maths less travelled

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Making a computer out of dominoes?

Posted on October 10, 2012

WhenImentionedcarryingoutcomputationalprocesseswitharoomfullofdominoes ,Iwasntkidding.Matt

Parkerisplanningtobuildadominocomputer attheManchesterScienceFestivalattheendofthemonth.The

ManchesterScienceFestivalbloghasanicewriteupexplainingtheproject.

HeresavideoofMattexplaininghowadominoANDgateworks (twochainsofdominoescomein,andone

goesout;theoutgoingdominoeswillfallonlyif bothincomingchainsdo).UnfortunatelyitseemsIcantembed

videosonthisblog,atleastnotwithoutgivingwordpresssomecash=(,soyoullhavetoactuallyclickthatlink

towatchthevideo,butitstotallyworthit(trustme).

IwishIcouldgoseeit,butitsabitfarforme.Any MathLessTraveledreadersintheUKwhocanactuallygowatchthecomputerinactionandreportback?

Posted in computation, links, video | Tagged computer, domino, festival, Manchester, Matt Parker, science | 3 Comments

Factorization diagramsPosted on October 5, 2012

InanidlemomentawhileagoIwroteaprogramtogenerate"factorizationdiagrams".Heres700:

Itseasytosee(Ihope),justbylookingatthearrangementofdots,thatthereare intotal.

HereshowIdidit.First,afewimports:afunctiontodofactorizationofintegers,anda librarytodrawpictures

(yes,thisisthelibraryIwrotemyself;Ipromisetowritemoreaboutitsoon!).

The Math Less Traveled

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> module Factorization where

>

> import Math.NumberTheory.Primes.Factorisation (factorise)

>

> import Diagrams.Prelude

> import Diagrams.Backend.Cairo.CmdLine

>

> type Picture = Diagram Cairo R2

TheprimeLayoutfunctiontakesanintegern(assumedtobeaprimenumber)andsomesortofpicture,and

symmetricallyarrangesncopiesofthepicture.

> primeLayout :: Integer -> Picture -> Picture

Thereisaspecialcasefor2:ifthepictureiswiderthantall,thenweputthetwocopiesoneabovetheother;

otherwise,weputthemnexttoeachother.Inbothcaseswealsoaddsomespaceinbetweenthecopies(equal

tohalftheheightorwidth,respectively).

> primeLayout 2 d

> | width d > height d = d === strutY (height d / 2) === d

> | otherwise = d ||| strutX (width d / 2) ||| d

Thismeanswhentherearemultiplefactorsoftwoandwecall primeLayoutrepeatedly,weendupwiththingslike

Ifwealwaysputthetwocopies(say)nexttoeachother,wewouldget

whichismuchclunkierandhardertounderstandataglance.

Forotherprimes,wecreatearegularpolygonoftheappropriatesize(usingsometrigIworkedoutonanapkin,

dontaskmetoexplainit)andpositioncopiesofthepictureatthepolygonsvertices.

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> primeLayout p d = decoratePath pts (repeat d)

> where pts = polygon with { polyType = PolyRegular (fromIntegral p) r

> , polyOrient = OrientH

> }

> w = max (width d)(height d)

> r = w * c / sin (tau / (2 * fromIntegral p))

> c = 0.75

Forexample,heresprimeLayout 5appliedtoagreensquare:

Now,givenalistofprimefactors,werecursivelygenerateanentirepictureasfollows.First,ifthelistofprime

factorsisempty,thatrepresentsthenumber1,sowejustdrawablackdot.

> factorDiagram' ::[Integer]-> Diagram Cairo R2

> factorDiagram' [] = circle 1 # fc black

Otherwise,ifthefirstprimeiscalledpandtherestareps,werecursivelygenerateapicturefromtherestofthe

primesps,andthenlayoutpcopiesofthatpictureusingthe primeLayoutfunction.

> factorDiagram' (p:ps)= primeLayout p (factorDiagram' ps) # centerXY

Finally,toturnanumberintoitsfactorizationdiagram,wefactorizeit,normalizethereturnedfactorizationinto

alistofprimes,reverseitsothebiggerprimescomefirst,andcall factorDiagram'.

> factorDiagram :: Integer -> Diagram Cairo R2

> factorDiagram = factorDiagram'

> . reverse

> . concatMap (uncurry $ flip replicate)

> . factorise

Andvoila!Ofcourse,thisreallyonlyworkswellfornumberswithprimefactorsdrawnfromtheset

(andperhaps ).Forexample,heres121:FollowFollow



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Arethere11dotsinthosecircles?13?Icantreallytellataglance.Andheres611:

Uhhwell,atleastitspretty!

Herearethefactorizationdiagramsforalltheintegersfrom1to36:

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Powersofthreeareespeciallyfun,sincetheirfactorizationdiagramsare Sierpinskitriangles!Forexample,heres

:

Po wersoftwoarealso fun.Heres :

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[ETA:asanonpointsout,thisfractalhasanametoo:Cantordust!]

Onelastone:104.

IwishIknewhowtomakeawebsitewhereyoucouldenteranumberandhaveitshowyouthefactorization

diagrammaybeeventually.

(Incaseyouwerewondering, .)

Posted in arithmetic, pictures, primes, programming, recursion | Tagged diagrams, fact orization, Haskell | 50 Comments

What I Do: Part 0Posted on October 4, 2012

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ThisisthefirstinaplannedseriesofpostsexplainingwhatIdoinmy"dayjob"asacomputersciencePhDstudent.

Theideaistowriteaseriesofpostsofincreasingspecificity,butallaimedatageneralaudience.

HaveyoueverwonderedwhatIactuallydoallday,otherthanwritethisblog?(Well,probablytheansweris

"no"since,asweallknow,peopleontheInternetdontactuallyhavereallives,inthesamewaythat

kindergartenteachersliveintheclosetintheirclassroom.)ButwhatIdowillactuallybequiteinterestingto

readersofthisblog,Ithink.

So,tostartoff:Iamacomputerscientist.Whatdoesthatmean?

WhatIdontdo

Letmebeginbysayingthat"computerscience"isactuallyaterriblenameforwhatIdo.Itsakintoan

astronomersayingtheystudy telescopescience ,oramicrobiologistsayingtheystudy microscopescience.Of

course,astronomersdontstudytelescopes,they usetelescopestostudystarsandsupernovas.Microbiologists

dontstudymicroscopes,they usemicroscopestostudycellsandDNA.AndIdontstudycomputers,I use

computerstostudywell,what?

Computation

Inabroadsense,whatcomputerscientistsstudyis computation,bywhichwemeanprocessesofsomesortthat

takesomeinformationandturnitintootherinformation.Questionsthatcanbeaskedaboutcomputation

include:

Whataredifferentwaysofdescribingacomputationalprocess?

Howcaninformationbestructuredtomakecomputationalprocesseseasiertowrite,moreefficient,ormore

beautiful?

Howcantwodifferentcomputationalprocessesbecompared?Whenisoneprocess"better"thananother?

Howcandifferentprocessesbecombinedintoalargerprocess?

Howcanwebesurethatsomeprocessreallydoeswhatwewantitto?

Whatsortsof"machines"canbeusedtoautomatecomputationalprocesses?

What(ifany)arethelimitationsofcomputationalprocesses?

Imsureotherquestionscouldbeaddedtothislist,butthesearesomeofthemostfundamentalones.

Noticethatnoneofthesequestionsinherentlyhaveanythingtodowithcomputers.A"computationalprocess"

couldbecarriedoutwithpilesofrocks,anabacus,paperandpencil(didyouknowthattheword"computer"

usedtorefertoapersonwhosejobitwastodocarryoutcomputationalprocesses?),a carefullysetuproomfull

ofdominoes,oracarefullysetuptesttubefullofDNA.Itsjustthatmoderncomputerscancarryout(most)

computationalprocessesmanyordersofmagnitudefasterthananyothermethodweknowof,sotheymake

exploringtheabovequestionspossibleinmuchdeeperwaysthantheywouldotherwisebe.Andindeed,the

mathematicalrootsofcomputersciencegobackmanyhundreds,eventhousandsofyearsbeforetheadventof

digitalcomputers.

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So,Istudycomputation.Butasyoucanseefromthelistofquestionsabove,thatsstillincrediblybroad.Infact,

myresearchfocusesonthefirsttwoquestionsinthelistabove.InPart1Illdescribethosequestionsinabit

moredetail.Also,Imhappytotrytoansweranyquestionsleftinthecomments!

Posted in computation, meta | Tagged computation, computer sc ience, whatido | 5 Comments

CoM and Relatively PrimePosted on September 17, 2012

Acouplethingstodrawyourattentionto:

The90thCarnivalofMathematicsisupoveratWalkingRandomly.Theresquitealotofcoolstuffinthis

edition,gocheckitout!

ThefirstepisodeofRelativelyPrimeisup!Thisisaseriesofeightshowsallaboutthestoriesbehind

mathematicsproducedbySamuelHansen.Ihaventactuallyhadachancetolistentothefirstepisodeyet,

butIknowheflewallovertheworldinterviewingmathematiciansforthissoitoughttobeinteresting!I

alsohelpedfunditonKickstartersoImquiteexcitedtoseethefruits.

Posted in links | Tagged Carnival of Mathematics, episode, podcast, show | 1 Comm ent

Three new booksPosted on August 23, 2012

Athree-for-onetoday!HerearethreebooksIwantedtomentiontoyou,dearreader,foronereasonoranother.

AWealthofNumbers

BenjaminWardhaugh

PrincetonPresskindlysentmeareviewcopyofthisbook.Asan anthologyofpopularmathematicswritingfrom

the1500

stothepresent,itsnotthesortofbookIusuallyreviewherebutitsnonethelessfascinating.Thefeaturedexcerptsare,byturnsgripping,dull,lucid,incomprehensible,hilarious,irrelevant,andfun.Ididntlearn

awholelotofnewmathematicsbyreadingthisanthology,butitgavemesomeeye-openingperspectiveonhow

peoplehavethoughtandwrittenaboutmathematicsoverthelast500years.

