Turnul din Hanoi - Anul Matematicii 2010 tiffany klocka

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Matematica şi jocul

Nota 4.68 / 35 voturi interes general Turnul din Hanoi

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Unul dintre jocurile aparent simple, dar cu un adânc substrat matematic, este binecunoscutul Turn din Hanoi. Acest joc a fost inventat de matematicianul francez Edouard Lucas şi a fost comercializat ca jucărie pentru toate vârstele încă din anul 1883. La început era confecţionat din lemn şi consta în câteva discuri de mărimi diferite şi 3 beţişoare. Provocarea este de a muta turnul format din discuri de pe un beţişor pe alt beţişor liber, urmând două reguli simple: la fiecare mutare se mută un singur disc şi niciodată nu se aşază un disc mai mare peste un disc mai mic. Bineînţeles, scopul este acela de a realiza mutarea turnului folosind cât mai puţine mutări.

Se spune că acest joc este inspirat de legenda Turnului lui Brahma, aflat într-un templu al oraşului indian Benares. Acest turn este format din 64 de discuri de aur, de a căror mutare se ocupă preoţii templului, respectând regulile de mai sus. Legenda spune că atunci când turnul discurilor de aur va fi complet transferat pe o altă tijă, templul se va prăbuşi iar lumea va lua sfârşit. Ceea ce ne face să ne întrebăm cu îngrijorare care este numărul minim de mutări în cazul turnului cu 64 de discuri... Pentru a răspunde la această întrebare, vom începe prin a cerceta ce se întâmplă în cazul jocului cu mai puţine discuri.

Să notăm cele 3 beţişoare folosind literele A, B, C. La început, turnul se află pe beţişorul A, iar noi vrem să îl mutăm pe beţişorul C. Vom nota cu n numărul discurilor, iar cu U n   numărul minim de mutări atunci când avem un turn format din n discuri.
Cu ajutorul jocului de mai jos, putem observa că:
• pentru n = 1, turnul se mută printr-o singură mutare, deci U 1   = 1;
• pentru n = 2, turnul se poate muta prin 3 mutări, deci U 2   = 3;
• pentru n = 3, turnul se poate muta prin 7 mutări, deci U 3   = 7.

Dar în cazul unui turn format din n discuri? Să ne gândim: în procesul de mutare a acestui turn, ajungem la momentul în care toate discurile înafară de cel mai mare sunt mutate pe un alt beţişor (B) decât cel iniţial (A), formând un turn cu n-1 discuri. Dacă presupunem că am ajuns la acest moment folosind numărul minim de mutări, înseamnă că până acum am făcut U n-1   mutări. În continuare, mutăm discul cel mai mare de pe A pe C, folosind o singură mişcare. Apoi nu ne rămâne decât să mutăm turnul cu n-1 discuri de pe B pe C, peste discul cel mai mare. Acest lucru poate fi făcut prin minim U n-1 mutări. Acum să numărăm câte mutări am făcut în total: U n   = U n-1   +1 + U n-1   = 2 U n-1   + 1
Astfel am obţinut o relaţie de recurenţă între numărul minim de mutări pentru un turn cu n discuri şi unul format din n-1 discuri, motiv pentru care acest joc este foarte des folosit ca exemplu pentru tehnici de programare care folosesc recurenţa.
Folosind procedeul numit inducţie matematică, se poate demonstra uşor că numărul minim de mutări în cazul turnului cu n discuri este: U n   = 2 n - 1.
Aşadar, revenind la Turnul lui Brahma, numărul minim de mutări este 2 64 - 1, adică un număr de 20 de cifre! Dacă 2 64 - 1  ar fi secunde, atunci asta ar însemna  circa  584 942 417 355 de ani.  Deci putem sta liniştiţi, sfârşitul lumii e departe: chiar dacă preoţii ar munci fără încetare şi ar face o mutare pe secundă, tot le-ar trebui multe mii de milioane de ani pentru a-şi termina treaba.

După ce matematicienii au epuizat de întors pe toate feţele acest joc, ei au descoperit şi alte variante, mult mai provocatoare din punct de vedere matematic. De exemplu, care ar fi numărul maxim de mutări prin care se poate transfera turnul de pe un beţişor pe altul, fără a reveni la o configuraţie anterioară? Dar dacă jocul ar avea 4 sau mai multe beţişoare în loc de 3? Acestea sunt doar câteva exemple care ne arată cum se poate transforma o problemă relativ simplă într-una de o mare complexitate matematică, prin modificarea minimă a unei singure variabile din enunţul problemei... dar până la urmă, în astfel de lucruri stă frumuseţea matematicii!

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Vietnam Military WATCH Issue

 NOTE: THE ONLY DIVE WATCH US MIL CONTRACTS WERE THE CANTEEN AND THE TORNEK. IF SOME ONE CAN PINT ANOTHER I WOULD BE HAPPY TO KNOW. FROM ROLEX TO SEA WOLF, MANY DIVES WERE UTILIZED BY NAVY DIVERS UDT ETC,  HELIO AND JET PILOTS ALSO UTILIZED MANY DIFFERENT WATCHES, BUT ONLY MIL SPEC WATCHES FROM[…]

ROCKS CURRENT OFFERS

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FREE ID AND WATCH CHECK

    THIS IS YOUR COMMUNITY! THOUGH GSW/MWB OPERATES AN AWESOME BRICK AND MORTAR STUDIO & GALLERY & 5 WEBSITES ON A DEDICATED SERVER WITH OVER 30,000 PICTURES OF RESTORATIONS, AND HAVE OFFERED OUR SERVICES SINCE 1989 [JOINED THE INTERNET 1992], WE NOT ONLY SELL WE ALSO INVEST IN DOZENS WATCHES FROM FELLOW WATCH SELLERS.[…]

Dial Restoration Is Costly Yet Necessary

TO RESTORE OR NOT RESTORE VINTAGE WATCH DIALS I like to say I am not a watchmaker .. rather a hacking master restoration expert restoring history. With over 35000 pictures, 1.2 mil Google requests every 15 days,+near 10,000 articles online. i am obsessive LOL. Thus I call upon only the best of the best. When[…]

CHRONOGRAPH A RETOUR GEISMAR JAPANESE WWII WATCHES

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EIGENTUM DER FLIEGERTRUPPEN ALPINA INVESTED BY ALPINA

      GSW PO BOX 2301 SUWANEE, GA, USA 30024 goldsmithworks.com 770.831.1257 – 1.800.438.6894 HTTP://FREE.TICKETS.COM Date/Report Number …..120811.LWW Item:  EIGENTUM DER FLIEGERTRUPPEN ALPINA               PURCHASED BY THE ALPINA COMPANY   ALPINA ca. 1915: 182597 WITH MILITARY BLACK DIAL SIGNED EIGENTUM DER FLIEGERTRUPPEN ALPINA WITH 60 MINUTE OUTER TRACK BORDER WHITE[…]

