Martin Gardner in the twenty-first century

312 Pages·2012·4.64 MB·English

by  Gardner

MAA-Gardner cover v7POD_Layout 1 11/19/12 4:10 PM Page 1
Martin Gardner
Martin Gardner in the Twenty-First Century
in the Twenty-First Century
Michael Henle and Brian Hopkins, Editors Michael Henle and Brian Hopkins, Editors
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Martin Gardner enormously expanded the field of n

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recreational mathematics with the Mathematical Games a
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columns he wrote for Scientific Americanfor over 25 years
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and the more than 70 books he published. He also had in
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a long relationship with the Mathematical Association e
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of America, publishing articles in the MAA journals e
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right up to his death in 2010. This book collects articles F
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Gardner wrote for the MAA in the twenty-first century, C
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together with other articles the MAA published from tu
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1999 to 2012 that spring from and comment on his work.
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Martin Gardner
in the
Twenty-First Century
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c 2012bytheMathematicalAssociationofAmerica,Inc.
(cid:13)
LibraryofCongressCatalogCardNumber2012954077
PrinteditionISBN:978-0-88385-913-1
ElectroniceditionISBN:978-1-61444-801-3
PrintedintheUnitedStatesofAmerica
CurrentPrinting(lastdigit):
10987654321
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Martin Gardner
in the
Twenty-First Century
edited by
Michael Henle
Oberlin College
and
Brian Hopkins
Saint Peter’s University
PublishedandDistributedby
TheMathematicalAssociationofAmerica
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Preface
MartinGardnerandtheMAAsharealonghistory.In1958,aroundthetimehestartedhis
famous “Mathematical Games” column forScientificAmerican, he submittedthefirst of
many problems to The American MathematicalMonthly. In 1982, as his column wound
down, Gardner’s first MAA article was publishedinThe Two Year CollegeMathematics
Journal.He wroteforMAAjournalstherest ofhislife, particularlyThe CollegeMathe-
maticsJournalandMathHorizons.Gardnercontributedtothelatteralmostannuallyfrom
itsfoundingin1993until2005.
Gardner’sprodigiouswritingactivitycontinuedrightuntilhisdeathin2010.Articles,
stories,problems,solutions,Quickies,andotherkindsofcontributionscontinuedtoflow.
HislastmathematicalarticletoappearinanMAAjournal,“L-trominoTilingofMutilated
Chessboards,” was the centerpiece of a special puzzle issue ofThe CollegeMathematics
Journalin2009,anditisincludedhere.
Earlyin2010,MathHorizonseditorsSteveAbbottandBruceTorrenceweresurprised
toreceive atypescriptmanuscript.Gardnerused atypewriterhiswholelife,never email.
Thesubmissionwasaccompaniedbyanote,“Isthisshortstorysomethingyoucanuse?I
wrotethemathcolumninScientificAmericanfor25years.IfmypieceisnotrightforMath
Horizons,there isnoneed tosend itback. Allbest,Martin.” There was notenoughtime
for the editorsto thank Martinfor his submission[6]. Fittingly,thisstory, “Superstrings
andThelma,”isthelastpieceinthiscollection.
Apartfromhisownwork,MartinGardner,byenormouslyexpandingthefieldofrecre-
ational mathematics, opened up vast mathematical tracts for exploration by others. This
was quitedeliberate. In an interview in The Two-Year College MathematicsJournal [1],
Gardnersaid,“I’mdefining[recreationalmathematics]intheverybroadsensetoinclude
anything that has a spirit of play about it.” Gardner, of course, had a refined and very
well-developed sense of play, one quality that made his pieces so enjoyable to read. In
almost everything he wrote, Gardner posed problems to challenge his readers, and they
responded.Hemaintainedanextensivecorrespondencewithmathematicians,bothprofes-
sionalandamateur.Theirworkfueledhisownpieces, butthenhiscorrespondentsturned
aroundandwrotetheirownarticles.
Oneconsequencewastheexpansionofrecreationalmathematicsintoamajorresearch
area(alsohelpedbythedevelopmentofthecomputerandthecorrespondingexpandedin-
terestindiscretemathematics)thatissuchafeatureofthecurrentmathematicallandscape.
TheCMJdevotedtheJanuary2012issuetopapersontopicsthatGardnerintroducedtothe
mathematical public.There were somany articlestoincludethathalfoftheMarch2012
issuecontinuedthetheme.
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vi Preface
Anotherconsequenceofthefloweringofrecreationalmathematicsisthisvolume.We
havecollectedMAAjournalarticlesstartingfrom1999ontopicsthatGardnerdeveloped.
SomearewrittenbyGardner,butmostarebyothers.ThetributearticlesfromtheJanuary
andMarch2012CMJissuesareallhere,buttheyconstitutelessthanhalfofthiscollection.
AlltheMAAjournalsarerepresented,MathematicsMagazineandtheMonthly,aswellas
CMJ and Math Horizons. The limitationto pieces published roughly in the twenty-first