DeadReckoning:CalculatingWithoutInstruments

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Theresalotofcontroversyovertheuseofcalculatorsandcomputersinmathclassrooms.Shouldtheybe

welcomedaslabor-savingdevicesthatallowstudentstoexploremathematicsinnewways,oreschewedas

crutchesthatturnstudentsintobutton-pushingautomatawithnorealunderstanding?Itsanimportantdebate,

butitseemstomethatthosewhoargueagainstcalculatorssometimeshaveanonethelessimpoverishedviewof

thealternative:justtheabilitytodothestandardWesternalgorithmsforthefourbasicarithmeticoperations,

andunderstandwhythealgorithmswork.

Thisbook(whichIreceivedasagiftaboutsixmonthsago)isntintendedtoenterthatdebatedirectlybutitis

afar-rangingsurveyofmethodsforpencil-and-paper(ormental)calculation.Therearegeneralalgorithms,of

course,(andnotjustforarithmetic,butforsquareroots,logarithms,trigfunctions)buttherearealsoallsorts

oftricksandspecialcaseswhicharisefrom,andengender,intimatefamiliaritywithnumbers.Isit"practical"?

Well,notreally.Butthatscertainlynotthepoint.Ihadsomuchfunreadingthroughthisbookandtryingoutall

thedifferentalgorithms:multiplyinginmyhead,computingsquarerootstomanydecimalplacesusingjusta

singlesheetofpaperHonestlyIdontrememberanyoftheactualalgorithmsanymore,butIcameawaywith

betternumbersenseandImayjustpullitoutandtrysomeofthealgorithmsagain.Eventuallysomeofthem

areboundtostick!

MathematicalLiteracyintheMiddleandHighSchoolGrades

FaithWallaceandMaryAnnaEvans

Imentionthisbookparticularlybecauseitcontains(inaboxonpage67)anessaybyyourstruly!Iwrote

somethingabouttheexperienceofwritingthisblogandinspiringreaderswithmathematicalbeauty.

Inolongerteachmiddleorhigh-schoolstudents,buttheideaofusingliteracytoinspireinterestinmathmakes

alotofsensetome(andofcourse,itsperfectlyapplicableatthecollegelevelaswell).Thereareallsortsof

interestingideasinheresomeofwhichIwillneveruse(discussingstatisticsandvotingvia AmericanIdol)but

othersofwhichIdlovetotrysomeday(usingbiographyorfictiontoinspiremathematicalinterest).

Posted in books, review | Tagged anthology, calculation, literacy | 2 Comments

Introduction to Mathematical Thinking with Keith Devlin

Posted on August 22, 2012

IjustlearnedfromDeniseatLetsPlayMath!thatKeithDevlinisgoingtobeteachingacourseonCoursera

calledIntroductiontoMathematicalThinking.Itsfreeandopentoanyonewithonlyabackgroundinhighschool

math.Lookslikeitshouldbequiteinterestingandperhapsofinteresttosomeofmyreaders!

Posted in links, teaching | Tagged Coursera, free, Keith Devlin, mathematical thinking, online | 1 Comment

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Visualizing nim-like gamesPosted on August 16, 2012

Inspiredbythecommentsonthispost,IvehadsomeideasbrewingforawhileImjustonlynowgetting

aroundtowritingthemup.

Thetopicisvisualizingwinningstrategiesfor"nim-like"games.WhatdoImeanbythat?Byanim-likegameImean

agameinwhichtwoplayerstaketurnsremovingobjectsfromsomepiles(subjecttosomerules),andthelastplayertoplayisthewinner(or,sometimes,theloser).

Acutevariant,duetoPaulZeitz(andintroducedtomebySueVanHattum ),istothinkofapetshopwith

differenttypesofpets;playerstaketurnsvisitingthepetshopandbuyingsomepets,untilthestoreisalloutof

pets.

Forgameswithonlytwopiles,orapetstorewithonlytwotypesofpets,playingthegamecanalsobethought

ofasmakingmovesonasquaregrid.The -coordinaterepresents,say,thenumberofXoloitzcuintli,andthe

y-coordinatethenumberofYaks;thesquarewithcoordinates meansthatthepetstorehas xolosand

yaksleft.Buyingsomexoloscorrespondstomovingleft;buyingyaksmeansmovingdown;buyinganequal

numberofeachmeansmovingdiagonallydownandleft;andsoon.

Hereareafewexamplesofnim-likegames:

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Inthegameofnim,youmayonlybuyonetypeofanimaloneachturn(butyoucanbuyasmanyasyou

want).Onagrid,youareallowedtomoveanydistanceleftordown(butnotboth).

Theinterestingthingisthatwecanvisualizethewinningstrategyforthisgameinthefollowingway.

(ZacharyAbelhasamuchmoredetailedexplanationofthisidea .)Awinningpositionisasquarethat

guaranteesawinthatis,ifitisyourturnandyouareonawinningsquare,then(assumingyoumakethe

rightmoveandcontinuetoplayperfectly)youwillwinthegame.Illindicatewinningpositionsbylightgreensquares,likethis: .Alosingpositionisapositionsuchthatyoucantwinnomatterwhatmoveyou

make(assumingyouropponentplaysperfectly).Illindicatelosingpositionsbydarkbluesquares,likethis:

.Infact,thewinningpositionsareexactlythosefromwhichthere existsatleastonelegalmovetoalosing

position;andthelosingpositionsarethosefromwhich everylegalmoveistoawinningposition.

Hereswhatitlookslikefornim:

Nottooexciting,butitmakessense.Ifthepetstorehasanequalnumberofeachtypeofpetremaining,the

firstpersontomoveisgoingtolose:theirmovewillresultinanunequalnumberofpets( i.e.asquareoffthe

mainbluediagonal),andalltheiropponenthastodoisbuyanequalnumberoftheothertypeofpetto

restorebalance.Ultimatelythelosingplayerwillbeforcedtocleanthestoreoutofonetypeofpet,andthe

otherplayerthenwinsbycleaningthestoreoutoftheothertype.

InWythoffsgame,youmaybuyanynumberofasingletypeofanimal, oranequalnumberofboth.Onagrid,

youareallowedtomoveanydistanceleft,down,orata45degreeangleleftanddown.

Thevisualizationforthisgame,ofcourse,ismuchmoreinteresting:

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Youcanreadallaboutthefascinatinganalysisofthisgame(anditsvisualization)onZacharyAbelsblog .

Inacomment,MaxdescribedagamefromtheInternationalOlympiadinInformatics:

Youstartwitharectangle,andyoucancutiteitherverticallyorhorizontallyatintegersizes,each

timekeepingthelargerpiece;thegoalistoobtainaunitsquare(sothatyouropponentcantmove).

Itsnotasobvious,butthisisalsoanim-likegame.Thetwodimensionsoftherectanglecorrespondtothe

numberofpetsoftwodifferenttypes.Forexample,a rectanglecorrespondsto10xolosand7yaks.

Therectanglemustbecuteitherverticallyorhorizontally,meaningthatyoucanonlybuyonetypeofpeton

agiventurn.Theinterestingtwististhatyoumustkeepthe largerpieceresultingfromacut,whichis

equivalenttosayingthatyoumaybuyanynumberofpetsbutonly uptohalfthenumberofpetsthestore

currentlyhas.Forexample,ifthestorehas xolosand yaks,youmaybuyupto xolos,orupto yaks.Buyinganymorethanthatwouldcorrespondtocuttingtherectangleandkeepingthesmallerpiece,whichis

notallowed.

Naturally,Iwonderedwhatthevisualizationofthisgamelookslike.Ifiguredithadtobesomething

interesting,andIwasntdisappointed!Hereitis(notethatthebottom-leftsquarerepresents here,

whereasinthevisualizationsfornimandWythoffsgameitrepresented ):

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Woah,neat!Itlooksasifthelosingsquaresfallalongdiagonallinesofslope forallintegern(thatis, , ,

, , ,andsoon)thoughthelinesdontallpassthroughtheorigin.Itsnotsurprisingthatthemain

diagonalconsistsofalllosingpositionsthestrategyisthesameasfornim;thefactthatonecanonlybuy

uptohalfofacertaintypeofanimaldoesntmakeanydifference.Ifitisyouropponentsturntomoveand

thestorehasanequalnumberofxolosandyaks,iftheybuyacertainnumberofyaksyoucanbuythesame

numberofxolos,andviceversa.Eventuallythestorewillhaveoneofeach,atwhichpointyouropponent

losessincetheycantbuyanymoreanimals(theywouldonlybeallowedtobuy,say,halfayak,butthepet

storehasthesensiblepolicyofnotchoppingpetsinhalftoomessy).

However,apparentlythereareadditionallosingpositionsotherthanthemaindiagonal.Forexample,

accordingtothegraph,ifthestorehas4xolosand19yaks,thenthenextplayertomoveisgoingtolose!

Youcanverifyforyourselfthatfromthispoint,theonlylegalmovesaretolightgreen(i.e.winning)

positions,andthatfromanyofthesepositionstheotherplayercanmakealegalmovetoanotherdarkblue

(i.e.losing)position,andsoon.

Somedirectionswecouldgofromhere:

Canyouthinkofanyvariantnim-likegamestoexplore?Ihaveaverygeneralprogramforcreatinggame

visualizationsliketheabove,soifyoudescribeavariantgameinthecommentsIwillbehappyto tryto

generateavisualizationforit.