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AWESOME-WWII-RESTORATIONS

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APPRAISING

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  GOLDSMITHWORKS PO BOX 2301 SUWANEE, GA, USA, 30024 +1.770.831.1257  – 1.800.438.6894 Watch For Sale Need Help ID Value? 35000 WATCH  IMAGES 25000 PAGES ONLINE 24/7 GOLDSMITHWATCHWORKS.COM GOLDSMITHWORKS.COM     MILITARYWATCHBOX.COM LSYF.COM TIMEFRAUD.COM          GOLDSMITHTRAINING.COM  1947 – 1949 WHAT MAKES A FINE WATCH FINE [this video tajke you into the most fabulous era of watch making[…]

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goldsmithworks and rock restoring seiko dive watch

WHOM DO YOU WANT TO RESTORE YOUR SEIKO WATCHES 1890 T0 1990 ,, Seiko is one of a few rare companies that made everything for their watches. From Cases, Dials, Movements and Tools, TO Oils, Grease, Repair Equipment, Straps, Bands, Bracelets, Crowns, everything was designed, created manufactured in house. SOME PARTS WERE PROBABLY IMPORTED SEIKO[…]

1940 WWII KOREA VIETNAM WYLER FLOATING BEZEL AIRMAN WATCH

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HISTORIC WEEMS LONGINES BEST

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DOES THE CASE HAVE TO BE CUSHION NO THE PERSHING

BEFORE A FORUM DODO BIRD SWEEPS IN TO “HELP” BY INFORMING US  IT AIN’T SO…. ALLOW ME TO INFORM YOU OF THE HISTORY OF THE 48 STAR AMERICAN SHEILD W/EAGLE AND PORCELAINE DIAL.   AN ELGIN POCKET WATCH -WITH PERSHING DIAL-   AFTER RESTORING OVER 200 PERSHING DIALED WATCHES, WE CONCLUDED THAT, THOUGH ELGIN PRODUCED[…]

WWII NAZI LUFTWAFFE HACKING SWEEP SECONDS ALPINA

THE RAREST WWII GERMAN WATCH A SMALL RUN OF THESE POCKET WATCHES WERE MADE IN GERMANY BY ALPINA   HONORING   HITLER.–\FEW SURVIVED  FEWER STILL ACTUALLR RUN DUE TO PARTS THIS ALPINA WAS/IS NOT SIMPLY A FASHION OR HONOR WATCH THE FACT THIS ALPINA HAS A MILITARY  HACKED SWEEP SECONDS HAND AND THIS RARE MILITARY GRADE[…]

WALTHAM GENUINE TRENCH

WALTHAM GENUINE TRENCH MILITARY ILLUMINATED 24HR VERSION WITH CRYSTAL GUARD RED 12, RED 24HR CHAPTER ORIGINAL HANDS INCLUDING FITTING  TAILED SUB SECONDS HAND OUTLINED  ILLUMINATED RAILROAD NUMERALS LIKE KITCHENER MILITARY STRAPS ONE TONGUE IS LONGER THE ONE THAT IS TO RIGHT/ABOVE WHEN WEARING ON LEFT ARM, THIS ALLOWED THE SOLDIER TO PLACE OVER CUFF, JACKET[…]

VIETNAM ERA RED CROSS HELIOS CUFF STLE

FACT. GOLDSMITHWORKS HAS BEEN ONLINE TRADING & RESTORING WATCHES SINCE 1992 SINCE JOINING EBAY TEN YEARS AGO, WE HAVE OFFERED OUR VINTAGE TIME EXCLUSIVELY ONLINE WITH EB WE HAVE ALSO POSTED OVER 150,000 IMAGES OF OUR RESTORATIONS PICASA -PHOTO BUCKET -GOOGLE FREE WARRANTY AND APPRAISAL VERIFYING THE AUTHENTICITY OF OUR TIME PIECES 13 WEBSITES PROVIDING[…]

CASA BLANCA BUCHERER FREE FRENCH NURSE

   CASA BLANCA BUCHERER FREE FRENCH NURSE AVAILABLE ON ETSY WITH FREE SHIPPING TODAY WORLD WIDE   CASA BLANCA BUCHERER FREE FRENCH NURSE THIS PIECE IS READY TO WEAR EVERY DAY TIME PIECE NOTE. CASABLANCA THE MOVIE I RECENTLY WATCHED THE MOVIE AND HIJACKED SOME FRAMES. WOW, PLUM FULL OF FREE FRENCH MATERIAL. IMPORTANT THEME[…]

German_WWII

GERMAN-MILITARY-WATCHES.pdf   GERMAN WWII ARMY SWEEP SECONDS INTELLIGENCE WATCH **************************************** BELOW IS PHOTO OF GENUINE DI-H ABOVE WAS POSTED ONLINE AS IS *************************************** I HAVE HAD ONLY TWO OF THESE IN 20 YEARS THE GERMAN MILITARY PROPERTY MARK”DI SERIAL H” WAS UTILIZED FOR THE ARMY INTELLIGENCE SERVICE MORE GERMAN WWII WATCH HISTORY BELOW ALL WATCHES[…]

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  <WE HAVE HAD ONLY 3 NEGATIVE & 2 NEUTRAL FEED BACKS IN 10 YEARS WITH 2500 TRANSACTIONS!!!!!!!! AND NO NEGATIVE/NEUTRAL FOR THE 8000 PRIVATE TRANSACTIONS < CHECK THIS FEED BACK!!!! TELL ME WHAT STORE EVER HAD THAT TYPE OF RECORD WHILE SELLING USED WATCHES OR USED CARS OR ANYTHING USED, OLD AND MECHANICAL–REGARDLESS OF[…]

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    FACTS CONCERNING MILITARY VS PRIVATE WWI THROUGH VIETNAM http://goldsmithwatchworks.com/military_issue_watch_or_not_1/ ACTUAL DIRECT WWII WATCHES RESTORED BY ROCK/GSW http://goldsmithwatchworks.com/military_issue_watch_or_not_2/ ACTUAL WWI/WWII JAPAN/GERMAN/SWISS WATCHES RESTORED BY ROCK/GSW http://goldsmithwatchworks.com/military_issue_watch_or_not_3/ ACTUAL WWII/VIETNAM/SWISS/US/RESTORED BY ROCK/GSW http://goldsmithwatchworks.com/military_issue_watch_or_not_4/   ALL TIME PIECES RESTORED BY GOLDSMITHWORKS all watches restored by GOLDSMITHWORKS  GOLDSMITHWORKS USED TO INVEST IN 60 TO 70% OF OUR BASE[…]

