Turnul din Hanoi - Anul Matematicii 2010 tiffany klocka

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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

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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<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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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[…]

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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

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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. 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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. 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TITUS FABRIQUE SOLVIL DES MONTRES PAUL DITISHEIM S.A. TOUCHON WITTNOUER & CIE. S.A. TOURIST ALLEMANN AD. FILS S.A. TRANSMARINE NICOLET WATCH S.A. TRAVELER FABRIQUE D’HORLOGERIE ZILA S.A. TREBEX MILEX ELEM WATCH CO. LTD. TRIVERA TRIVERA WATCH, WERNER TRIEBOLD-RIS TRIWERA TRIVERA WATCH, WERNER TRIEBOLD-RIS TUDOR MONTRES ROLEX S.A. ULESSE NARDIN NARDIN ULESSE S.A. ULTRASONIC ENICAR S.A. UNITAS MANUFACTURES D’HORLOGERIE RAYMOND A. S.A. UNIVERSAL CHRONOGRAPHES UNIVERSAL UNIVERSAL MANUFACTURES DES MONTRES PERRET & BERTHOUD S.A. UNVER JOHN BRINGOLF & CO VACHERON & CONSTANTIN VACHERON & CONSTANTIN VALORY COMPAGNIE DES MONTRES VALORY S.A. VENDOME VENDOME WATCH – CORCELLES VENUS LES FILS DE PAUL SCHWARZ ETIENNE VINCA VINCA S.A. WATCH FACTORY VIVAX FABRIQUE D’HORLOGERIE ED. KUMMER S.A. VOGIS MATINA WATCH LTD. VULCAIN VULCAIN WATCH FACTORY W.TRIEBOLD W. TRIEBOLD WATCH FACTORY WALDORF ARTHUR SCHWAR – MICAL WATCH CO. INC. WARINES WARINES WATCHES A.GERBER S.A. WATERMASTER FABRIQUE D’HORLOGERIE ZILA S.A. WATEX AD. OBRECHT S.A. WEGA WEGA WATCH S.A. WHITE STAR FABRIQUE WHITE STAR – WEISS & CO. WILKA WILKA WATCH FACTORY WILLE FRERES NATIONAL S.A. MANUFACTURE DE MONTRES WITTNAUER WITTNOUER & CIE. S.A. WYLER FABRIQUE DES MONTRES WYLER S.A. ZEDON ZEDON WATCH CO. ZELIA FABRIQUE SOLVIL DES MONTRES PAUL DITISHEIM S.A. ZENITH FABRIQUE D’HORLOGERIE ZENITH S.A. ZENO EBOSA S.A. EIGELDINGER & CIE. S.A. ZILA FABRIQUE D’HORLOGERIE ZILA S.A. ZODIAC MONTRES ZODIAC S.A.

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Designing Elijah Snake Ring by rockster
Here is a Custom silver Ring hand made with 2 CZ stones set tin the eyes. This ring is size 7 and is 1″ 1/4 wide. It looks very good on finger. An eye catcher for sure. This ring can be cast in Gold as well. We also have a top of the[…]
GM Pendant Conversion by rockster
Gm Pin Pendant Conversion Thank you Mr Richardson for allowing me to convert your GM pin to a pendant. Mr Richardson is celebrating 30 yrs of service from General Motors. He contacted me asking me to convert this Sterling Silver Pin to a Pendant that he could wear with a chain. If you[…]
Benrus Dial O Rama Jump Hou… by rockster
1973 Caravelle 666 & 1… by rockster
CHECK OUT THESE 2 AWESOME CARAVELLE DIVER WATCHES- UPCOMING RESTORATIONS 1973 CARAVELLE 666 DIVER 1973 Caravelle 666 Diver Notice the LollyPop Second Sweeps & Raised Markers 1973 Caravelle 666 Diver- 1973 Caravelle 666 Diver Winder signed “C” […]
Elijah Reedy Speaks by rockster
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 by rockster
WWII MILWWII MILITARY ITARY Vintage Time Re-Made In America ™ THIS GSW TRADE MARK IS NOW A NEW INFORMATION AND WATCH SERVICES WEBSITE. CLICK ABOVE FOR TRANSPORTATION. BE SURE YOU BOOK MARK THE SITE GOLDSMITHWORKS GOLDSMITHWATCHWORKS MILITARYWATCHBOX State Licensed Precious Metal, Gemstone & 2nd Hand Dealer Turning Your Old Jewelry Into Something New ™ Vintage[…]
PICS by rockster
WWII MILITARY WATCH RESTORA… by rockster
Vintage Time Re Vintage Time Re-Made In America ™ THIS GSW TRADE MARK IS NOW A NEW INFORMATION AND WATCH SERVICES WEBSITE. CLICK ABOVE FOR TRANSPORTATION. BE SURE YOU BOOK MARK THE SITE GOLDSMITHWORKS GOLDSMITHWATCHWORKS MILITARYWATCHBOX State Licensed Precious Metal, Gemstone & 2nd Hand Dealer Turning Your Old Jewelry Into Something New ™ Vintage Time[…]
JAPANESE GERMAN BRITISH MIL… by
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[…]
Feb 2017 Ready To Ship by
Feb 2017 Ready To Ship wyler dive w/internal bezel zodiac british military craw ford ladies adjusted Ready To Ship Feb 1, 2017 2 wwi pockets Feb 2017 Ready To Ship Save

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