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What is Recreational ?

Definitionby example: paradigms of topics, people and publications.

CHARLESW. TRIGG 2404 LoringStreet San Diego, CA 92109

Mindfulof the way Tom Sawyergot his fence whitewashed, I asked a numberof mathematcians (includingthe speakers at theMiami University Recreational Mathematics Conference) to answerthe question,"What is recreationalmathematics?" The responsewas gratifyingand illuminating.Many willrecognize their brushmarks in thefollowing discussion. For those who detect untouched areas on thefence, there are plentyof brushesavailable. One thingis immediatelyclear: defining"recreational mathematics" is not recreational.The difficulttask of defining"mathematics" is not simplifiedby thequalifying "'recreational". ".Recreation"is definedin [1] as "a pastime,diversion, , or otherresource affording relaxationand enjoyment". One indulgesin recreationto re-create oneself, to relaxfrom work-a-day pursuits,and to clearand refreshthe mind before returning it to itsregular occupation. The closely relatedword "" is definedin [1] as "exerciseor actionby wayof amusementor recreation". Leonard[21 asserts that "all creativeactivity begins with play, to whichwill is thenapplied, as muchin scienceand as insports. The Pythagoreantheorem and the motdel of the double helix are at leastas proportional,fit, elegant and admirableas the one-handedjump shotand the perfectlyexecuted doubleplay. Moreover, thev are ". Indeed, W. F. White[31 once observed that 4' is onie of thefields of appliedmathematics". "Mathematicalwork is highlysatisfying. So is mathematicalplay. And, as mostoften is thecase, one is apt to workmuch harder at anyform of play,mental or physical,than one wouldfor nere remuneration.Mathematical activity, more than any other, gives scope for the exercise of that faculty whichhas elevatedman above othercreatures" [4]. It is humannature to enjoythose things that can be donewell. It is also humanto resist mandatory tasks.At theUniversity of Michiganthe late Norman Anning used to takeadvantage of thishuman frailtyby offering an intriguingnon-credit problem at theend of each class section. The problemmay or maynot have been related to thecourse content, but it stuck in the long-time memories of many of hisstudents. Some rather dry souniding problems can be recastinto very nice recreational settings. "Mathematicscan provideenjoyment for a varietyof reasons - meetingthe reeds of thosewho seek recreation,while giving satisfaction to thosewho are intriguedby solvingproblems, close reasoningand unexpectedsolutions" [5]. The allureof problemsolving may be a matterof ego satisfaction.The joy of triumphover an opponentcan be equalledor surpassedby theglow of findinga solutionto a toughmathematics problem,even though it mayturn out thatthe result had been publishedin an obscure19th century journal.That manyenjoy such challenges is attestedto bY the popularityof problemsections in variousmagazines. There the student who works all theproblems in the textbooks just for the joy ofit can furthertest his mathematicalmettle. Decades ago RichardBellman, by thenan accomplished ,told me thathe had cut his mathematicaleye-teeth on thechallenge problems in SchoolScience and 'Iathematics. Challengeproblems also can widenthe mathematicalhorizons of thosewhose mathematical developmenthas notbeen blockedby unfortunatecircumstances, even thoughthey do notfollow