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      FACTS CONCERNING MILITARY VS PRIVATE WWI THROUGH VIETNAM http://goldsmithwatchworks.com/military_issue_watch_or_not_1/ ACTUAL DIRECT WWII WATCHES RESTORED BY ROCK/GSW http://goldsmithwatchworks.com/military_issue_watch_or_not_2/ WWI/WWII JAPAN/GERMAN/SWISS WATCHES  BY ROCK/GSW http://goldsmithwatchworks.com/military_issue_watch_or_not_3/ ACTUAL WWII/VIETNAM/SWISS/US/RESTORED BY ROCK/GSW http://goldsmithwatchworks.com/military_issue_watch_or_not_4/   MARCH 01,2015 CLICK BELOW FOR DAVID’S VINTAGE RESTORED DIRECT TO PUBLIC/SOLDIER MILITARY WATCHES NOT ALL OF THESE ARE MILITARY, BUT THE MAJORITY ARE[…]

OnePatronFiftyWWII-Restorations

      OCT 30TH 2014   MARCH 01,2015 CLICK BELOW FOR DAVID’S VINTAGE RESTORED MILITARY WATCHES PDF! ROCK IS NOW RESTORING/SERVICING THIS COLLECTION THE FOLLOWING WATCHES ARE FROM ONE PATRON’S [DAVID] COLLECTION WILL UPDATE AS EACH TIME PIECE OF THIS INCREDIBLE EDUCATION PDF STORED POCKET TRENCH CONVERSIONS VIETNAM US GOV ISSUE WATCHES GERMAN SUBMARINE[…]

Elijah Reedy Speaks

Elijah Reedy Speaks I am Elijah Reedy, head of Goldsmith Watch Works. What better way to introduce myself than a story from 2008 about a German Alpina WWII military wrist watch that Rock invested in back in 2008. I witnessed that investment. Rock was pretty excited, the battle went back and forth to the wire[…]

WWII MILITARY WATCH RESTORATION COLLECTION

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JAPANESE GERMAN BRITISH MILITARY HEAVEN

  JAPANESE GERMAN BRITISH MILITARY HEAVEN   ROCK’S PARADISE   VINTAGE & MILITARY WATCHES & FINE JEWELRY KM & PANZER PARA JAPANESE WWII NAVY SEIKOSHA SUB TIMER RCAF WWII TRENCH TRENCH WYLER GERMAN WWII WWII HELBROS 24HR JAPANESE WWII CHRONO SEIKOSHA CHRONO WWII NAVY SUB TIMER WWII SEIKOSHA JULY 24 2003 WWII-IMPERIAL-JAPAN-SEIKOSHA-SEIKO-LUME-DIAL-MILITARY-WATCH JULY 24 2003[…]

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WATCH MANUFACTURERS & RESTORATION

Published on: November 4, 2013
Categories: AAA HELP ME, AUDIO VIDEO, BOREL, BRITISH, BUY-SELL, ELECTRIC, GOLDSMITHWORKS, HAMILTON, HATTORI-SEIKO, JAPANESE, LADIES, MENS, MILITARY, NEW OLD STOCK, NEWS, OMEGA, POCKET WATCH, PUBLICATIONS, REPAIR, RESTORATION, ROCK SPEAKS, ROCKs CUSTOM JEWELRY, TOP QUARTZ

 

PRICES ABOVE ARE OUTDATED. THESE WERE THE PRICES FROM 2004 THRU 2006
SINCE THEN PRICES OF PARTSD WATCHES MOVEMENTS AND OTHER FEES HAVE SKYROCKETED YET WE ARE STILL THE BEST AND HAVE THE LOWEST PRICES

AVERAGE IS 225 TO 275 MOST WITH RESTORED AND ILLUMINATED DIALS
THIS PRICE IS NOT JUST SERVICE OR OVERHAUL
THESE PRICES ARE FOR COMPLETE RESTORATION
MOST INCLUDE CASE DIAL MOVEMENT STRAP
AND WARRANTY