Whataboutnim-likegameswiththree(ormore)piles?Tovisualizethestrategiesforthese,wehavetouse

three(ormore!)dimensions.Ihopetoeventuallycomeupwithawaytodothis,atleastforthreedimensions.Ihappentoknowthatnimismuchmoreinterestinginthreedimensionsthanintwo!

Canyouprovethatthevisualizationoftherectangle-cuttinggamereallylooksliketheabove?Canyoucome

upwithanicewaytocharacterizethewinningstrategy,oratleastthelosingpositions?

Posted in games, pattern, pictures | Tagged nim, strategy, visualization, Wythoff | 4 Comments

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Searchable tiling databasePosted on August 2, 2012

Justalinktodaycheckoutthisawesome tilingdatabase!Itsgottonsofbeautifulplanetilings(with

informationandfurtherreadingabouteachone)andmanywaystosearchthroughthedatabase.Itsagreatway

tofindexamplesofparticularsymmetriesortypesoftilingsbutitsalsofuntojustoohandahhoverrandom

entriesfromthedatabase.

IwishIcouldrememberwhereIfirstcameacrossthis.

Posted in geometry, group theory, links, pattern, pictures | Tagged beauty, database, search, symmetry, tiling | 2 Comments

Book Review: The Enigma of the Spiral WavesPosted on July 14, 2012

TheEnigmaoftheSpiralWaves(SecretsofCreationVolume2)

wordsbyMatthewWatkins,picturesbyMattTweed

MatthewWatkinsandMattTweedhavedoneitagain!Ipreviouslywrotea(verypositive)

reviewofVolumeI thisbookisjustasengaging,ifnotmore.Itexplainsthe Riemann

Hypothesisoneofthebiggest,mostmysteriousopenquestionsinmathematicstodayingreatdetail.But,asonemightexpectafterreadingVolumeI,itremainsthoroughly

accessible,eventothosewithnotmuchmathematicalbackground.Asamathematical

writer,Ifinditincrediblyinspiring:itshowsthatwithenoughtimeandhardwork,itis

possibletoexplainverytechnicalideasinawaythatisaccessiblebutstilldetailedand

accurate.

So,whatistheRiemannHypothesis?Simplyput,itstatesthatthesolutionstothefunction

(where isacomplexnumber)allliealongacertainline.Butthatmakesitsoundboring,likesayingthataroller

coasterisamodeoftransportwithwheels.Beforereadingthisbook,though,Ididntknowmuchmorethan

that.Ihadntthefaintestidea whyitissointerestinganddeep,orwhyanyonewouldthinkthat solvingitwould

beworthonemilliondollars .Thisbookexplainsallthatandmore.ItturnsoutthattheRiemannHypothesisis

intimatelylinkedtothenatureoftheprimenumbers,whichare(still!)quitemysterious.Theyaredefinedby

suchasimplerule,butseemtobehavesoerratically!Whatsgoingon?

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Highlyrecommendedforanyonewhowantsaglimpseintooneofthemostfascinatingandmysteriousopen

questionsinmodernmathematics!

Posted in books, open problems, primes, review | Tagged Reimann Hypothesis, s piral, waves | 1 Comment

Blockly

Posted on June 28, 2012

ItseemsthatGoogleisdevelopingagraphicalprogramminglanguagecalledBlockly,inspiredbyScratchbut

web-based,withtheabilitytocompiledowntoJavaScript,Dart,orPython(orrawXML,soyoucanprocessit

further).IcantsayImallthatexcitedaboutthelanguageitselfnothingnewthere,justthesameoldtired

imperativeprogrammingbutitsureisfun!Giveitatrycanyousolvethemaze ?Howbigofaprogramdoyou

need?

Posted in challenges, programming | Tagged Blockly, Google, graphical, maze, programming | 7 C omments

Picture this

Posted on June 13, 2012

Picturethis!isaverycoolinteractivethingy,madebyJasonDavies,intendedtogetstudents(oranyone,really)

thinkingaboutsomeinterestingmath.Goplayaroundwithitandseeifyoucanansweranyofthelisted

questions(oranyotherquestionsyoumightcomeupwithyourself).Itturnsouttobequiteintimatelyrelatedto

somethingIvewrittenaboutbeforebutIwontspoilitbysayingwhat(atleast,notyet=).

Posted in challenges, links, pattern, pictures | Tagged interactive, picture, rectangles, this | 6 Comments

How to explain the principle of inclusion-exclusion?Posted on June 11, 2012

Ivebeenremissinfinishingmy seriesofpostsonacombinatorialproof.Istillintendto,butImustconfessthat

partofthereasonIhaventwrittenforawhileisthatImsortofstuck.Thenextpartofthestoryistoexplain

thePrincipleofInclusion-Exclusion,butIhaventyetcomeupwithacompellingwaytopresentit.SoperhapsI

shouldcrowd-sourceit.IfyouknowaboutPIE,howwouldyoumotivateandpresentit?Ordoyouknowofany

linkstogoodpresentations?

Posted in combinatorics, meta | Tagged exclusion, inclusion, PIE, presentation | 7 Comments

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Wythoffs game at Three-Cornered ThingsPosted on June 10, 2012

IvereallybeenenjoyingZacharyAbelsseriesofpostsonWythoffsgame[WythoffsGame:RedorBlue?;A

GoldenObservation;TheFibonacciestString;WythoffsFormula],overonhisblog Three-CorneredThings .The

Fibonaccinumbersshowupinthestrangestplaces!

Moregenerally,ifyouhaventseenZacharysblogbefore,gocheckitout.Ifyouenjoymyblog,Ithinkyoull

enjoyhistoo.

Posted in fibonacci, games, links | Tagged Wythoff's game | 7 Comments

Fibonacci multiples, solution 1Posted on June 9, 2012

Inapreviouspost,Ichallengedyoutoprove

If evenlydivides ,then evenlydivides ,

where denotesthe thFibonaccinumber( ).

Heresonefairlyelementaryproof(thoughitcertainlyhasafewtwists!).Picksomearbitrary andconsider

listingnotjusttheFibonaccinumbersthemselves,buttheirremainderswhendividedby .Forexample,lets

choose ,sowewanttolisttheremaindersoftheFibonaccinumberswhendividedby .

Herearethefirst17Fibonaccinumbers:

Andherearetheirremainderswhendividedby ,representedgraphically(red=1,orange=2,blankgap=

0):

Remainders of the first 17 F ibonacci numbers mod 3

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Andindeed,aswewouldexpectifthetheoremistrue,everyfourthremainderiszero: isevenlydivisibleby

.Butthereseemstobeabitmorethanthatgoingonthepatternofremainderswegotisdefinitelynot

random!Letstry .Herearetheremainderswhenthefirst21Fibonaccinumbersaredividedby5:

The first 21 Fibonacci numbers mod 5

Hmm.Everyfifthremainderiszero,asweexpectedbuttheotherremaindersdontseemtofollowanice

patternthistime.

ordothey?Actually,ifyoustareatitlongenoughyoullprobablyfindsomepatternsthere!Nottomention

thatIhaventreallyshownyouenoughofthesequence.Herearetheremaindersofthefirst41Fibonacci

numberswhendividedby5:

First 41 Fibonacci numbers mod 5

Aha!Soitdo esrepeatafterall.Wejusthadntlookedfarenough.

Andjustf orfun,lets includesomeFibonaccinumbersmod .Theserepeatmuchmorequicklythanfor .

Fibonacci numbers mod 8

OK,nowthatwevetriedsomespecificvaluesof ,letsthinkaboutthismoregenerally.Whenlistingthe

remaindersoftheFibonaccinumbersdividedby ,theinitialpartofthelistwilllooklike

becausealltheFibonaccinumbersbefore areofcourselessthan ,sowhenwedividethemby theyare

simplytheirownremainder.Next,ofcourse, leavesaremainderofzerowhendividedbyitself.Thenwhat?

Well, bydefinition,soitsremaindermod is .OK,sofarwehave

Infact,theruleforfindingthenextremainderinthesequencewillbethesameastheruledefiningtheFibonacci

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andnow,itseems,wearestuck!Whatdowegetwhenweadd ?Whoknows?

HereiswhereIwilldosomethingsneaky.Iamgoingtoreplacethesecondcopyof by ,likethis:

Huh!?Whatdoesthatevenmean?Surelywecanneveractuallygetanegativenumberasaremainderwhen

dividingby .Well,thatstrue,butfromnowon,insteadofstrictlywritingtheremainderwhen isdividedby

,Illjustwritesomethingwhichis equivalentto modulo .Thisisallthatreallymatters,sinceIjustwant

toseewhichpositionsareequivalenttozeromodulo .

So,theclaimisthat .Whyisthat?Well, ,andthenwecan

justsubtract frombothsides.

Thenwhat? ;then ,andsoon:wegettheFibonacci

numbersinreverse,withalternatingsigns!

Forexample,herearethefirstfewFibonaccinumbersmod8again,butaccordingtotheabovepattern,with

negativenumbersindicatedbydownwardspointingbars(andtheoriginalbarsshownmostlytransparent,for

comparison):

Seehowthecolorsofthebarsrepeatnow,butrunningforwardsthenbackwards?

If iseven,then willbepositive(asyoucanseeintheexampleabove);thatis,weget

Afterthispointweget ,andsoon,andthewholepatternrepeatsagain,asweveseen.

Whatif isodd?Then willbenegative,sowehavetogothroughonemorecyclewitheverythingnegated:

Thisexplainswhywehadtolookatalongerportionoftheremaindersmod5beforetheyrepeated.Heresmod

5again,withnegativesshowngraphically:

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Thecolorsofthebarsstillrepeat:forwards,thenbackwardsbutalternatingupanddown,thenforwardsbutall

upsidedown,thenbackwardsandalternating(buttheotherway).Butattheendwerebackto ,sothewhole

thingwillrepeatagain.

