nine) cards into a pile and thenthere are two piles on the table with oneof the desired cards at the bottom of eachpile. You are all set for the grand finale.

Proof Without Words

magic techniques (e.g., double cuts) toget each card to the bottom. As a result,the third volunteer's card is at the bot-tom of the deck, the second volunteer'scard is one up from the bottom, and thefirst volunteer's card is two up from thebottom. Peel cards off the bottom of thedeck - without altering their order -one for each letter of the name of themagician called out, as you spell out bothwords in full. Hand the resulting packetof cards to the first volunteer and ask thatthe longer of the two names (HOUDINIin our example) be spelled out as cardsare dealt into a pile, before dropping theremainder on top. Now give the cards tothe second volunteer and finally to thethird volunteer for two more deals. Thethree chosen cards are now on the topof the packet of cards, with the order re-versed, and you are all set to conclude in

triumph.

The third effect uses the fact that afterthree deals of the type described, not onlydoes the bottom card rise to the top, thenext to last card becomes the second cardfrom the top, the one above that becomesthe third card, and so on:

Finally, we suggest that during all deal-ing the cards are held low, close to thetable, so as to justify fully the title of thisarticle.

This is the real triple scoop:

Claim 3: If k cards from n cards are dealtout into a pile, reversing their order, andthe remaining n - k are dropped on topas a unit, and this process is repeatedtwice more, then provided that n ?: k ?:n/2, the original k bottom cards becomethe top k cards, in reverse order.

Calm Mulcahy (colm@spelman.edu) has

taught at Spelman College since 1988. He

is currently the chair of the department ofmathematics there. The number of syl-

lables in his first name is strictly between 1

and 2. He is a member of the FOCUS edi-

torial board and will be contributing a

regular "Card Calm" to FOCUS Online.

Why are all of the above claims valid forany nand k with n ~ 2k ~ 2n? It's cer-tainly easy to see if n = k (reversing all ofthe cards each time), and almost as easyto see if n = 2k (reversing exactly half of

To perform the third trick, ask each ofthree volunteers to pick a card at random.

9