ROCK PHOTO GALLERY GSW

GoldSmithWorks.Com  | GoldSmithWatchWorks.Com | GoldSmithTraining.Com | eBay.GoldSmithWorks.Com
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GSWW SERVICES – REPAIRS – RESTORES & SELLS THE FOLLOWING BRANDS Brand Make r ADIX MONTRES ORFINA S.A. ADMES ADMES S.A.R.L. AERNI CHARLES AERNI S.A. AGASSIZ WITTNOUER & CIE. S.A. AGON AGON WATCH   – ROBERT TIEBOLD AIRMASTER FELCA WATCH S.A. AJAX LES FILS DE PAUL JOBIN ALCAR MONTRES ORFINA S.A. ALEURE ANCIENNE FABRIQUE GEORGES PIAGET & CO. S.A. ALIENOR UHLMANN & CIE. ALMEDY ALMEDY S.A. ALPHA LES FILS DE PAUL SCHWARZ ETIENNE ALPINA ALPINA UNION HORLOGERE S.A. ALPROSA ENICAR S.A. ALTUS MONTRES ALTUS S.A. ALVICA FABRIQUE D’HORLOGERIE WILLY CHOFFAT & CO. AMATA MATINA WATCH LTD. AMIDA AMIDA S.A. AMYRAL LOOPING S.A. ANGELUS ANGELUS – STOLZ FRERES S.A. ARBU ARBU WATCH ARCADIA FLEURIER WATCH CO. ARCALA MONTRES ZODIAC S.A. ARDATH ARDATH WATCH CO. ARIOS MUROS, FABRIQUE ERWIN TRIEBOLD ARISTEX FABRIQUE D’HORLOGERIE ED. KUMMER S.A. ARISTON FABRIQUE D’HORLOGERIE ED. KUMMER S.A. ARLUX ARLY WATCHES ARLY ARLY WATCHES ARNAND MUROS, FABRIQUE ERWIN TRIEBOLD AROLA DANIEL ET ROCHAT, ALFRED GIRARDET ARSA MANUFACTURE D’HORLOGERIE A.REYMOND S.A. ARTA FABRIQUE D’HORLOGERIE LIGA S.A. ARUGO BADER & HAFNER S.A. ASCANIA ASCANIA WATCH CO. F.HOFFMANN ASTER FABRIQUE D’HORLOGERIE LA CHAMPAGNE S.A. ASTIN ASTIN WATCH S.A. ASTON-VILLA FELCA WATCH S.A. ATIMA BUREN WATCH COMPANY S.A. ATLANTA UHLMANN & CIE. ATLANTIC FABRIQUE D’HORLOGERIE ED. KUMMER S.A. ATMOS FABRIQUE D’HORLOGERIE LE COULTRE & CIE. AUDEMARS, PIGUET & CO. AUDEMARS, PIGUET & CO. S.A. AUREOLE AUREOLE WATCH CO. AVIA FABRIQUE D’HORLOGERIE DE MONTRES DEGOUMOIS S.A. B.W.C. BUTTES WATCH CO. S.A. BAUME & MERCIER BAUME & MERCIER S.A. BELECO BRAILOWSKY MAX – BELECO WATCH BELETA BRAILOWSKY MAX – BELECO WATCH BELLTONE FABRIQUE D’HORLOGERIE ZILA S.A. BELTEX BELTEX S.A. BELVIL BELVIL WATCH CO. S.A. BERNA BERNA WATCH FACTORY LTD. BIRSECLOCK COURT WATCH CO. S.A. BLANCPAIN RAYVILLE S.A. BLITA BOILLAT FRERES S.A. BOLSA BOLSA WATCH CO. BOMA WEGA WATCH S.A. BOREL ERNEST BOREL & CIE. S.A. BOUDOIRES BRADUX S.A. BOVET BOVET FRERES & CO. S.A. BOVET FRERES BOVET FRERES & CO. S.A. BRADUX BRADUX S.A. BREITLING G.LEON BREITLING LTD. BREMON G.LEON BREITLING LTD. BRIGHT MONTRES ZODIAC S.A. BRINA JOHN BRINGOLD BRITIX BRITIX WATCH CO. S.A. FACTORY LENGNAU BUECHE-GIROD V.BUECHE-GIDOT BULLA MANUFACTURE DE MONTRES BULLA – EMILE JUILLARD S.A. BUREN BUREN WATCH COMPANY S.A. BUREN GRAND PRIX BUREN WATCH COMPANY S.A. BUSER   BUSER FRERES & CIE. S.A. BUSER FRERES BUSER FRERES & CIE. S.A. BUSGA BUSGA S.A. MONTRES BUT BUTTES WATCH CO. S.A. BUTEX BUTTES WATCH CO. S.A. CALAN MUROS, FABRIQUE ERWIN TRIEBOLD CALENDERMETO FABRIQUES MOVADO CALENDOGRAF FABRIQUES MOVADO CALENDOMATIC FABRIQUES MOVADO CALENDOPLAN FABRIQUES MOVADO CALENDOR MONTRES OCTO S.A. CALISTON FABRIQUE D’HORLOGERIE NISUS S.A. PERY CAMY FABRIQUE D’HORLOGERIE CAMY – STROUN FRERES S.A. CANDINO CANDINO WATCH CO. LTD CAPITOL BUSGA S.A. MONTRES CARINA E.KIRCHHOFER & CIE. HENO WATCH CAROLINE ENICAR S.A. CATARINA FELCA WATCH S.A. CELESTOGRAF FABRIQUES MOVADO CERTINA KURTH FRERES S.A. FABRIQUE DE MONTRES CHANCE FABRIQUE D’HORLOGERIE PERFECTA S.A. CHITA FELCA WATCH S.A. CHRONO FABRIQUE D’HORLOGERIE PERFECTA S.A. CINY CINY WATCH CO. LE NOIRMONT CLIMAX MILEX ELEM WATCH CO. LTD. COLGOR COLGOR WATCH – ROB. BRANDT COLMA BUSGA S.A. MONTRES COLOMBIA UHLMANN & CIE. CONSUL FABRIQUE D’HORLOGERIE CHARLES VIRCHAUX CORIS PAUL BUHRE S.A. CORONA FABRIQUE D’HORLOGERIE CORONA S.A. CORTEBERT MANUFACTURE D’HORLOGERIE CORTEBERT CORUM RIES, BANNWART & CO. COURT COURT WATCH CO. S.A. CREDOS MONTRES CREDOS S.A. CROCUS NICOLET WATCH S.A. CYMA CYMA WATCH CO. S.A. CYMA-AMIC CYMA WATCH CO. S.A. CYPAR LES FILS DE PAUL JOBIN DACAL MONTRES DACAL, CALAME DANIEL & CIE. DAMAS BEGUELIN & CO. DATIP MONTRES ORFINA S.A. DATUM DATUM WATCH FACTORIES LTD. DAVOZ MONTRES ORFINA S.A. DE SULLENS FABRIQUE D’HORLOGERIE ROGER PUTHOD DECO FREY & CO. S.A. DELANO DELANO S.A. DELBANA DELBANA WATCH CO. DELBOS DELBANA WATCH CO. DESERTROUTER FELCA WATCH S.A. DILA FREY & CO. S.A. DIRECTIME HOGA WATCH LTD. DIVINE BRAILOWSKY MAX – BELECO WATCH DOGMA FABR.D’HORL. ARTHUR DORSAZ & CO.