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This content downloaded from 131.91.4.38 on Wed, 24 Jun 2015 13:09:30 UTC All use subject to JSTOR Terms and Conditions mathematicsprofessionally. They have the tools to pursuean inexpensiveand stimulating recreational activity.Their laboratory and shop consist of pencil and paper. At times,this recreation may re-create thework of others, but the possibility of discovering new relationships is always present. Howard Eves oncecompiled a longlist of original articles that had been inspired by the problem departments of the AnmericanMathematical Monthly. Mathenaticsaffords an ever-newnever-boring avocation both before and afterretirement where itis an effectiveweapon against vegetating. The humaninterest section of the Otto DLinkel Memorial ProblemBook [6] liststhe names of a nurseryman,a lawyer, a ballisticsexpert, a dentist,a steelworks manager,a retiredtelephone engineer, an au4omobiledealer, an insuranceinspector, a patent examiner,and a clergymanwho were contributors to the problem departments of theMonthly. It is of morethan passing interest that the problemdepartments of thejournals of the twocollege level mathematicalfraternities (Pi Mu Epsilonand Kappa Mu Epsilon)are editedby a practicingdentist and a practicingattornev - Dr. Leon Bankoffin thePi Mu EpsilonJournal and KennethWilke in The Pentagon. In the less austeredays of the problemdepartments of the Monthly,the late NormanAnning proposedthe problem [7] to "findthe element of likeness in: (a) simplifyinga fraction, (b) powdering theniose, (c) buildingnew steps on thechurch, (d) keepingemeritus professors on campus,(e) putting B, C,D, in thedeterminant I a a2 3 a3 1 a a2 B a3 I a C D a3 1

The publishedsolution remarked that the value, (1 - a4)3, ofthe determiinant was independentof the valuesof B, C, andD, so theirinsertion merely changes the appearance of the determinant and not its value."Thus, the element of likenessin (a), (b), (c), (d), and (e) is thatonly the appearance of the principalentity is changed.The sameelement appears also in: (f)changing the name-label of a rose, (g) changinga decimalinteger to thescale of 12,(h) gildinga lily,(i) whitewashinga politician, and (j) grantingan honorarydegree". Anning sent the solver a cartoon,clipped from the Saturday Review, of an Indianwatching the cloud of an atomicblast. The caption:"I wishI had said that". 0therirreverent contributors to theproblem departments are thosewith risible nom-de-plumes, such as ALICE MALICE, POLLY TOPE, NEV R. MIND, ZAZU KATZ, and ALFRED E. NEUMANN of MU ALPHA DELTA FRATERNITY. My favoritesare NOSMO KING and BARBARA SEVILLE. Stillother recreational support of the tenet that mathematics is too important to taketoo seriously are: thesporadic appearances of Professor Euclide Paracelso Bombasto Umbugio, the priceless lyrics ofTom Lehrer, Leo Moser'sverse [8, 91, the Mathematical Swifties ("The angleis lessthan 90, Tom notedacutely") in the1964 American Mathiematical Monthly, the Show Me's ("Showme a manwho couintson his fingersand I'll show you a digitalcomputer") in the Journalof Recreational Mathematics,and thevaried offerings in thatdelightful new Canadian periodical, Eureka [101.The disdainfulmay say thatin mathematicsa little humor goes the wrongway. Few willdisagree with the classification of mathematicalhumor, poems, anagrams, rebuses, word equations,cross-number , acrostics, and cryptarithmsas purely recreational, although they havesome educational use. Some mayquestion whether they qualify as mathematics.However, to excludethe anecdotes of Eves' In MathematicalCircks would be to excludea portionof history of mathematicsas well.But whatof othertopics? In the preambleto his excellentdiscussion of "NumberGames and Other Mathematical Recreations",[11] WilliamL. Schaafremarks, "Mathematical recreations comprise puzzles and gamesthat vary from naive amusements to sophisticatedproblems, some of whichhave never been solved.They may involve , , , theory of numbers,, ,