Inanycase,whether isoddoreven,thesepatternsofremainderswillkeeprepeatingforever,witha always

occurringevery positionsthatis,at ,so willalwaysbedivisibleby !

Thereareotherwaystoprovethisaswell;perhapsIllexplainsomeoftheminafuturepost.Itturnsoutthat

theconverseisalsotrue:if evenlydivides ,then mustevenlydivide .Idontknowaproofoffthetopof

myhead,butmaybeyoucanfindone?

Posted in fibonacci, modular arithmetic, number theory, pattern, pictures, proof, sequences | Tagged divisibility, fibonacci, proof, remainders | 5 Comments

Nature by NumbersPosted on June 6, 2012

Thishasbeenmakingtheroundsofthemathblogosphere(blathosphere?),butincaseyouhaventseenityet,

checkoutCristbalVilasawesomeshortvideo, NaturebyNumbers.EspeciallyappropriategiventhatIhave

beenwritingaboutFibonaccinumberslately(Illpostasolutionto thechallengesoon).

Posted in fibonacci, golden ratio, links, video | Tagged fibonacci, nature, numbers, phi, v ideo | 1 Comment

Fibonacci multiplesPosted on May 15, 2012

Ihaventwrittenanythinghereinawhile,buthopetowritemoreregularlynowthatthesemesterisoverI

haveaseriesoncombinatorialproofstofinishup,somebookstoreview,andafewotherthingsplanned.Butto

easebackintothings,heresalittlepuzzleforyou.RecallthattheFibonaccinumbersaredefinedby

.

Canyoufigureoutawaytoprovethefollowingcutetheorem?

If evenlydivides ,then evenlydivides .

(Incidentally,theexistenceofthistheoremconstitutesgoodevidencethatthecorrectdefinitionof is ,not

.)

Forexample, evenlydivides ,andsureenough, evenlydivides . evenlydivides ,andsure

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Iknowoftwodifferentwaystoproveit;thereareprobablymore!NeitheroftheproofsIknowisparticularly

obvious,buttheydonotrequireanydifficultconcepts.

Posted in arithmetic, challenges, f ibonacci, number theory, pattern | Tagged divisibility, fibonacci | 12 C omments

Carnival of Mathematics 86

Posted on May 8, 2012

Welcometothe86th CarnivalofMathematics! issemiprime,nontotient,andnoncototient.Itisalsohappysince and .Infact,itisthe

smallesthappy,nontotientsemiprime(theonlysmallerhappynontotientis68whichis,ofcourse,86in

reversebut68isnotsemiprime).

However,themostinterestingmathematicalfactabout86(inmyopinion)isthatitisthelargestknowninteger

forwhichthedecimalexpansionof containsnozeros!Inparticular, .

Althoughnoonehasprovedit isthelargestsuch ,every upto (whichisquitealot,althoughstill

slightlylessthanthetotalnumberofintegers)hasbeencheckedtocontainatleastonezero.Theprobability

thatanylargerpowerof2containsnozerosisvanishinglysmall,givensomereasonableassumptionsaboutthe

distributionofdigitsinbase-tenexpansionsofpowersoftwo.

Eighty-sixisalsoapparentlysomesortofslangterminAmericanEnglish,butitreallyhasnothingtodowith

math,sowhocares?Onwardtothecarnival!Ihadalotoffunreadingallthesubmissions,andhavedecidedto

organizethemsomewhatthematicallythoughtheydontalwaysfitperfectly,sodontassumeyouwontbe

interestedinapostjustbecauseofmycategorization!

Art

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ChristianPerfecthasstartedaseriesofpostsonthethemeofArtyMaths,withlinkstoartisticimagesand

videoswithamathematicalbent.Aboveisacoolexample,somesortofstellatedpolyhedronmadeoutof money

byKristiMalakoff(youcanfindmorehere).

KatieStecklessubmittedalinktoRobbyIngebretsensblogpostFirstDigital3DRenderedFilm(from1972)and

MyVisittoPixar.Katiesays,

Thisispossiblytheearliestexampleofacomputeranimation,andoneofitstwocreators,Edwin

Catmull,whowentontofoundPixar,iscreditedwithhavingwork[ed]out[the]mathtohandle

thingsliketexturemapping,3Danti-aliasingandz-buffering.Fascinatingtothinkhehadtoinventall

ofthatinordertodothis!

Robbysblogpost(andtheextensivecommentsonit)givealotmorecontextandfascinatingdetails.And,of

course,youcanwatchthevideoitself!

MikeCroucherofWalkingRandomlywritesaboutsomecoolmathematically-themedstainedglasswindows,and

wonderswhetheranyoneknows ofanyot hers.

Statistics/dataanalysis

ArthurCharpentierofFreakonometricswritesaboutNonconvexity,andplayingindoorpaintball:ifabunchofpeopleinanonconvexplayingareaareallholdingwaterpistolsandshootattheclosestperson,whodoesntget

wet?

KatieStecklessubmittedalinkto Data:itshowstoresknowyourepregnant ,anarticlebyMatthewLaneof

MathGoesPop!Everwonderhowcompaniescanpredictvariousthingsaboutyou(suchaswhetheryouare

pregnant!)basedonyourbrowsinghabitsandotherpubliclyavailabledata?Thisarticleexplainssomeofthe

basicmathunderlyingthissortofdatamining.

JohnCookofTheEndeavouranswersthequestion:Whatisrandomness?inRandomisasrandomdoes .Itturns

outthatthebestanswermightjustbetoavoidansweringatall!

Geometry

AugustusVanDusenof thinkingmachineblogsubmittedaposttitledSuperellipse,saying

IreadanarticleaboutSergelstorg,aplazainStockholm,beinganexampleofasuperellipse.WhenI

lookedupsuperellipseonWolframmathworld,InoticedthattheareaformulainvolvedgammaFollowFollow



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functions.Ithendecidedtoderivetheresultmyselftoseeifitcouldbesimplifiedandhowitwould

reducetothefamiliarformulafortheareaofanellipse.

FrederickKohofWhiteGroupMathematicssharesageometricsolutiontoanoptimizationproblem thatdoesnt

initiallyseemlikeithasanythingtodowithgeometry.

ZacharyAbelofThree-CorneredThings haswrittenaseriesofthreeexcursionsintothemiraculousand

interconnectedworkingsofthehumbletriangle:ManyMorleyTriangles,SeveralSneakyCircles,andThree-

CorneredDeltoids.ThesearesomeofmypersonalfavoritesfromthismonthsCarnival:chock-fullofsurprising

mathematicsandbeautifulillustrationsandanimations!

Teaching

ColinWrightwritesTheTrapeziumConundrum:howshouldatrapezium(akatrapezoidifyourefromtheUS)

bedefinedwithexactlyonepairofparallelsidesoratleastonepairofparallelsides?Moregenerally,howare

definitionsarrivedatandagreedupon?Theanswermaydependontheaudience.

KarenG.ofschoolaramamusesupontherelationshipbetweenlanguageandlearningplacevalueinherpost

LookingtoAsia.

OnherblogMathMamaWrites,SueVanHattumwritesabout LinearAlgebra:LeadingintotheEigenStuff.Sue

says,Imteachinglinearalgebraforthefirsttimeinoveradecade.Thathasmeantrelearningitadelightful

experience.

PaulSalomonofLostInRecursionwritesExponentsandtheScaleoftheUniversea21stCenturymathlesson ,

afunstoryabouthowaninitiallydrylessononexponentsturnedintoaremarkablelearningexperience.

Fun

AlistairBirdsubmittedalinktoEnormousIntegers,saying,

Itsstillacommonenoughmisconceptionthatpuremathematicsresearchmustbeaboutlargerand

largernumbers,butitsstillnicetosometimesplayuptothisstereotype,asJohnBaezsblogposton

AzimuthaboutEnormousIntegersdoes.Commentsareworthalooktoo.

PatBallewwritesonPatsBlogaboutPandigitalPrimes:exploringpandigitalprimesandfindingouthowhandy

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Computerprogrammingskillsmightbe.

Quick,whatcomesnextintheseries ?Theanswer,asexplainedbySteven

Landsburgonhisblog,TheBigQuestions,maysurpriseyou!(ThankstoKatieStecklesforthesubmission, via

AlexandreBorovik.)

PaulSalomonofLostInRecursiondisplaysTheLostinRecursionRecursion.Canyoufigureoutwhatsgoing

onwithoutgettinglostintheTheLostinRecursionRecursionrecursion?

StuffThatDidNotFitInAnyOtherCategoryButIsStillAwesome

ColinBeveridgeofFlyingColoursMathssubmittedSecretsoftheMathematicalNinja:Thesurprisingintegration

ruleyoudontgettaughtinschool ,andwrites,

WhenIstumbledacrossthisrule,myreactionwaswhoa.Itsquick,itsextremelydirty,andits

surprisinglyaccurate.Thekindofthingthemathematicalninjadreamsof.

AndrewTaylorwritesaguestpost,ElectoralReformsandNon-TransitiveDice,onTheAperiodical,explaining

WhyChoosingaVotingSystemisHardintermsofasetofnontransitivedice.

PeterRowlett,of TravelsinaMathematicalWorld,opinesinhispost,Whatanicejobyouhave,thatapopular

rankinglistingmathematicianasoneofthetoptenbestjobsshouldntjustbeacceptedandrepeated

uncritically.

InherarticleHowcultureshapedamathematician,CarolClarkgivesaglimpseintothelifeandbackgroundof

mathematicianSkipGaribaldi.Shewrites:

Mathematiciansseetheworlddifferentlythanme.Iinterviewedamathematiciantogetaglimpseof

thatview,andlearnedhoweverythingfromfinearttopopularfilmsandbooksplayedaroleinshaping

thatview.