(CLEMENCE FRERES) DOM CHARLES DOM DOMATIC CHARLES DOM DOUPHINE SPERA S.A. DOXA MANUFACTURE DES MONTRES DOXA DRAGA DRAGA WATCH – RENE BLOCH DREFFA DREFFA WATCH CO. DRIVA DRIVA WATCH CO. DROZ FABRIQUE D’HORLOGERIE DROZ & CO. MONTRES TAVALOR DUBOIS FRERES FABRIQUE D’HORLOGERIE DUBOIS FRERES S.A. DUCTOR MILDIA WATCH CO. LTD. DUKE MUROS, FABRIQUE ERWIN TRIEBOLD DULUX RENE GINRAT MONTRES DULUX DYNAMO ARLY WATCH S.A. E.WENGER E.WENGER WATCH CO. EBEL FABRIQUE EBEL S.A. EBELLO FABRIQUE EBEL S.A. EBERHARD & CO. EBERHARD & CO. EBOSA EBOSA S.A. EIGELDINGER & CIE. S.A. EDISSA WEGA WATCH S.A. EDOX ERA WATCH CO. LTD. – C.RUEFLI-FLURY & CO. EFFEM F.MEYER FABRIQUE D’HORLOGERIE ELDOR ELDOR WATCH CO. S.A. ELECTION ELECTION – GRAND PRIX ELEM MILEX ELEM WATCH CO. LTD. ELFARO NOUVELLES FABRIQUES LE PHARE S.A. ELOGA FABRIQUE D’HORLOGERIE ELOGA ELUXA BRADUX S.A. ELVOS MONTRES ORFINA S.A. ENICAR ENICAR S.A. ERA ERA WATCH CO. LTD. – C.RUEFLI-FLURY & CO. ERGUEL FABRIQUE D’HORLOGERIE ERGUEL S.A. ERHUB PAUL BUHRE S.A. ERMETO FABRIQUES MOVADO ERNEST BOREL ERNEST BOREL & CO. S.A. ESCULAPE HY.MOSER & CIE. S.A. ESKA KOCHER SYLVAN & CO. ETERNA ETERNA S.A. ETNA ETNA WATCH CO. EXACTO SCHLUP & CO. S.A. WATCH FACTORY EXACTUS MONTRES EXACTUS S.A. EXCELSIOR PARK EXCELSIOR PARK FAMBO MUROS, FABRIQUE ERWIN TRIEBOLD FAVRE-LEUBA COMPAGNIE DES MONTRES FAVRE-LEUBA S.A. FELCA FELCA WATCH S.A. FELINA FELCA WATCH S.A. FERANES FEREX S.A. FEREL FEREX S.A. FEREX FEREX S.A. FESTIVA MONTRES OSTARA S.A. W.RIS FIDEA NICOLET WATCH S.A. FINORA MONTRES ORFINA S.A. FIRSTLINE SINDACO S.A. FLORA FABR. D’HORLOGERIE LES FILS DE PAUL JOBIN FLUCANO FLUCANO WATCH W.FLUCK FORTIS FORTIS FRECO FREY & CO. S.A. FRENCA BUSER FRERES & CIE. S.A. FREY FREY & CO. S.A. FRIDOX FRIEDLI FRERES S.A. FRIEDLI FRIEDLI FRERES S.A. FRISARD RAYVILLE S.A. FULTON FULTON WATCH CO. LTD. GALAN MUROS, FABRIQUE ERWIN TRIEBOLD GASTON CAPT COLUMBIER GENEVA SPORT GENEVA SPORT WATCH CO. GEORGES BRANDT GEORGES BRANDT S.A. GERMINAL FABRIQUE GERMINAL S.A. GIGANTIC FABRIQUE D’HORLOGERIE GIGANTIC GIRARD-PERREGAUX GIRARD PERREGAUX GIRO LUNETTE FABRIQUE D’HORLOGERIE PERFECTA S.A. GLORIOUS MONTRES ZODIAC S.A. GLORIUS MONTRES ZODIAC S.A. GLYCINE MANUFACTURE D’HORLOGERIE GLYCINE S.A. GRACE MONTRES ZODIAC S.A. GRAL MUROS, FABRIQUE ERWIN TRIEBOLD GRANDIA JEAN GRANDY WATCH FACTORY GRANFINA MONTRES ORFINA S.A. GRUEN GRUEN WATCH CO. GUBELIN GUBELIN GUINARD GUINARD WATCH CO. S.A. HAFIS F. SUTER & CIE. HAMEL SPERA S.A. HAREOL AUREOLE WATCH CO. HEBDOMAS SCHILD & CIE. S.A. HELIOS FABRIQUE D’HORLOGERIE HELIOS S.A. HELOISA HELOISA S.A. HELVETIA HELVETIA WATCH CO. HENO HENO WATCH   E.KIRCHHOFER & CIE. HENRI DUVOISIN & CIE. DUVOISIN VVE HENRI & CIE. HERODIA MONTRES HERODIA HEUER ED.HEUER & CO. HIPCO HIPCO S.A.R.L. HISLON FABRIQUE D’HORLOGERIE ZILA S.A. HOGA HOGA WATCH LTD. HOMIS HOMIS WATCH CO. HOSAM HOGA WATCH LTD. HOVER H.HOFER LTD. HOVERTA H.HOFER LTD. HUGEX HUGA S.A. HUGUENIN A.HUGUENIN FILS S.A. HY MOSER & CIE. HY.MOSER & CIE. S.A. I.W.C. FABRIQUE D’HOROLOGERIE E.HOMBERGER-RAUSCHENBACH ILEX MONTRES ILEX S.A. J.WEBER-CHOPARD IMACO IMACO S.A. WATCH FACTORY INCAFLEX FABRIQUE DES MONTRES WYLER S.A. INREA CHARLES AERNI S.A. INTEGRA FABRIQUE D’HORLOGERIE ED. KUMMER S.A. INTERNATIONAL WATCH CO.   FABRIQUE D’HOROLOGERIE E.HOMBERGER-RAUSCHENBACH INVENTIC FABRIQUE D’HORLOGERIE ED. KUMMER S.A. INVICTA INVICTA LIMITED ITANA ITANA WATCH CO. LTD. J.P.MATTHEY J.P.MATTHEY JAEGER FABRIQUE D’HORLOGERIE LE COULTRE & CIE. JAEGER-LE COULTRE FABRIQUE D’HORLOGERIE LE COULTRE & CIE. JEAN RICHARD JEAN RICHARD S.A. JENCO FABRIQUE D’HORLOGERIE JENNY & FREY S.A. JONA BUSGA S.A. MONTRES JORIN MONTRES ORFINA S.A. JOSMAR JOSMAR WATCH S.A. JOSY MANUFACTURES DE PENDULETTES ROSEMONT JUVENIA FABRIQUE JUVENIA, DIDISHEIM-GOLDSCHMIDT FILS & CIE. KINGMATIC FABRIQUES MOVADO KOWAL NOUVELLES FABRIQUES LE PHARE S.A. KULM CUANILLON & CIE. L.A.LEUBA LOUIS A.LEUBA S.A. LA PALME PERRET & CIE. LAKEMASTER MONTRES OCTO S.A. LANCO LANGENDORF WATCH CO. LANCO-MATIC LANGENDORF WATCH CO. LANCYL LANGENDORF WATCH CO. LANDI FABRIQUE D’HORLOGERIE CHARLES VIRCHAUX LAREINE MUROS, FABRIQUE ERWIN TRIEBOLD LAREX LANGENDORF WATCH CO. LAVINA MANUFACTURE D’HORLOGERIE LAVINA S.A. LE CASTEL WERMEILLE & CIE.   / J.P.SCHLUNEGGER LE COULTRE LE COULTRE & CIE. S.A. LE PHARE NOUVELLES FABRIQUES LE PHARE S.A. LEMANIA FABRIQUE D’HORLOGERIE LUGRIN S.A. LEMANIA LEMIEUX SPERA S.A. LEONIDAS LEONIDAS WATCH FACTORY LEVRETTE FABRIQUE D’HORLOGERIE CH.WILHELM & CO. S.A. LIGA FABRIQUE D’HORLOGERIE LIGA S.A. LIGHTHOUSE NOUVELLES FABRIQUES LE PHARE S.A. LOGIC OCTO S.A. LONGEAU ENICAR S.A. LONGINES CAMPAGNIE DES MONTRES LONGINES S.A. LONVILLE LANGENDORF WATCH CO. LOOPING LOOPING S.A. LORETTO FABR. D’HORLOGERIE LES FILS DE PAUL JOBIN LOTUS HENZI-SCHAFFTER C. MANUFACTURE D’HORLOGERIE LOUIS ENGEL LOUIS A.LEUBA S.A. LOVARY LOVARY S.A., HY MOSER & CIE. S.A. LUGANA WEGA WATCH S.A. LUGRAN MONTRES CREDOS S.A. LUMINOUS MONTRES ZODIAC S.A. LUNESA HENZI-SCHAFFTER C. MANUFACTURE D’HORLOGERIE LUNORA HENZI-SCHAFFTER C. MANUFACTURE D’HORLOGERIE LUSINA PAMM FRERES, MONTRES LUSINA S.A.R.L. LUXOR MONTRES LUXOR S.A. MAGDA BOVET FRERES & CO. S.A. MARBEN MARCEL BENOIT S.A. MARDON MARDON WATCH LIMITED MARION HENZI-SCHAFFTER C. MANUFACTURE D’HORLOGERIE MARQUISE LABORATORIES OXYDOR MARTEL MARTEL WATCH CO. S.A. MARVIN COMPAGNIE DES MONTRES MARVIN S.A. MATHEY-TISSOT E.MATHEY-TISSOT & CO. S.A. MATINA MATINA WATCH LTD. MEDANA MEYER & STUDELI S.A. MEDIATOR MEDIATOR WATCH CO. S.A. MELIPOL MUROS, FABRIQUE ERWIN TRIEBOLD MEMOVOX LE COULTRE & CIE. S.A. MENTOR BADER & HAFNER S.A. MENTORA BADER & HAFNER S.A. MENTORES BADER & HAFNER S.A. METRIC METRIC S.A. MICAL ARTHUR SCHWAR – MICAL WATCH CO. INC. MICOSA MIDGET WATCH CO. S.A. MIDGET MIDGET WATCH CO. S.A. MIDO MIDO WATCH COMPANY – FORMERLY G.SCHAEREN & CO MILDIA FABRIQUE DES MONTRES MILDIA S.A. MILEX ELEM MILEX ELEM WATCH CO. LTD. MINERVA MANUFACTURE D’HORLOGERIE MINERVA SPORT S.A. MIRAMAR FABRIQUE D’HORLOGERIE MIRAMAR MOEBIUS H.MOEBIUS & FILS MOERIS FABRIQUE DES MONTRES MOERIS MONDIA FABRIQUE D’HORLOGERIE PAUL VERMOT & CO. S.A. MONDIA-MATIC FABRIQUE D’HORLOGERIE PAUL VERMOT & CO. S.A. MONSTRO WEGA WATCH S.A. MONTANUS FABRIQUE D’HORLOGERIE SCHINDLER & MOENIG MONTBRILLIANT G.LEON BREITLING LTD. MONTILIER MONTILIER WATCH CO. S.A. MOVADO FABRIQUES MOVADO MOVASCOPE FABRIQUES MOVADO MULCO MULCO S.A. MULFI HENRI MULLER & FILS MULTIALARM PAUL BUHRE S.A. MURIAL MURIAL WATCH FACTORY – ARLY WATCH S.A. MURINDIS MUROS, FABRIQUE ERWIN TRIEBOLD MURINDO MUROS, FABRIQUE ERWIN TRIEBOLD MURONDIS MUROS, FABRIQUE ERWIN TRIEBOLD MUROS MUROS, FABRIQUE ERWIN TRIEBOLD NATIONAL MANUFACTURE DE MONTRES NATIONAL S.A. (CHS-LEON SCHMID & CIE.) NATIONAL WATCH MANUFACTURE DE MONTRES NATIONAL S.A. (CHS-LEON SCHMID & CIE.) NAVIS FRIEDLI FRERES S.A. NEUCHATEL & CO. NEUTACHEL WATCH CO. LTD. NICE MONTRES ZODIAC S.A. NICOLET NICOLET WATCH S.A. NICOLET WATCH NICOLET WATCH S.A. NIDOR BUSER FRERES & CIE. S.A. NIGA G.GAGNEBIN & CIE. S.A. NIKE NICOLET WATCH S.A. NILEG ONSA WATCH, H. GILOMAN-RITTER NISUS FABRIQUE D’HORLOGERIE NISUS S.A. PERY NITELLA NITELLA WATCH LTD. NITON NITON WATCH S.A. NIVADA NIVADA S.A. NIVIA NIVIA FABRIQUE D’HORLOGERIE S.A. NOREXA FABRIQUE D’HORLOGERIE NOREXA NORMANA NORMANA WATCH CO. OBERON OBERON WATCH S.A. OCTO MONTRES OCTO S.A. OCTUS OCTUS WATCH S.A. OGIVAL FABRIQUE D’HORLOGERIE OGIVAL S.A. OLMA NUMA JEANNIN S.A. OLOR OLOR WATCH S.A. OLYMPOS MONTRES ZODIAC S.A. OLYMPUS MONTRES ZODIAC S.A. OMEGA OMEGA WATCH CO. S.A. LOUIS BRANDT & FRERE ONSA ONSA WATCH, H. GILOMAN-RITTER ONTIME CYPRESS MUROS, FABRIQUE ERWIN TRIEBOLD OPTIMA OPTIMA, FABRIQUE D’HORLOGERIE ALB.GROSSENBACHER OPUS FABRIQUE D’HORLOGERIE ED. KUMMER S.A. ORANJE LES FILS DE PAUL JOBIN ORANO MONTRES ORANO S.A. ORATOR MANUFACTURE D’HORLOGERIE SCHILD & CO. S.A. ORFINA MONTRES ORFINA S.A. ORIS ORIS WATCH CO. S.A. OSTARA MONTRES OSTARA S.A. W.RIS P.BUHRE PAUL BUHRE S.A. P.W.C. FABRIQUE D’HORLOGERIE PERFECTA S.A. PALLAS PALLAS WATCH CO.LTD. PALMA PERRET & CIE. PANDAN MUROS, FABRIQUE ERWIN TRIEBOLD PAREX MILEX ELEM WATCH CO. LTD. PATEK, PHILIPPE & CO. PATEK, PHILIPPE & CIE. S.A. PAUL BUHRE PAUL BUHRE S.A. PAUL CHOFFAT HOGA WATCH LTD. PAUL JOBIN FABR. D’HORLOGERIE LES FILS DE PAUL JOBIN PERFECTA FABRIQUE D’HORLOGERIE PERFECTA S.A. PERRET PERRET & CIE. PERY PERY WATCH ET MONTRE SIGMA PHENIX SOCIETE HORLOGERE DE PORRENTRUY PHENIX WATCH CO. PIAGET   ANCIENNE FABRIQUE GEORGES PIAGET & CO. S.A. PIAGET & CO. ANCIENNE FABRIQUE GEORGES PIAGET & CO. S.A. PIERCE MANUFACTURE DES MONTRES PIERCE S.A. PIERPONT SAUTER FRERES & CIE S.A. PONTIFA PONTIFA M.T. STAUFFER JEUNE PRECIMAX MANUFACTURE D’HORLOGERIE PRECIMAX S.A. PREXA MONTRES PREXA S.A. PRIMALUX ARTHUR SCHWAR – MICAL WATCH CO. INC. PROGRESS ARTHUR SCHWAR – MICAL WATCH CO. INC. PRONTO PRONTO WATCH CO. L.MAITRE & FILS S.A. QUEENMATIC FABRIQUES MOVADO R.JAQUET JAQUET RENE RADO SCHLUP & CO. S.A. WATCH FACTORY RAMADES MUROS, FABRIQUE ERWIN TRIEBOLD