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This content downloaded from 131.91.4.38 on Wed, 24 Jun 2015 13:09:30 UTC All use subject to JSTOR Terms and Conditions matrices,, (dealing with problems ofarrangements ordesigns), , symboliclogic, or . Any attempt toclassify this colourful assortment ofmaterial isat bestarbitrary". However, in his Bibliography ofRecreational Mathematics, [121 Schaaf does impose a classificationbymeans of the chapter headings: arithmetical recreations, as recreation, geometricrecreations, topological recreations, magic squares and related configurations, Pythagorean recreations,recreations inantiquity, combinatorial recreations, manipulative recreations, miscellane- ous recreations,mathematics in related fields, and recreations in the classroom. Mostof themathematics books with "Recreational", "Play", "Amusement", "Fun", "Diver- sions","", "", or"Pastimes" in their titles are problem oriented. Predominant amongthe topicsdealt with in suchbooks are cryptarithms,magic squares, dissections and ,decanting liquids, measuring, weighing, packing, shortest paths, calendars, the census, slidingmovement, chess, dominoes, cards, river crossing, match arrangements, networks, permuta- tions,combinations, diophantine equations, number properties, and various . In hisdetailed discussion of the first recreational mathematics book, that written by Bachet [131, Dudley[14] remarked that "problems of a recreationalnature appear on Babyloniantablets 3500 yearsold" and "the Egyptian Rhind papyrus of about 1650 B.C. contains" a "problem that must have beenmade up for the fun of it". Dudley also reports the existence of other recreational problems in variousworks of the 16th century and earlier. Thoseinterested in theevolution of recreationalmathematics through the ages will find an excellenthistorical account in [111,and a verygood treatment ofan internationalcross-section of recreationalbooks in thePostscript of O'Bierne'sPuzzles and Paradoxes[151. The classicfour volumesof Lucas [161 now appear in a Frenchpaperback edition. English translations areavailable of therecreatlonal works of the Russians Domoryad [171 and Kordemsky [181, the Polish Steinhaus [191, theDutch Schuh [201, and the Belgian Kraitchik [211. Kraitchikwas the editor of the defunct Sphinx (1931-1939), a magazine devoted to recreational mathematics.ItsAmerican counterpart, Recreational Mathematics Magazine (1961-1964), was also short-lived,although its successor, Journal of Recreational Mathematics (1968-), is flourishing. Thecontemporaries Dudeney in England and Loyd in the United States were prolific inventors of mathematicalpuzzles which appeared in variousperiodicals. They were great rivals and did not hesitateto borrow from each other. Some of their efforts are preserved in[22, 23, 241. Somewhat later HubertPhillips under the nom de plume Caliban contributed problems to Englishperiodicals, some of whichare collected in [251. Oneof the great stimuli of interest inrecreational mathematics is Ball's comprehensive book [261 nowin its12th edition. Another great stimulus is theScientific American column "Mathematical Games"written by the dean of contemporary recreational mathematics, . A list,by title,of his skillfully written columns, beginning with "Flexagons" in December 1956, appears in [12]. Muchof the column material has been assembled in his many books, such as [27,281. For a multitude ofother worthy volumes see [12]. Someconsider that for mathematics to be recreationalthere must be someelement of playor gamesinvolved. But many who abhor games get great pleasure from other branches of mathematics. It maybe a matterof definitionof "games";and ifwe acceptGoodman's statement [291 that "Mathematicsis the greatest ever invented by man," we areback where we startedfrom in searchof a definition. Formany individuals, as they approach the limit of their abilities, mathematics loses its fun aspect. Whena topicis undeveloped, itis recreational tomany. As the theory isdeveloped and becomes more abstract,fewer persons find it recreational.Of course,there are somewho get their pleasure by concentratingona singletopic, pursuing itto the extent of current knowledge, and then trying toadd somethingnew. In manycases, what starts out to be purelyrecreational develops into an extensive disciplineof serious mathematics such as nowexists in number theory, topology, combinatorics, and gametheory.

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This content downloaded from 131.91.4.38 on Wed, 24 Jun 2015 13:09:30 UTC All use subject to JSTOR Terms and Conditions Manyconsider mathematics recreational ifit is sufficientlyelementary to be understandableby the non-mathematician.Recreational mathematics is "somethingyou can explainto a businessman sittingnext to youon a flightfrom Chicago to Cincinnati".It is "a pieceof mathematics that is both subtleenough to interestthe professional mathematician and simpleenough to be accessibleto the man-in-the-street". Othersconsider recreational mathematics from more of an academicslant: it is "'a mathematician's holiday"that falls outside the bounds of standard school and university mathematics. "Learning what othershave done is notrecreational, doing it yourself is". Recreationalmathematics is that which one workson "withoutthought of practicality,generality or academicrewards". "A recreationaltopic losesits standing when one squeezesfrom it an articleto enhancethe publication list." Stillothers approach the topicfrom the standpointof personalenjoyment. Surely the great amateursOmar Khayyam,,Blaise Pascal, and PierreFermat considered all mathematicsto be fun.Indeed, many professional consider all mathematicsto be pleasurable.Richard Guy has said, "It alwaysfascinates me thatpeople are willingto pay me for doingwhat I woulddo forenjoyment in anycase." Recreationaltastes are highlyindividualized, so no classificationofparticular mathematical topics as recreationalor notis likelyto gainuniversal acceptance.

