Waldschmidt M - Some Arithmetic Problems Raised by Rabbits Cows and the DaVinci Code.ppt

simonfraseruniversity vancouvercampus harbourcenter segalcentersfuvancouver515 westhastings atrichards somearithmeticproblemsraisedbyrabbits cowsandthedavincicode michelwaldschmidtuniversit p etm curie parisvi july12 2006cnta9 http www math jussieu fr miw versionr vis ele14 07 2006 narayana scowsmusic tomjohnsonsaxophones danielkientzyrealization michelwaldschmidt http www math jussieu fr miw narayanawasanindianmathematicianinthe14th century whoproposedthefollowingproblem acowproducesonecalfeveryyear begininginitsfourthyear eachcalfproducesonecalfatthebeginingofeachyear howmanycowsaretherealtogetherafter forexample 17years whileyouareworkingonthat letusgiveyouamusicaldemonstration thefirstyearthereisonlytheoriginalcowandherfirstcalf long short thesecondyearthereistheoriginalcowand2calves long short short thethirdyearthereistheoriginalcowand3calves long short short short thefourthyeartheoldestcalfbecomesamother andwebeginathirdgenerationofnaryana scows long short short short long short year thefifthyearwehaveanothermothercowand3newcalves year 2 3 4 5 thesixthyearwehave4productivecows 4newcalves andatotalherdof13 thesixthyear 4productivecows 4long 9youngcalves 9short total 13cows 13notes 17thyear 872cows narayana scowsmusic tomjohnsonsaxophones danielkientzyrealization michelwaldschmidt http www math jussieu fr miw narayanawasanindianmathematicianinthe14th century whoproposedthefollowingproblem acowproducesonecalfeveryyear begininginitsfourthyear eachcalfproducesonecalfatthebeginingofeachyear howmanycowsaretherealtogetherafter forexample 17years whileyouareworkingonthat letusgiveyouamusicaldemonstration thefirstyearthereisonlytheoriginalcowandherfirstcalf long short thesecondyearthereistheoriginalcowand2calves long short short thethirdyearthereistheoriginalcowand3calves long short short short thefourthyeartheoldestcalfbecomesamother andwebeginathirdgenerationofnaryana scows long short short short long short thefifthyearwehaveanothermothercowand3newcalves thesixthyearwehave4productivecows 4newcalves andatotalherdof13 archimedescattleproblem archimedes thecattleproblemofarchimedesaskstodeterminethesizeoftheherdofthegodsun thisproblemamountstofindtwointegersxandysuchthatthesquareofxminusasuitablemultipleofthesquareofyis1 x2 410286423278424y2 1 archimedescattleproblem thereareinfinitelymanysolutions thesmallestonehasxwith206545digits thisproblemwasalmostsolvedbyagermanmathematician a amthor in1880 whocommented assumethatthesizeofeachanimalislessthanthesizeofthesmallestbacteria takeasphereofthesamediameterasthesizeofthemilkedway whichthelighttakestenthousandyearstocross thenthisspherewouldcontainonlyatinyproportionoftheherdofthegodsun numberofatomsintheknownfiniteuniverse wheniwasyoung 1060atoms afewyearslater longback 1070 nowadays solutionofarchimedesproblem h c williams r a germanandc r zarnke 1965 pell fermatequation x2 dy2 1 brahmagupta 628 x2 92y2 1 smallestsolution11512 92 1202 1 bhaskharaii 1150 x2 61y2 1 smallestsolutio61 2261539802 1 narayana xivthcentury x2 103y2 1 smallestsolution2275282 103 224192 1 fermat 1601 1665 1151 1151 132480192 120 120 1324800 fibonacci leonardodipisa pisa italia 1175 1250liberabaci 1202 modelizationofapopulation firstmonth thirdmonth fifthmonth sixthmonth secondmonth fourthmonth adultpairs youngpairs sequence 1 1 2 3 5 8 thefibonaccisequence 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 1 1 2 1 2 3 2 3 5 3 5 8 5 8 13 8 13 21 13 21 34 21 34 55 theoryofstablepopulations alfredlotka assumeeachpairgeneratesanewpairthefirsttwoyearsonly thenthenumberofpairswhoareborneachyearagainfollowthefibonaccirule arctictrees incoldcountries eachbranchofsometreesgivesrisetoanotheroneafterthesecondyearofexistenceonly thedavincicodefiveenigmastobesolved 1thefirstenigmaasksforputtingintherightordertheintegersofthesequence13 3 2 21 1 1 8 5 thisreorderingwillprovidethekeyofthebankaccount 2anenglishanagramodraconiandevil ohlamesaint 3afrenchanagramsacroixgravel heure inthebookwrittenbydanbrownin2003onefindssome weak cryptotechniques thedavincicodefiveenigmastobesolved continued 4afrenchpoemtobedecoded lcaltseessegasedtomxueivnusnade talc ellimafastinu riuqsreilpmetselrap in bet tale l v raressuovhsabtaceva 5anoldwisdomwordtobefound