RAYVILLE RAYVILLE S.A. RECONVILIER SOCIETE HORLOGERE RECONVILIER RECORD RECORD WATCH CO. RECTA MANUFACTURE D’HORLOGERIE RECTA S.A. RECTORY MONTRES ZODIAC S.A. REFORM SCHILD & CIE. S.A. REGIONAL MUROS, FABRIQUE ERWIN TRIEBOLD RELIDE FABRIQUE D’HORLOGERIE RELIDE S.A. WALTER TRIEBOLD REMO AUREOLE WATCH CO. RENOVA FABRIQUE D’HORLOGERIE LIGA S.A. REPCO REPCO WATCH – PIERRE NICOLET REUSSER REUSSER FRERES S.A. REVIDA FABRIQUE D’HORLOGERIE LIGA S.A. REVOX AMIDA S.A. REVUE FABRIQUE D’HORLOGERIE THOMMEN S.A. RILA FABRIQUE D’HORLOGERIE WILLY CHOFFAT & CO. RIST-MATE FELCA WATCH S.A. RIVAL BUREN WATCH COMPANY S.A. RIVO RICHARD VOGT ROAMER MEYER & STUDELI S.A. ROBERT CART FABRIQUE D’HORLOGERIE ROBERT CART S.A. RODANA RODANA S.A. RODANIA RODANIA S.A. ROFINA MONTRES ORFINA S.A. ROINEK NICOLET WATCH S.A. ROLEX MONTRES ROLEX S.A. ROLNIK NOUVELLES FABRIQUES LE PHARE S.A. ROSEMONT MANUFACTURES DE PENDULETTES ROSEMONT ROSKOPF   MANUFACTURE DE MONTRES NATIONAL S.A. (CHS-LEON SCHMID & CIE.) ROSKOPF PATENT MANUFACTURE DE MONTRES NATIONAL S.A. (CHS-LEON SCHMID & CIE.) ROSKOPF WILLIE FRERES MANUFACTURE DE MONTRES NATIONAL S.A. (CHS-LEON SCHMID & CIE.) ROTARY FILS DE MOISE DREYFUSS ROTODATOR PAUL BUHRE S.A. ROTOGRAPH PAUL BUHRE S.A. ROVAC ARTHUR SCHWAR – MICAL WATCH CO. INC. ROYCE MONTRES KOCHER S. & CO. ESKA S.T.B. FABRIQUE D’HORLOGERIE DE ST-BLAISE S.A. SABINA PAUL – VIRGILE MATHEZ S.A. SAINT-BLAISE FABRIQUE D’HORLOGERIE DE ST-BLAISE S.A. SALTOFIX LOVARY S.A., HY MOSER & CIE. S.A. SANDOZ HENRY SANDOZ & FILS. SARWA ARTHUR SCHWAR – MICAL WATCH CO. INC. SATURNE NICOLET WATCH S.A. SAVILLON ENICAR S.A. SCHILD SCHILD & CIE. S.A. SEA-HORSE FABRIQUE D’HORLOGERIE ZILA S.A. SEAPEARL ENICAR S.A. SELLITA SELLITA WATCH CO. S.A. SELZA SELZA WATCH CO. S.A. SHELL AUREOLE WATCH CO. SHERPA ENICAR S.A. SHERPAS ENICAR S.A. SIBEL SORNA WATCH E.SCHENKER SIGMA PERY WATCH ET MONTRE SIGMA SILEX NICOLET WATCH S.A. SILVANA FABRIQUE D’HORLOGERIE SILVANA S.A. SIMON FABRIQUE D’HORLOGERIE PERFECTA S.A. SIMOS EBOSA S.A. EIGELDINGER & CIE. S.A. SINDACO SINDACO S.A. WATCH FACTORY SINDACO SWISSLINE SINDACO S.A. WATCH FACTORY SINEX MONTRES SINEX S.A. SLOGAN CHS. VERMOT & CO. SOLORA FABRIQUE D’HORLOGERIE LIGA S.A. SOLVIL FABRIQUE SOLVIL DES MONTRES PAUL DITISHEIM S.A. SOLVIL & TITUS FABRIQUE SOLVIL DES MONTRES PAUL DITISHEIM S.A. SOLVOX EBOSA S.A. EIGELDINGER & CIE. S.A. SORINA SORNA WATCH E.SCHENKER SORNA MONTRES ROSKOPF EN TOUS GENRES – E.SCHENKER SORNADAT SORNA WATCH E.SCHENKER SORNANA SORNA WATCH E.SCHENKER SPERA SPERA WATCHES SPORT CUANILLON & CIE. SPORTMASTER FELCA WATCH S.A. STADION FABRIQUE D’HORLOGERIE LIGA S.A. STEADY NICOLET WATCH S.A. STRAND EBOSA S.A. EIGELDINGER & CIE. S.A. STUDIO VULCAIN WATCH FACTORY SUBLIMA MONTRES PREXA S.A. SULLY WATCH FABRIQUE D’HORLOGERIE S.SCHAYA SULTANA CIE DES MONTRES SULTANA S.A. SUMART SUMART – ROGER UEBELHART WATCH STYLISTS SUPERATIC G.BENGUEREL FILS SUPERTEST ENICAR S.A. SUPREMO MANUFACTURE DE MONTRES NATIONAL S.A. (CHS-LEON SCHMID & CIE.) SUTER SUTER WATCH FACTORY SW WERMEILLE & CIE. SWISSHOUSE ARTHUR SCHWAR – MICAL WATCH CO. INC. TALANTA UHLMAN & CIE. TEAM NICOLET WATCH S.A. TEAM-STEADY NICOLET WATCH S.A. TECHNICIAN FABRIQUE D’HORLOGERIE ZILA S.A. TECHNOS GUNZINGER FRERES S.A. WELSCHENROHR TEDDY EIGELDINGER & CIE S.A. TELIX TELIX WATCH – TENNENBAUM & CIE. TELOR ARTHUR SCHWAR – MICAL WATCH CO. INC. TEMPOMATIC FABRIQUES MOVADO TEMPORIS NOUVELLES FABRIQUES LE PHARE S.A. TENOR TENOR S.A. TENSO ARTHUR SCHWAR – MICAL WATCH CO. INC. TEXA FELCA WATCH S.A. TIMES FABRIQUE D’HORLOGERIE ED. KUMMER S.A. TIMOR TIMOR WATCH CO. J.BERNHEIM CO. TISSOT FABRIQUE D’HORLOGERIE CHS. TISSOT & FILS TITONI FELCA WATCH S.A. 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    ;strong><a href="http://www.onsaleroom.com/bedding-sets-c-69.html ">Bedding Sets</a></strong><br> <strong><a href="http://www.onsaleroom.com/bedding-sets-c-69.html ">Cheap Bedding Sets</a></strong><br> <strong><a href="http://www.onsaleroom.com/bedding-sets-c-69.html ">Cheap Bedding</a></strong><br> <strong><a href="http://www.onsaleroom.com/bedding-sets-c-69.html ">Luxury Bedding</a></strong><br> <strong><a href="http://www.onsaleroom.com/bedding-sets-c-69.html ">Bedding Sets for Sale</a></strong><br> decadery John   04:40 am, 04 Aug 2012 <strong><a href="http://www.dressenter.com/aline-weddin tiffany klocka Turnul din Hanoi - Anul Matematicii 2010