ThepreviousCarnivalofMathematicswashostedatTravelsinaMathematicalWorld;nextmonth,the87th

CarnivalwillbehostedbyMr.Chaseat RandomWalks,sostartgettingyoursubmissionsreadynow!Forlistsof

pastandfuturecarnivals,instructionsonsubmitting,andanswerstofrequentlyaskedquestions,seethe main

CarnivalofMathematicssite.ThenextMathTeachersatPlaycarnivalisalsocomingupsoon,witha submissionFollowFollow



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deadlineofthisFriday.

Posted in links | Tagged Carnival of Mathematics | 3 C omments

IllbehostingtheCarnivalofMathematics,andthesubmissiondeadlineiscomingupsoonTuesday,May1.Pleasesubmitsomething!Itcouldbe

somethingyouwrote,orsomethingsomeoneelsewrotethatyouenjoyed.Allmathematicsrangingfromelementarytoadvancediswelcome.

Posted on April 23, 2012

Book review: In Pursuit of the Traveling SalesmanPosted on April 7, 2012

Asmathematicalproblemsgo,thetravelingsalesmanproblem(TSP)is

araregem:itissimultaneouslyofgreattheoretical,historical,and

practicalinterest.Onthetheoreticalfront,itisawell-knownexampleof

theclassofNP-completeproblems,whichlieattheheartofthe million-

dollarPvsNPquestion(whichIstillintendtoexplainatsomepoint).

Historically,ithasbeenstudiedforalmost200years(givenasufficientlyinclusivedefinitionofstudy),andhasoccupiedaplaceinthepublic

consciousnessforatleastthelast50.Andthisgreathistoricalinterestis

partlyduetotheproblemspracticalsignificance.

So,whatisit?Givenasetofpointsintheplane(or,moregenerally,aset

ofpointswithdistancesspecifiedsomehowbetweeneachpairof

points),theproblemistodeterminethe shortestpathwhichvisitsevery

pointexactlyonceandthenreturnstothestartinglocation.Ofcourse,in

onesensethisiseasy:justlistallpossiblepathsandcomputethe

lengthofeach.Note,however,thatforasetof points,thereare

(thatis, )possiblepathsthatvisiteverypointonceandthenreturntothestart.Evenwithonly

points,thatsawhopping possiblepathsifyoucouldcomputethelengthof

onetrillionpathseverysecond,itwouldstilltake280millionmillenia(thats years,slightlylongerthan

Ivebeenalive)tocheckallofthem!Andthatsonly pointsinpractice,peoplewanttosolvetheTSPforsets

ofpointsmuchlargerthan .Sothepointisnotjusttosolvetheproblem;therealquestionis,canitbe

solvedefficiently?

Amazingly,nooneknows!Butthathasntstoppedpeoplefromcomingupwithextremelycleveralgorithmsthat

seemtoworkwellinpractice,onverylargesetsofpoints( i.e.thousands,oreventensofthousandsof

points)eventhoughtherearepathologicalinputsforwhichthesealgorithmsdoessentiallynobetterthan

justtryingeverypath.SothesealgorithmsconstituteagoodsolutiontotheTSPf romapracticalpointofview

butnotatheoreticalone!

WilliamCooksnewbook,InPursuitoftheTravelingSalesman:MathematicsattheLimitsofComputation,doesa

wonderfuljobpresentingthehistoryandsignificanceoftheTSPandanoverviewofcutting-edgeresearch.Itsa

beautiful,visuallyrichbook,fullofcolorphotographsanddiagramsthatenlivenboththenarrativeand

mathematicalpresentation.Anditincludesawealthofinformation(perhapsevenabit toomuchattimes;IgotFollowFollow



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lostinafewplaces).Butitactuallybillsitself,partly,asanintroductiontocutting-edgeideasinTSP

researchandIthinkoverallitsucceedsadmirably,explainingideasinwaysthatareaccessiblebutnot

patronizing.Readthisbookifyouwantafun,beautifullyillustratedintroductionto(thisonefascinatingpiece

of)theedgeofhumanknowledge!

Posted in books, computation, geometry, open problems, review | Tagged book review, salesman, TSP. traveling | 6 Comments

New Carnival of MathematicsPosted on April 5, 2012

TheCarnivalofMathematicshasbeenrevived!AbigthankstoMikeCroucherofWalkingRandomlyfor

organizingitforthepastfewyears,andto KatieSteckles,ChristianPerfect,andPeterRowlettfortakingover.

Thelatestedition,carnival#85(wow,hasitreallybeengoingthatlong?)isnowupatPeterRowlettsblog,

TravelsinaMathematicalWorld.Lotsofcoolstuffthere,sobesuretocheckitoutifyouhaventalready.

IllbehostingCarnival#86here,so pleasesubmitsomething!ThedeadlineisMay1st,andIllpostthecarnival

sometimetheweekafterthat.

Posted in links, meta | Tagged Carnival of Mathematics

Making our equation countPosted on March 3, 2012

[Thisispost#4inaseries;previouspostscanbefoundhere: Differencesofpowersofconsecutiveintegers ,

Differencesofpowersofconsecutiveintegers,partII ,Combinatorialproofs.]

Werestilltryingtofindaproofoftheequation

whichexpressesthef actthatacertainarithmeticprocedurealwaysseemstoresult,strangelyenough,inthe

factorialof .LasttimeIintroducedtheideaofusingacombinatorialproof,andgaveasimpleexampleinvolving

abinomialcoefficientidentity.

Inorderforthisideatoyieldanyfruit,weneedawaytointerpretthevariouspiecesoftheequationas counting

something.Letsgooverthepiecesonebyoneanddiscusssomewaystointerpretthemcombinatorially.

Factorial,permutations,andmatchings

Letsstartwiththeright-handside, .Thisoneisnottoohard: countsthenumberofpermutationsofobjects,thatis,thenumberofdifferentwaystotake distinctobjectsandarrangetheminanorderedlist.Why

isthat?Well,thereare objectswecouldchoosetoputfirst;oncewevemadethatchoice,thereare

remainingobjectswecouldchoosetogosecond;then choicesforthethirdobject,andsoon,foratotalof

choices.Forexample,herearethe differentpermutationsofsize :

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However,theresanotherwaytothinkaboutpermutationswhichwillcomeinhandylater.Namely,wecanthink

ofapermutationasamatchingbetweentwosetsofsize .Youknow,likethosepuzzlesthatgivetwo

side-by-sidelistsandsaydrawalinematchingeachcartooncharacterwiththeirfavoritecheese!(or

whatever).Likethis:

Herewehaveamatchingbetweentwosetsofsize .Eachdotontheleftismatchedwithexactlyonedotonthe

right,andviceversa.

Whyarematchingsanotherwayofthinkingaboutpermutations?First,itsnottoohardtoseethattherearealso

matchingsbetweentwosetsofsize :wehave possiblechoicesofwhattomatchthefirstelementwith;

thenthereare choicesleftoverforwhattomatchthesecondelementwith,andsoon.

Butwecanalsoseeacorrespondencebetweenpermutationsandmatchingsmoredirectly.Startbylabelingthe

dotsontheleftofamatchingwithconsecutivenumbers:

Now,imagineeachnumbertravelingalongthecorrespondingrededgeuntilitreachesthedotontheother

side.Likethis:

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Seehowthe traveleddownthesteepedgetoendupatthefourthdotfromthetop;the traveledacrossthe

horizontaledgetostayinthesameposition;the traveleduptothetop;andsoon.

Whatwegetoutisalistofthenumbersfrom16insomeorder;inthisexampleweget .Inother

words,wecanviewamatchingasalittlephysicalmachinefortakingalistofobjectsandputtingtheminto

someparticularorder.

Hereareallthepermutationsofsize again,thistimevisualizedasmatchings.

Now,atthispointIamverytemptedtogooffonatangentexploringgrouptheory,symmetrygroups,andall

sortsofotherstuff,butIshallrestrainmyself(fornow!).

Binomialcoefficients

Anotherpieceoftheequationisthebinomialcoefficient .Butofcoursewealreadyknowwhatbinomial

coefficientscount isthenumberofwaystochoose thingsoutof ,thatis,thenumberofsize- subsetsof

asize- set.(Ialsotalkedaboutthislasttime .)

ExponentiationandfunctionsWhatabout ?Whatdoesthatcount?Itturnsoutthatexponentiationcorrespondstocountingfunctions:in

particular, isthenumberoffunctionsfromasetofsize toasetofsize .Whyisthat?Well,foreachofthe

elementsofthedomain,wehave choicesforwhereafunctioncouldsendit,andeachofthesechoicesis

independentsothetotalnumberofchoicesis .

Forexample,hereareallofthe functionsfromasize- settoasize- set:

Hmmmthislooksfamiliar!Notethatsomeofthesefunctionsarematchings,andsomearent.Perhapsyoure

startingtogetaninklingnowwhyIintroducedtheideaofpermutationsasmatchings

Allthepiecesarealmostinplacenow.TheonepieceoftheequationwestillhaventyettalkedaboutisthatFollowFollow



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mysterious .Itcertainlydoesntmakesensetointerpretthatasthenumberoffunctionsfromasetofsize

toasetofsizenegativeone,becauseofcoursethereisnosuchthingasasetwithanegativesize.Sohow

canweinterpretitcombinatorially?TheanswerliesinsomethingcalledthePrinc ipleofInclusion-Exclusion(or

PIEforshort),whichwillbethesubjectofmynextpost!