References

[1 The Random House Dictionaryof the English Language. Random House, New York, 1966 [21 JohnLeonard, Review of Michael Novak's The Joyof ,The New York Times Book Review,June 13, 1976, p. 6. [31 H. E. Dudeney, 536 Puzzles and Curious Problems,Charles Scribner'sSons, New York, 1967. p. xiii. [41 Harry Langman, Play Mathematics,Hafner, New York, 1962, p. 5. [51 Roland Sprague, Recreation in Mathematics,Dover, New York, 1963, foreword. [6] The Otto Dunkel MemorialProblem Book, Amer. Math. Monthly,64 (August-September1957), Part 11, 8. [7] Norman Anningand C. W. Trigg,An InvariantDeterminant, Amer. Math. Monthly,60 (February1953) 115. [8] W. E. Mientka,Professor Leo Moser - Reflectionsof a Visit,Amer. Math. Monthly,79 (June-July1972) 609-614. [9] Leo Moser, Some MathematicalVerses Amer. Math. Monthly,80 (October 1973) 902. 1101 EUREKA, published monthly(except Julyand August), Algonquin College, 200 Lees Ave., Ottawa, Ontario, KIS OCS, Canada, yearlyrate $6.00. (111 W. L. S., Encyclopedia Britannica,15th ed., 1974, volume 13, pp. 345-357. [12] W, L. Schaaf,A Bibliographyof RecreationalMathematics, vol. 1, 2 (1970), vol. 3 (1973), National Council of Teachers of Mathematics,Reston, Virginia. [13] Claude-Gaspar Bachet, ProblrmesPlaisants et Delectables qui se Font par les Nombres,Lyon, 1612. [14) UnderwoodDudley, The FirstRecreational Mathematics Book, J. RecreationalMathematics, 3 (July1970) 164-165. [151 T. H. O'Beirne, Puzzles and Paradoxes, Oxford,New York, 1965, pp. 209-229. [16] M. E. Lucas, Rcr6ations Math6matiques,4 vols. Gauthier-Villars,1882-1894; new edition, Blanchard, Paris, 1960. [17) A. P. Domoryad, MathematicalGames and Pastimes,Macmillan, New York, 1964. [181 B. A. Kordemsky,The Moscow Puzzles, Scribner's,New York, 1972. [19] H. Steinhaus,Mathematical Snapshots, Oxford, New York, 1960. [201 Fred. Schuh, The Master Book of MathematicalRecreations, Dover, New York, 1968. [21] Maurice Kraitchik,Mathematical Recreations, Dover, New York, 1953. [22) H. E. Dudeney, The CanterburyPuzzles, Dover, New York, 1958. [23) , Amusementsin Mathematics,Dover, New York, 1958. [241 MartinGardner, Mathematical Puzzles of , Dover, New York, vol.1 (1959), vol. 2 (1960). [251 HubertPhillips, S. T. Shoveltonand G. S. Marshall,Caliban's ProblemBook, Dover, New York, 1961. (26] W. W. R. Ball, MathematicalRecreations and Essays (rev, by H. S. M. Coxeter), 12thed. Universityof Toronto Press, 1974. [271 MartinGardner, The Unexpected Hanging, Simon and Schuster,New York, 1969. [28) , MathematicalCarnival, Knopf, New York, 1975. [291 A. W. Goodman, The of Math, Macmillan,New York, 1965, p. 13.

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