    Turnul din Hanoi - Anul Matematicii 2010 tiffany klocka

    Ştiri

    18 Mar 2011 16:42 Cea mai “cool” oră de mate îşi anunţă câştigătorii!
    22 Feb 2011 12:43 Concursul Cea mai ”cool” oră de mate s-a terminat!
    09 Feb 2011 14:18 Cea mai ”cool” oră de mate gata de final

    Matematica şi jocul

    Nota 4.68 / 35 voturi interes general Turnul din Hanoi

    Comenteaza Trimite unui prieten Imprima

    Unul dintre jocurile aparent simple, dar cu un adânc substrat matematic, este binecunoscutul Turn din Hanoi. Acest joc a fost inventat de matematicianul francez Edouard Lucas şi a fost comercializat ca jucărie pentru toate vârstele încă din anul 1883. La început era confecţionat din lemn şi consta în câteva discuri de mărimi diferite şi 3 beţişoare. Provocarea este de a muta turnul format din discuri de pe un beţişor pe alt beţişor liber, urmând două reguli simple: la fiecare mutare se mută un singur disc şi niciodată nu se aşază un disc mai mare peste un disc mai mic. Bineînţeles, scopul este acela de a realiza mutarea turnului folosind cât mai puţine mutări.

    Se spune că acest joc este inspirat de legenda Turnului lui Brahma, aflat într-un templu al oraşului indian Benares. Acest turn este format din 64 de discuri de aur, de a căror mutare se ocupă preoţii templului, respectând regulile de mai sus. Legenda spune că atunci când turnul discurilor de aur va fi complet transferat pe o altă tijă, templul se va prăbuşi iar lumea va lua sfârşit. Ceea ce ne face să ne întrebăm cu îngrijorare care este numărul minim de mutări în cazul turnului cu 64 de discuri... Pentru a răspunde la această întrebare, vom începe prin a cerceta ce se întâmplă în cazul jocului cu mai puţine discuri.

    Să notăm cele 3 beţişoare folosind literele A, B, C. La început, turnul se află pe beţişorul A, iar noi vrem să îl mutăm pe beţişorul C. Vom nota cu n numărul discurilor, iar cu U n   numărul minim de mutări atunci când avem un turn format din n discuri.
    Cu ajutorul jocului de mai jos, putem observa că:
    • pentru n = 1, turnul se mută printr-o singură mutare, deci U 1   = 1;
    • pentru n = 2, turnul se poate muta prin 3 mutări, deci U 2   = 3;
    • pentru n = 3, turnul se poate muta prin 7 mutări, deci U 3   = 7.

    Dar în cazul unui turn format din n discuri? Să ne gândim: în procesul de mutare a acestui turn, ajungem la momentul în care toate discurile înafară de cel mai mare sunt mutate pe un alt beţişor (B) decât cel iniţial (A), formând un turn cu n-1 discuri. Dacă presupunem că am ajuns la acest moment folosind numărul minim de mutări, înseamnă că până acum am făcut U n-1   mutări. În continuare, mutăm discul cel mai mare de pe A pe C, folosind o singură mişcare. Apoi nu ne rămâne decât să mutăm turnul cu n-1 discuri de pe B pe C, peste discul cel mai mare. Acest lucru poate fi făcut prin minim U n-1 mutări. Acum să numărăm câte mutări am făcut în total: U n   = U n-1   +1 + U n-1   = 2 U n-1   + 1
    Astfel am obţinut o relaţie de recurenţă între numărul minim de mutări pentru un turn cu n discuri şi unul format din n-1 discuri, motiv pentru care acest joc este foarte des folosit ca exemplu pentru tehnici de programare care folosesc recurenţa.
    Folosind procedeul numit inducţie matematică, se poate demonstra uşor că numărul minim de mutări în cazul turnului cu n discuri este: U n   = 2 n - 1.
    Aşadar, revenind la Turnul lui Brahma, numărul minim de mutări este 2 64 - 1, adică un număr de 20 de cifre! Dacă 2 64 - 1  ar fi secunde, atunci asta ar însemna  circa  584 942 417 355 de ani.  Deci putem sta liniştiţi, sfârşitul lumii e departe: chiar dacă preoţii ar munci fără încetare şi ar face o mutare pe secundă, tot le-ar trebui multe mii de milioane de ani pentru a-şi termina treaba.

    După ce matematicienii au epuizat de întors pe toate feţele acest joc, ei au descoperit şi alte variante, mult mai provocatoare din punct de vedere matematic. De exemplu, care ar fi numărul maxim de mutări prin care se poate transfera turnul de pe un beţişor pe altul, fără a reveni la o configuraţie anterioară? Dar dacă jocul ar avea 4 sau mai multe beţişoare în loc de 3? Acestea sunt doar câteva exemple care ne arată cum se poate transforma o problemă relativ simplă într-una de o mare complexitate matematică, prin modificarea minimă a unei singure variabile din enunţul problemei... dar până la urmă, în astfel de lucruri stă frumuseţea matematicii!

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