Posted in combinatorics, pictures | Tagged binomial coefficients, combinatorics, functions, m atching, permutation | 3 Comments

Combinatorial proofsPosted on February 17, 2012

Continuingfromapreviouspost,wefoundthatifwebeginwith thpowersof consecutiveintegersand

thenrepeatedlytakesuccessivedifferences,itseemslikewealwaysendupwiththefactorialof ,thatis,

.Wethenderivedanalgebraicexpressionfortheresultoftheiterativedifferenceprocedure.So

thegoalnowistoprovethat

thatis,

Now,itspossible(probable,infact)thatthiscanbeprovedbypurelyalgebraicmeans.Ifyoucomeupwith

suchaproofIwouldlovetoseeit!ButImustconfessthatIspentseveralhoursbangingmyheadagainstit

(algebraicallyspeaking)withoutmakinganyprogress.EventuallyIturnedtotheideaofacombinatorialproof.

WhatdoImeanbythat? Combinatoricsisthesubfieldofmathematicsconcernedwith counting.Theessenceofa

combinatorialproofistoshowthattwodifferentexpressionsarejusttwodifferentwaysofcountingthesame

setofobjectsandmustthereforebeequal.Ive describedsomecombinatorialproofsbefore,incountingthe

numberofwaystodistributecookies.

Asanothersimpleexample,considerthebinomialcoefficient identity

Itscertainlypossibletoprovethisalgebraically,byexpandingoutthebinomialcoefficients(using ),

butwecangiveamoreelegantproof,basedonthefactthat isthenumberofwaystochooseasubsetof

thingsoutofasetof things.Forexample,herearethe waystochoosethreethingsoutasetoffive:

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Considerthefirstelementofthesize- set.Everysubsetofsize eitherincludesthisfirstelement,oritdoesnt.

Thenumberofsize- subsetswhichdonotincludethefirstelementis ,sincethatsthenumberofwaystochoose thingsfromtheremaining elements.Thenumberofsize- subsetswhichdoincludethefirst

elementis ,becausetheycorrespondtochoosing oftheremaining things.Therefore

.

Heresanillustrationofhowthisworksintheparticularcasewhen and :

Noticehowthetensubsetsfromabovehavebeensplitintotwogroups:thefirstgroup,ontheleft,arethose

thatdontincludethefirstelement;youcanseethateachofthemcorrespondstooneofthe size- subsets

oftheremainingfourelements.Thesecondgroup,ontheright,arethosethatdoincludethefirstelement;each

correspondstooneofthe size- subsetsoftheremainingfourelements.

Sothatstheideaofacombinatorialproof.Andwewanttodosomethingsimilarfortheidentitywearetrying

toprovealthough,ofcourse,itsgoingtobeabitmoredifficult!

Youmighthavefuntryingtothinkaboutwhatacombinatorialproofof ourtargetequationmightlooklike;

althoughifyoudonthavemuchexperiencewithcombinatoricsyoumayhavetroublecomingupwithwhatsorts

ofthingsthetwosidesoftheequationmightbecounting!ThatswhatIlltalkaboutinmynextpost.

Posted in combinatorics, pictures, proof | Tagged binomial coefficients, combinatorial proof, identity | 12 Comm ents

Differences of powers of consecutive integers, part IIPosted on February 16, 2012

Ifyouspentsometimeplayingaroundwiththeprocedurefrom Differencesofpowersofconsecutiveintegers

(namely,raise consecutiveintegerstothe thpower,andrepeatedlytakepairwisedifferencesuntilreaching

asinglenumber)youprobablynoticedthecuriousfactthatitalwaysseemstoresultinafactorialinthe

factorialof ,tobeprecise.

Forexample:

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Severalcommentersfiguredthisoutaswell.Doesthisalwayshappen?Ifso,canweproveit?

Letsstartbythinkingaboutwhathappenswhenwedothesuccessive-differencingprocedure.Ifwestartwith

thelist ,thenweget .(Iwanttokeepthelettersinorder,whichiswhyIwrote insteadof .

Insteadofsubtractingthefirstvaluefromthesecond,wecanthinkofitasaddingthenegationofthefirstvalue

tothesecond.)Ifwestartwith ,weget

.

(Thenegationof is ;addingthisto yields .)From weget

Doyouseeanypatternsyet?Letsdoonemore.Fromtheabovecalculationwecanalreadyseethatdoingfour

iterationson willresultin (doyouseewhy?).Doingonefinaliteration

givesus

.

Hmm.Letsmakeatable.

Result

1

2

3

4

5

Iincludedonemorerow(whichyoucanverifyifyoulike).Nowdoyouseeapattern?Thecoefficientsseemto

betakenfromPascalstriangle,butwithalternatingsigns!

Infact,itsactuallynottoohardtoseewhythishappens.Ateachstepwetaketwooffsetcopiesoftheprevious

row(by"of fset"I meanthatthelettersareshif tedbyone,like and )andaddthenegationof

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thefirsttothesecond.Sincethesignsarealternating,wereallyendup addingabsolutevaluesofthe

coefficients.Probablythebestwaytoreallyseethisisthroughanexample:

Noticehowweflipallthesignsinthefirstrow,sothattheymatchthesignsinthesecondrow.Butthisis

exactlyhowPascalstriangleisgeneratedeachrowisthesumofthepreviousrowwithitself,offsetbyone.

Now,intherealproblem,wedontstartwith ,butwiththe thpowersof consecutiveintegers.

Letscallthefirstinteger ,sothesequenceofconsecutiveintegersis .Giventhis,wecan

nowwritedownanexpressionforwhatweendupwithafterdoingtheiterateddifferenceprocedure:

Letsbreakthisdownabit.Weknowthatwegetasumof terms;thatsthe part(youcanreadmore

aboutsigmanotationhere ).Welluse toindextheterms.Wealsoknowthatthetermsalternatesign,sowe

needtoinclude raisedtosomepowerinvolving ;thelasttermisalwayspositive,soweuse ,whichis

equalto when .Thebinomialcoefficient givesusthe thentryonthe throwofPascalstriangle.Finally,ofcourse, isthe thpowerofoneoftheintegerswestartedwith.

Theclaim,therefore,isthat

AndIwillproveittoyou,withprettypictures,aspromised!

Posted in arithmetic, iteration, pascal's triangle | Tagged binomial coefficients, consecutive, difference, integers, powers | 3 Comments

1717 4-coloring with no monochromatic rectanglesPosted on February 9, 2012

Quick,whatsspecialaboutthefollowingpicture?

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AsjustannouncedbyBillGasarch,thisisa gridwhichhasbeenfour-colored(thatis,eachpointinthe

gridhasbeenassignedoneoffourcolors)insuchawaythatthereareno monochromaticrectangles,thatis,no

fourgridpointsformingthecornersofanaxis-alignedrectangleareallofthesamecolor.Forexample,ifwe

changethetop-leftgridpointtored,wecanseeseveralmonochromaticrectanglespopup:

OrheresanotherversionwhereIrandomlypickedagridpointinthemiddle,changeditscolor,andsureenough,

moremonochromaticrectanglesresult:

Asyoucantryverifyingforyourself(andasIalsoverifiedusingacomputerprogram),therearenosuchmonochromaticrectanglesinthefour-coloringatthetopofthispost!(Ifyouwanttoplaywiththefour-coloring

yourself,hereitisinasimpledataformat.)

Forseveralyearsnooneknewifthiswaspossible,andBillhadofferedaprizeof$289(thats ,ofcourse)to

anyonewhocouldfindsuchafour-coloring.TheprizewillbecollectedbyBerndSteinbachandChristian

Posthoffyoucanfindmoredetailsin Billspost.Nooneyetknowsexactlyhowtheyfoundtheirfour-coloring,FollowFollow



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butapparentlytheywillbepresentingapaperaboutitinMay.Illtrytowritemoreaboutitthen(ifI

understanditatall)!

Ifyoureinterestedinreadingmoreaboutthehistoryandmathbehindthisproblem(andtogetsomeintuition

forwhyitisdifficult),takealookatthesepostsbyBrianHayesonhisblog,bit-player: The1717challengeand

17x17:Anonprogressreport.IalsorememberseeingHeresafuninteractiveappletwhereyoucanplayaround

withtheproblem,createdbyMartinSchweitzer.

Posted in open problems, pattern, people, pictures | T agged 17x17, four-coloring, graph, grid, monochromatic, rectangles | 5 Comments

Book review: Nine Algorithms that Changed the FuturePosted on February 4, 2012

NineAlgorithmsthatChangedtheFuture:theIngeniousIdeasthatDriveTodays

Computers,byJohnMacCormick.PrincetonUniversityPress,2012.

Imoftenwaryofbookswrittenforgeneralaudiencesontechnicaltopics.Itsquite

difficulttowriteinawaythatisbothaccessibletoawideaudienceandtechnically

accurate.Manysuchbooksendupsacrificingaccuracyinthenameofaccessibility,

tryingtoconveyjusttheintuitionorgeneralsenseofsometopic,butoftenend

upgivingpeoplethewrongideainstead.

Iwasquitehappytofind,therefore,thatJohnMacCormicknailsit:9Algorithms

thatChangedtheFutureistechnicallyrightonthemoney,butmanagestoexplain

thingsinwaysthatarebothunderstandableandfun.Wanttounderstandhow

Googlerankssearchresults?OrhowAmazonmanagestoneverloseormessupyourorderinformation,even

thoughtheygethundredsofthousandsoforderseachdayand(asweallknow)networksandharddrivesare

unreliable?Everwonderhowyoucanordersomethingovertheinternetwithoutyourcreditcardnumberbeing

stolen?Orhowzipisabletomakeyourfilessmaller,seeminglybymagic?Evenifyouhaveneverwondered

aboutthesethings,perhapsIhavemadeyouwonderaboutthemjustnow.Andthatsexactlythepointofthis

book:therearequiteafewingeniousalgorithmicideasthatmostofusrelyon everydaythatwerarelyor

neverevenstoptowonderabout.