centuryisapracticalone.Evenso,somepuzzlecollections,longerarticles,andpiecesless
directlylinkedtoMartinGardnerhavebeenomitted.
The 41pieces collectedhere are groupedaroundcommon themes, such as geometry,
numbertheoryandgraphtheory,andcards andprobability.Flexagons,thetopicofGard-
ner’sfirstScientificAmericancolumn,areseentobeassociatedwithCatalannumbersand
together merit their own section. Geometric tiling and various “magic” number puzzles
are all about“Making Things Fit,” and there are enough other puzzles and games to fill
anothersection.
Gardner’s interests ranged far beyond mathematics. A fan of magicians and magic
tricksfromchildhood(“Iwastealotoftimeonit”[2]),hewroteseveralbooksonmagic.
HeannotatedLewis Carroll’sAlicebooksandotherclassics, andproducedtwonovelsof
hisown.Othertopicsincludedphilosophy,religion,literatureandpseudo-science,leading
tosome70books.
The lastsectionofthisvolumehighlightssome ofthese otherfacets ofMartinGard-
ner’s wide-ranging interests. It includes two short stories by Gardner and several other
piecesthatdemonstratehissupportforamateurmathematicians,hisloveofplay(aboutan
AprilFool’sjokeheplayedonhisScientificAmericanreaders),andhisinterestindebunk-
ingfalsescience.AlsoincludedisGardner’sreviewofa2004novelinwhichanimportant
characterseemstobebasedonhim.
Our hope is that this volume will play a role in perpetuating the memory of Martin
Gardner, modest celebrity, larger-than-life character, self-confessed amateur as a mathe-
matician, popularizerofrecreationalmathematics inthe broadestsense, prolificandbril-
liantwriter.Giventhelastingimpressionhe made onseveral generationsofmathematics
enthusiastsofallbackgrounds,weareconfidentthattheMAAandotherswillbepublish-
ingarticlesinspiredbyGardner’sworkforalongtime.
Gardner, likethereaders ofthisbook,lovedmathematics. We closethispreface with
Gardner’sownwordsonhisbackground,thecommunity,andwhyheenjoyedthefieldso
much(from[5],[4],and[3],respectively).
Itooknomathincollege.I’mlikeapersonwholovesmusicbutcan’thum
a tuneorplayan instrument.Myunderstandingofmathdoes notgobeyond
a minimalunderstandingofcalculus. I hasten toaddthatI considerthisone
reasonforthesuccess ofmyScientificAmerican column.Ihadtoworkhard
tounderstandwhateverIwroteabout,andthismadeitmucheasierformeto
writeinawaythatreaderscouldunderstand.
When I first startedthe column, I was not intouch withany mathemati-
cians,andgraduallymathematicianswhowerecreativeinthefieldfoundout
aboutthecolumnand begancorrespondingwithme. Somymostinteresting
columnswerecolumnsbasedonmaterialthatIgotfromthem,soIowethem
abigdebtofgratitude.
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Preface vii
[I enjoy mathematics] because it has a strange kind of unearthlybeauty.
Thereisastrongfeelingofpleasure,hardtodescribe,inthinkingthroughan
elegantproof,andevengreaterpleasureindiscoveringaproofnotpreviously
known.
Aclnowledgments
The editors are grateful for the encouragement and hard work of the MAA publications
staff:Directorof PublicationsIvars Peterson, who conceived the book,ProductionMan-
agerCarolBaxter,whodesignedit,andespeciallyElectronicPublicationManagerBeverly
JoyRuedi,whosetthebookintype.
Bibliography
[1] AnthonyBarcellosandMartinGardner,AConversationwithMartinGardner,Two-YearCollege
Math.J.10(1979)233–244.
[2] DonAlbersandMartinGardner,OntheWayto“MathematicalGames”:PartIofanInterview
withMartinGardner,CollegeMath.J.36,(2005)178–190.
[3] DonAlbers andMartin Gardner,“MathematicalGames”andBeyond:Part II of anInterview
withMartinGardner,CollegeMath.J.36(2005)301–314.
[4] Colm MulcahyandMartin Gardner,An Interview with Martin Gardner,CardColm, October
2006,availableatwww.maa.org/columns/colm/cardcolm200610.html
[5] MichaelHenleandMartinGardner,InterviewwithMartinGardner,CollegeMath.J.40(2009)
158–161.
[6] BruceTorrenceandStephenAbbott,ToOurReaders,MathHorizons18(2010)2–4.
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Contents
Preface v
I Geometry 1
1 TheAsymmetricPropeller 3
MartinGardner
Gardner,payingtributetodentistandgeometerLeonBankoff,discussessomeofhisunpub-
lishedresultsandconcludeswithanopenquestion.
2 TheAsymmetricPropellerRevisited 7
GillianSaenz,ChristopherJackson,andRyanCrumley
ThreeUniversityofTexasstudentsusedynamicgeometrysoftwaretoconfirmBankoff’sre-
sultsandresolveGardner’squestion.
3 BracingRegularPolygonsAsWeRaceintotheFuture 11
GregW.Frederickson
AproblemGardnerpublishedin1963continuestospurgeneralizationsandimprovedsolu-
tionsaroundtheworld.
4 APlatonicSextetforStrings 19
KarlSchaffer
Theprofessoranddancecompanyco-directordetailsstring polyhedraconstructionsfor ten
participants.
5 PrinceRupert’sRectangles 25
RichardP.JerrardandJohnE.Wetzel
A17thcenturypuzzlethatGardnerposedinhigherdimensionsisheresolvedinthecaseof
three-dimensionalboxes.
II NumberTheory andGraphTheory 35
6 TranscendentalsandEarlyBirds 37
MartinGardner
GardnermovesfromLiouvilletoan“innocentbuttotallyuselessamusement”thatnonetheless
capturedtheattentionofSolomonGolomb.
7 Squaring,Cubing,andCubeRooting 39
ArthurT.Benjamin
Theprofessorand“mathemagician,”inspiredasahighschoolstudentbyGardner’stricksfor
mentalcalculations,extendssomeofthemhere.
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