answerfor5 sophia sophieneveu thedavincicodethebankaccountkeyinvolvingeightnumbers thesearetheeightfirstintegersofthefibonaccisequence thegoalistofindtherightorderatthefirstattempt therightanswerisgivenbyselectingthenaturalordering 1 1 2 3 5 8 13 21 thetotalnumberofsolutionsis20160 theeightnumbersofthekeyofthebankaccountare 13 3 2 21 1 1 8 5 primitivelanguages with3lettersa b c selectthefirstletter 3choices onceitisselected completewiththe2wordsinvolvingthe2remainingletters hencethenumberofwordsis3 2 1 6 namelyabc acb bac bca cab cba givensomeletters howmanywordsdoesoneobtainifoneuseseachletterexactlyonce with1lettera thereisjustoneword a with2lettersa b thereare2words namelyab ba threeletters a b c sixwords 3 2 1 6 c abc acb bac bca cab cba 3 2 1 firstletter second third word a b c d b c a c d a b d a b c d c d b d b c c d a d b d a d a b b c a c a b a c d c d b c b d c d a c a d b d a b a c b c a b a abcd abdc acbd acdb adbc adcb bacd badc bcad bcda bdac bdca cabd cadb cbad cbda cdab cdba dabc dacb dbac dbca dcab dcba fourletters a b c d thesequence13 3 2 21 1 1 8 5 inthesameway with8letters thenumberofwordsis8 7 6 5 4 3 2 1 40320 herethedigit1occurstwice thisiswhythenumberoforderingsisonlyhalf 20160 thedavincicode 2anenglishanagramdraconiandevil ohlamesaintthemonalisaleonardodavinci 3afrenchanagramsacroixgravel heurelaviergeauxrochers thedavincicode continued dansunvieuxmotdesagesseestlacl quir unitsafamille clat elat teb nieparlestempliersavecatbashvousserar v l e 4afrenchpoemtodecode lcaltseessegasedtomxueivnusnade talc ellimafastinu riuqsreilpmetselrap in bet tale l v raressuovhsabtaceva utiliserunmiroirpourd chiffrerlecode useamirrorfordecoding exponentialsequence firstmonth secondmonth thirdmonth fourthmonth numberofpairs 1 2 4 8 numberoffibonaccirabbitsafter60months 1548008755920 13digits beetlelarvasbacteriaeconomy exponentialgrowth numberofpairsofmiceafter60months 1152921504606846976 19digits sizeofnarayana sherdafter60years 11990037126 11digits howmanyancestersdowehave sequence 1 2 4 8 16 beesgenealogy beesgenealogy numberoffemalesatagivenlevel totalpopulationatthepreviouslevelnumberofmalesatagivenlevel numberoffemalesatthepreviouslevel sequence 1 1 2 3 5 8 1 0 1 1 1 2 1 2 3 2 3 5 3 5 8 0 1 1 phyllotaxy studyofthepositionofleavesonastemandthereasonforthemnumberofpetalsofflowers daisies sunflowers aster chicory asteraceae spiralpaterntopermitoptimalexposuretosunlightpine cone pineapple romanescocawliflower cactus leafarrangements http www unice fr leml coursjdv tp tp3 htm universit denice laboratoireenvironnementmarinlittoral equiped accueil gestiondelabiodiversit phyllotaxy phyllotaxy j kepler 1611 usesthefibonaccisequenceinhisstudyofthedodecahedronandtheicosaedron andthenofthesymmetryoforder5oftheflowers st phanedouadyetyvescouderlesspiralesv g taleslarecherche250 janvier1993 vol 24 pluckingthedaisypetals iloveyoualittlebitalotwithpassionwithmadnessnotatall sequenceoftheremaindersofthedivisionby6ofthefibonaccinumbers 1 1 2 3 5 2 1 3 4 1 5 0 5 firstmultipleof6 144 eneffeuillantlamarguerite jet aimeunpeubeaucouppassionn ment lafoliepasdutout helovesmehelovesmenot divisionby6ofthefibonaccinumbers 8 6 1 2 13 6 2 1 21 6 3 3 34 6 5 4 55 6 9 1 89 6 14 5 144 6 24 0 iloveyou 1alittlebit 2alot 3withpassion 4withmadness 5notatall 0 divisionby2ofthefibonaccinumbers 8 2 4 0 13 2 6 1 21 2 10 1 34 2 17 0 55 2 27 1 89 2 44 1 144 2 72 0 helovesme 1helovesmenot 0 geometricconstructionofthefibonaccisequence 1 2 5 8 1 2 3 thisisanicerectangle anicerectangle asquare 1 1 1 goldenrectangle sides 1and condition thetworectangleswithsides1and forthebigone 1and1forthesmallone havethesameproportions goldenrectangle sides 1and solution isthegoldennumber1 618033 1 1 1 618033 equation 2 1 0 thegoldenrectangle 1 1 1 3 1 togofromthelargerectangletothesmallone divideeachsideby 1 2 ammonite nautilusshape spiralsinthegalaxy thegoldennumberinart architecture aesthetics keesvanprooijenhttp www kees cc gldsec html regularpentagonsanddodecagons nbor7 gif 2cos 5 penrosenon periodictilingpatternsandquasi crystals g m thickrhombusg thinrhombusm proportion diffractionofquasi crystals g om tried unchampdelavandehttp math unice fr frou lavande htmlfran oisrouvi re nice doublyperiodictessalation lattices cristallography thegoldennumberandaesthetics thegolden