Forexample,Iactuallylearnedsomethingnew:Iknewaboutpublic-keycryptographybuthadneverreally

knownmuchaboutDiffie-Hellmankeyexchange,whichiswhatallowsyourwebbrowsertotalkto,say,

Amazonsserverssecurelyeventhoughtheyhavenevercommunicatedbefore.Its likehavingasecret

conversationincodewithapen-palwhomyouvenevermet,eventhoughlotsofpeoplearereadingyourmail.

Howcanyoueveragreeonasecretcodeinthefirstplacewithoutthepeoplereadingyourmailfindingout(and

hencebeingabletoreadallyoursubsequentcodedmessages)?Soundsimpossible,doesntit?Butitturnsout

thatitispossible,withsomecleverideas,whichMacCormickskillfullyexplainsusingafunmetaphorabout

mixingcolorsofpaint.

Eachchapterstartsoutverysimply,graduallybuildingupmorecomplexexamplesuntilyoureachafull

understandingofthealgorithmbeingexplained.AlongthewayMacCormickintroducesthetrickstheclever,FollowFollow



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centralideasthatmakeeachalgorithmwork.Thewritingisexcellent:clear,precise,andfun.Ihighly

recommendthisbooktoanyonecuriousabouttheingeniousmathematicalandalgorithmicideasunderlying

someoftodaysmostubiquitoustechnology.

Posted in books, computation, review | Tagged algorithms, history, J ohn MacCormick

Computing with decadic numbersPosted on January 30, 2012

[Thisistheninth,and,Ithink,finalinaseriesofpostsonthe decadicnumbers(previousposts:Acuriosity,An

invitationtoafunnynumbersystem ,Whatdoes"closeto"mean?,Thedecadicmetric,Infinitedecadicnumbers,

Morefunwithinfinitedecadicnumbers,Aself-squarenumber,u-tube).]

Inapreviouspost,wefoundadecadicnumber

withthecuriouspropertythatitisitsownsquare,eventhoughitisobviouslynotzeroorone.Wethenderiveda

moreefficientalgorithmforgeneratingthedigitsof .HeressomeHaskellcode(explainedinthepreviouspost)

whichimplementsthemoreefficientalgorithm,whichIincludeherejustsothatthispostwillbeavalidliterate

Haskellfileinitsentirety.

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> {-# LANGUAGE TypeSynonymInstances

> , FlexibleInstances

> #-}

>

> module Decadic2 where

>

> import Control.Monad.State

>

> -- State for incrementally constructing u_n.

> -- Invariant: curT = 10^n; un^2 = pn*curT + un

> data UState = UState { pn :: Integer

> , un :: Integer

> , curT :: Integer

> }

> deriving Show

>

> -- u_1 = 5; 5^2 = 25 = 2*10 + 5

> initUState = UState 2 5 10

>

> uStep :: State UState Int

> uStep =do

> u p t

> let d = p `mod` 10 -- next digit

> u' = d * t + u -- prepend the next digit to u

> p' =(p + 2*d*u + d*d*t) `div` 10 -- see above proof

>

> put (UState p' u' (10*t))-- record the new values

>

> return $ fromInteger d -- return the new digit

>

> type Decadic =[Int]

>

> u :: Decadic> u = 5 : evalState (sequence $ repeat uStep) initUState

Toroundthingsout,Idliketoshowoffsomeofthecoolthingswecandowiththis.First, asweknow,its

possibletodoarithmeticwithdecadicnumbers.Soletsimplementit!

Additionofdecadicnumbersisdonejustlikeadditionoftheusualdecimalnumbers:weaddcorresponding

places(i.e.,lineupthenumbersoneundertheotherandthenaddincolumns).

> plus :: Decadic -> Decadic -> Decadic

First,wehavespecialcasesforzero,representedbytheemptylistofdigits:inthosecaseswejustreturnthe

othernumberunchanged.

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> plus [] n2 = n2

> plus n1 [] = n1

Next,toaddadecadicnumberwhosefirstdigitisxtoadecadicnumberwhosefirstdigitis y,wejustaddxandy

andthencontinueaddingrecursively.

> plus (x:xs)(y:ys)=(x+y) : plus xs ys

Ofcourse,werenotdone:thisdoesntdoanycarrying.Insteadofmodifyingour plusfunctiontodocarrying,we

justwriteafunctionnormalizewhichmakessureeveryplaceinadecadicnumberisbetween and ;itwillcome

inhandyformorethanjustaddition.

> normalize :: Decadic -> Decadic

Thenormalizefunctionsimplycallsarecursivehelperfunctionnormalize'whichkeepstrackofthecurrent"carry".

Thestartingcarry,ofcourse,iszero.

> normalize = normalize' 0

Tonormalizezero(theemptylist)whenthecurrentcarryiszero,justreturntheemptylist.

> where normalize' 0 [] = []

Withanonzerocarryandtheemptylist,wesimplyextendthelistwithaspecialzerodigitandcontinue

normalizing.

> normalize' carry [] = normalize' carry [0]

Inthegeneralcase,weaddthecurrentcarrytothenextdigitx,andcomputethequotientandremainderwhen

dividingthissumbyten.Theremainderisthenextdigitd,andthequotientisthenewcarrywhichgetspassed

alongrecursively.

> normalize' carry (x:xs)= d : normalize' carry' xs

> where(carry', d)=(carry + x) `divMod` 10

Andnowformultiplication,whichisbasedontheobservationthatzerotimesanythingiszero,andinthe

generalcaseFollowFollow



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.

> mul :: Decadic -> Decadic -> Decadic

> mul [] _= []

> mul _ [] = []

> mul (x:xs)(y:ys)= x*y : (map (x*) ys + (xs * (y:ys)))

Finally,wedeclareDecadictobeaninstanceofthe Numclass,whichallowsustousedecadicnumbersinthesame

waysthatwecanuseothernumerictypes:

> instance Num Decadic where

Toaddormultiplydecadicnumbers,usetheplusandmulfunctionsandthennormalize.

> n1 + n2 = normalize (plus n1 n2)

> n1 * n2 = normalize (mul n1 n2)

Tonegateadecadicnumber,subtractthelastdigitfrom10andtherestofthedigitsfrom9.

> negate [] = []

> negate (x:xs)= normalize $ (10-x) : negate' xs

> where negate' [] = repeat 9

> negate' (x:xs)=(9-x) : negate' xs

Finally,toconvertanintegerintoadecadicnumber,puttheintegerintoalistofoneelementand normalize.

> fromInteger = normalize . (:[]) . fromInteger

So,letstryit!Wellwantawaytodisplaydecadicnumbers:

> showDecadic :: Decadic -> IO ()

> showDecadic d = putStrLn . dots $ digits

> where d' = take 31 d

> dots | length d' | otherwise =("..." ++)

> digits = concat . reverse . map show . take 30 $ d'

Normaldecimalintegerscanalsobeusedasdecadicnumbers:

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*Decadic2> showDecadic 7

7

Heres :

*Decadic2> showDecadic u

...106619977392256259918212890625

Andheres ;ithadbetterbethesameas !

*Decadic2> showDecadic (u^2)

...106619977392256259918212890625

Well,lookslikeitsthesameforthefirst30digitsatleast!Wecanalsocompute .Remember,if then

,so shouldbeanotherself-squarenumber.Rememberhowwethoughttheremightbeaself-squarenumberendingin ?Well,thisisit!

*Decadic2> showDecadic (1-u)

...893380022607743740081787109376

*Decadic2> showDecadic ((1-u)^2)

...893380022607743740081787109376

Finally,wecancheckthat :

*Decadic2> showDecadic (u * (1-u))

...000000000000000000000000000000

Ifyourecall,thisisinsomesensethefundamentalreasonwhythedecadicnumbersactsofunny,becauseithas

zerodivisors:pairsofnumbers(like and ),neitherofwhichiszero,whoseproductisnonethelesszero.

Now,ifyouremember,fromevenfurtherback,whatgotusintothiswholedecadicmessinthefirstplace:

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Inthatfirstpost,Isaid

ImanagedtoextendthispatternforafewmoredigitsbeforeIgotbored.Doesitcontinueforeveror

doesiteventuallystop?Isthereanydeepermathematicalexplanationlurkingbehindthissupposed

curiosity?Whatssospecialabout ?Dopatternslikethisexistforotherfunctions?

Well,bythispointIhopeitsclearthatthereisindeedadeepermathematicalexplanationlurking!Theequation

admitsthesolutions and ,butdoesitadmitanyotherdecadicsolutions?Noticethatgiven

,whichhas and assolutions,then (and )arealsosolutions:

.

Sointhiscaseweget

asasolution(theothersolutionisnotadecadicinteger).

Toimplementit,weneedawaytohalvedecadicnumbers(Illletyouworkoutwhatsgoingonhere):

> halve :: Decadic -> Decadic

> halve [] = []

> halve t@(s:_)

> | odd s = error "foo"

> | otherwise = halve' t

> where

> halve' [] = []

> halve' [x]=[x `div` 2]

> halve' (x:x':xs)=(x `div` 2 + adj) : halve' (x':xs)

> where adj | odd x' = 5

> | otherwise = 0

Andnowwecandefine

> q = halve (3*u - 1)

*Decadic2> showDecadic q

...159929966088384389877319335937

*Decadic2> showDecadic (2*q^2 - 1)

...159929966088384389877319335937

Woohoo!Thisclearlyshowsthatthepatterndoes,infact,continueforever.Italsoshowsusthat is

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notparticularlyspecial:anyquadraticfunctionthatfactorsas ,attheveryleast,willleadtoapattern

likethis,andprobablylotsofotherequationsdotoo.

Ifyoureinterestedinreadingmore,hereswhereIgotsomeofmyinformation .Forexample,youcanread

abouthowthereisanothernumber ,definedbystartingwith anditerativelyraising

tothefifthpower(justaswedefined bystartingwith andsuccessivelysquaring),suchthat .Iteven

seemsthattheauthorofthatpage,GrardMichon,hasrecentlyaddedadiscussionofthisveryproblem,promptedbymyblogposts!Isnttheinternetgreat?

Posted in arithmetic, programming | Tagged computing, decadic, numbers | 2 Comments

Differences of powers of consecutive integersPosted on January 29, 2012

PatrickVennebushofMathJokes4MathyFolks recentlywroteaboutthefollowingprocedurethatyields

surprisingresults.Choosesomepositiveinteger .Now,startingwith consecutiveintegers,raiseeach

integertothe thpower.Thentakepairwisedifferencesbysubtractingthefirstnumberfromthesecond,the

secondfromthethird,andsoon,resultinginalistofonly numbers.Dothesamethingagain,resultinginnumbers,andrepeatuntilyouareleftwithasinglenumber.

Forexample,supposewechoose ,andstartwiththefiveconsecutiveintegers .Weraisethem

alltothefourthpower,givingus

Nowwetakepairwisedifferences: ,then ,andsoon,andwegetthe

newlist

Repeatingthedifferenceproceduregives

OK,soweget .Sowhat?

Well,ifyoutryenoughexamples,youmaynoticeasurprisingpattern.Illletyouplaywithitforawhile.Over

thecourseofafewfuturepostsIllexplainthepatternandprovethatitalwaysholdsbuttheproofwillbea

reallycoolcombinatorialone,withprettypictures!

Posted in arithmetic, pattern | Tagged consecutive, difference, integers, powers, surprising | 16 Comments

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> uStep :: State UState Int

> uStep =do

> u p t

> let d = p `mod` 10 -- next digit

> u' = d * t + u -- prepend the next digit to u

> p' =(p + 2*d*u + d*d*t) `div` 10 -- see above proof

>

> put (UState p' u' (10*t))-- record the new values

>

> return $ fromInteger d -- return the new digit

So,didwegainanything?Asaconcretecomparison,letsseehowlongittakestocompute .Usingourfirst,

simplecode,ittakes7.2seconds:

*Decadic> :set +s

*Decadic> length . show $ us !! 1000010001

(7.23 secs, 208746872 bytes)

(Ijusthaditprintthe numberofdigitsof toavoidwastingatonofspaceprintingouttheentirenumber.)

Andusingournewandhopef ullyimprovedcode?

*Decadic> length . show . un . flip execState initUState

$ replicateM_ 10000 uStep

10001

(1.55 secs, 225857080 bytes)

Only1.5seconds!Nice!

TheothernicethingaboutuStepisthatitspitsoutonedigitof atatime,whichwecanusetodefine asan

(infinite)listofdigitsasifthedigitswerecomingoneatatimedowna"tube",a -tube,onemightsaygetit,

a tubehehnevermind.

> type TenAdic =[Int]

>

> u :: TenAdic

> u = 5 : evalState (sequence $ repeat uStep) initUState

*Decadic> reverse $ take 20 u

[9,2,2,5,6,2,5,9,9,1,8,2,1,2,8,9,0,6,2,5]

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Nifty!Nexttimetherealfunbegins,asIshowoffsomecoolthingswecandowithourshinynew

implementationof .

Posted in computation, c onvergence, infinity, iteration, modular arithmetic, number theory, programming | Tagged decadic, Haskell, idempotent, streaming, u | 2

Comments

Herbert Wilf: 13 June 1931 7 January 2012Posted on January 9, 2012

Iwassadtolearnthat HerbertWilfdiedyesterday.Long-timereadersofthisblogmayrememberhimasoneof

thediscoverersofthe Calkin-Wilftree,whichIwroteaboutinaten-partseriesofposts( 1,2,3,4,5,6,7,8,9,

10).

The Calkin-Wilf Tree

Healsowrotegeneratingfunctionology,atextbookaboutgeneratingfunctions,atopicnearanddeartomyheart

(whichIhopetowriteaboutheresomeday).

AlthoughhewasanemeritusprofessorattheUniversityofPennsylvania(whereIamcurrentlydoingmyPhD)I

sadlynevergotachancetomeethim.

Posted in people | Tagged Herbert Wilf | 1 Comment

A self-square numberPosted on January 4, 2012

[Thisistheseventhinaseriesofpostsonthedecadicnumbers(previousposts:Acuriosity,Aninvitationtoa

funnynumbersystem,Whatdoes"closeto"mean?,Thedecadicmetric,Infinitedecadicnumbers,Morefunwith

infinitedecadicnumbers).Iknowit'sbeenawhilesinceI'vewrittenonthistopic,soifyou'vebeenfollowing

along,youmightwanttogobackandrefreshyourmemory.]

Finally,aspromised,Icanshowyouthestrangenumber uwhichisitsownsquare(butwhich isntzeroorone!).Upuntilnowallthedecadicnumbersweveconsideredhavebeenequivalenttofamiliarrationalnumbers,but

zeroandonearetheonlyrationalnumberswhicharetheirownsquare;clearlyumustbesomethingquite

different!

Assumingthatsuchaucouldexistandassumingitsadecadicinteger,thatis,hasnodigitstotherightofthe

decimalpointletsthinkforaminuteaboutwhat ucouldpossiblybe.Forexample,whatcouldits lastdigitbe?FollowFollow



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Areyouseeingapattern?Letsmakeatableoftheresultssofar:

1 5 25

2 25 625

3 625 390625

4 0625 390625

5 90625 8212890625

WhydidIputsomenumbersinbold?Well,hopefullyyouvenoticedbynowthateachnumberintheleft-hand

columnalwaysseemstobeasuffixofthesquareofthepreviousnumber.Soperhapsthenextdigitwillbe8?

Sureenough, endsin .

Willthisalwayswork?Yes,infact,itwill,andhereswhy.Letslet denotethelast digitsof (so ,

,andsoon).Oncewehavefound ,wecansetupamodularequationtofindthenextdigit(thisisjustageneralizationofwhatwedidearlier):

Now, isclearlydivisibleby sot hattermgoesaway.Butwhatabout ?Itseemsthatweonly

knowitisdivisibleby ,notnecessarilyby .Ah,butwait!Weknowthat endswith ,andhenceis

divisibleby ;combiningthiswiththe givesusanotherfactorof !Sothistermgoesawaytoo,andweareleft

with

No w, (bydefinition),sosubtracting fromb othsidesleavesamultipleo f intheplaceof

(namely, withtherightmost digitssettozero).Butwecanalsogetridofallthedigitstotheleftofthe

stbecauseweareworkingmod .Dividingby wefindthat mustbeequaltothat stdigitof

.

Soherestheprocedure:startingwith ,define

Thatis,squarethecurrentnumberoflength andtakethelast digitstogetthenextnumber.Theaboveproofshowsthat

Ateverystepwewillhaveanumber whosesquareendswiththedigitsof ;

thisprocedurewillalwayswork;and

thisproceduregivestheuniquesequenceof withthispropertywhenstartingwith !

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Sowehave

andsoon.

Sowhatis ?Itissimplythelimitofcarryingoutthisproceduretoinfinity!

Weknowthatanysuffixof ,whensquared,yieldssomethingendingwithitself.Soitmakessense(althoughit

takesabitofimagination!)thatsquaring itselfyields again.

Posted in arithmetic, infinity, iteration, modular arithmetic, proof | Tagged decadic, idempotent, self, square | 12 Comments

Four-figure offerPosted on December 1, 2011

Thisjustarrivedinmyinbox:

MynameisBeckyRaymond,ImaDomainBrokerageConsultantworkingonbehalfoftheownerof

traveled.comtosellthispremiumasset.

WhilesearchingonlineIcameacrossyourdomainmathlesstraveled.com;sincebothdomainshave

listingsunderarelatedkeywordIthoughtperhapsyourcompanymaybeinterestedinacquiring

traveled.com?Ifthisdomainisofinteresttoyou,pleasesubmitanofferinthefourfigurerangeand

abovetoqualifyasapotentialbuyer.

Surething,Becky!Hereismyfour-figureoffer:

Fig. 1. A graph of the first 1000 hyperbinary numbers.

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Fig 2. Hasse diagram for the subsets of a five-element set.

Fig. 3: Proof that .

Fig. 4: Complete set of 15-bracelets.

Ihopeyouwillagreethatthisisaveryfinesetoffigures.Icouldprobablyaddacouplemorefigurestomyoffer

ifthatbecomesnecessary.

Ilookforwardtobeingtheproudowneroftraveled.com,theplacetogoforallyourmathematicaltravelneeds!

Posted in humor, meta | Tagged domain, four figure, offer | 7 Comments

Sigmas and sums of squaresPosted on November 29, 2011

CommenterRachelrecentlyasked,

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Howwouldyoufindthesumof ?

Seehereforanexplanationofsigmanotationinthiscaseitdenotesthesum

Ofcourse,foranyparticularvalueof wecanjustpluginvaluesanddoabunchofadding.ButIinterpretRachelsquestiontomeancanwefindanalgebraicexpressionintermsof whichletsuscomputethesum

morequicklythanactuallyaddingtheindividualterms?

Letstry!

First,weobservethat

Why?Ifyouthinkaboutitabit,youcanseethatthesametermsshowupontheleftandtheright,justinadifferentorder:theleftsideis whereastherightsideis .

Sinceadditionisassociative( )andcommutative( )theseareequal.

Next,weobservethat

,

thatis, .Thisisbecausemultiplicationdistributesoveraddition.

So,wehave .

isjust copiesof ,soitisequalto .Ivewrittenbeforeabout ;itisequalto .Sofar,we

have

.

Whataboutthatpesky ?Canitbesimplifiedatall?

Itcan,andIknowofafewdifferentwaystofigureitout.Illshowyouone

maths less travelled

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