Matematika U Rubikovoj Kocki

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Matematika U Rubikovoj Kocki SveuˇciliˇsteJ.J. Strossmayera u Osijeku Odjel za matematiku Sveuˇciliˇsninastavniˇckistudij matematike i informatike Marina Bariˇsi´c Matematika u Rubikovoj kocki Diplomski rad Osijek, 2011. SveuˇciliˇsteJ.J. Strossmayera u Osijeku Odjel za matematiku Sveuˇciliˇsninastavniˇckistudij matematike i informatike Marina Bariˇsi´c Matematika u Rubikovoj kocki Diplomski rad Mentor: doc. dr. sc. Ivan Mati´c Komentor: dr. sc. Ljerka Juki´cMati´c Osijek, 2011. Sadrˇzaj 1. Uvod 1 2. Stoˇ je Rubikova kocka? 2 2.1. Povijest Rubikove kocke . 2 2.2. Varijacije Rubikove kocke . 4 2.2.1. Rubik 360 . 4 2.2.2. Sudokocka . 4 2.2.3. Square one . 5 2.2.4. Pyraminx . 5 2.2.5. Oktogonalna prizma . 5 2.2.6. Magiˇcnakugla . 5 2.2.7. Oblo . 6 2.3. Izgled i grada Rubikove kocke . 6 2.4. Broj konfiguracija Rubikove kocke . 7 2.5. Algoritmi za rjeˇsavanje Rubikove kocke . 8 2.6. Natjecanja . 9 3. Matematika u Rubikovoj kocki 11 3.1. Oznake i notacija . 11 3.2. Grupa i podgrupe Rubikove kocke . 13 3.2.1. Grupa Rubikove kocke . 13 3.2.2. Podgrupe . 15 3.3. Generatori . 15 3.4. Simetriˇcnagrupa . 16 3.4.1. Parnost . 18 3.5. Homomorfizam grupa . 19 3.6. Djelovanje grupe . 20 3.7. Konfiguracije Rubikove kocke . 21 3.7.1. Valjane konfiguracije Rubikove kocke . 24 3.8. Graf . 25 3.8.1. Cayley graf . 25 3.8.2. Boˇzanskialgoritam . 26 3.9. Strategije za rjeˇsavanje Rubikove kocke . 29 3.9.1. Komutatori . 29 3.9.2. Konjugacija . 30 3.9.3. Strategije . 30 3.10. Rubikova kocka u danaˇsnjoj kulturi . 33 4. Saˇzetak 36 5. Summary 37 6. Zivotopisˇ 38 1 1. Uvod U ovom diplomskom radu obradena je zabavna slagalica Rubikova kocka i njezina pri- mjena u matematici. Diplomski rad je podijeljen u dva poglavlja: ˇstoje Rubikova kocka i matematika u Rubikovoj kocki. U prvom poglavlju dan je odgovor na pitanje ˇsto je Rubikova kocka. Opisan je njezin nastanak te ˇzivotopis njezinog tvorca Ern¨oaRubika. Detaljno je opisana struk- tura kocke i predstavljene su mnoge njezine varijacije koje su nastale tijekom vremena. Cilj igre s Rubikovom kockom je sloˇzitiRubikovu kocku u poˇcetnipoloˇzaj, pa su u pr- vom poglavlju opisane razliˇcitemetode za pomo´cpri slaganju te su prikazani rezultati i zanimljivosti s natjecanja u slaganju kocke koja se ˇcestoodrˇzavaju. U drugom poglavlju opisana je matematika koja se nalazi u Rubikovoj kocki te razliˇcitipojmovi koji se vrlo lagano mogu objasniti uz pomo´cRubikove kocke: grupa, podgrupa, generatori, cikliˇcka grupa, homomorfizam grupa, parnost i graf su samo neki pojmovi koji pokazuju vaˇznostminijaturne igraˇcke. Takoder, u radu su opisane konfiguracije Rubikove kocke, koje su konfiguracije valjane i dane su neke strategije kako ju rijeˇsitiuz pomo´cmatematike. A ˇstoje boˇzanskialgoritam i boˇzanskibroj? Odgovori se mogu prona´ciu ovom radu. 2 2. Stoˇ je Rubikova kocka? Rubikova kocka je mehaniˇcka igraˇcka oblika kocke koja je nastala 1974. godine u Budimpeˇsti.Tijekom osamdesetih godina proˇslogstolje´cabila je najtraˇzenijai najpo- pularnija igraˇcka. Do 2009. godine prodano je preko 350 milijuna primjeraka, a s njom se igralo viˇseod jedne petine svjetskog stanovniˇstva. Smatra se da je Rubikova kocka najpoznatija igraˇcka na svijetu. U medicinskom rjeˇcnikumogu se na´cipoj- movi "Rubikov zglob" i "Kockin palac" koji oznaˇcavaju bolesti koje nastaju opsesivnim rjeˇsavanjem Rubikove kocke, a od 1980. godine u Americi postoji grupa lijeˇcenih "kockara" nazvana "Cubaholics" koja pomaˇzeljudima koji su ovisni o rjeˇsavanju Ru- bikove kocke. 2.1. Povijest Rubikove kocke Rubikovu kocku sluˇcajno je izumio Ern¨oRubik, inˇzenjer arhitekture i profesor na fakul- tetu primjenjenih umjetnosti u Budimpeˇsti(Slika 2.1.). Slika 2.1. Ern¨oRubik Ern¨oRubik je roden 13. srpnja 1944. godine u Budimpeˇsti,Madarska. Njegova majka je bila umjetnica i pjesnikinja, a otac strojarski tehniˇcari avio inˇzenjer. Ern¨oje 1967. godine diplomirao na Fakultetu tehniˇckihznanosti u Budimpeˇsti,nakon ˇcega je zavrˇsiosrednju ˇskolu dekorativne umjetnosti interijera. Od 1971. do 1975. godine radio je kao arhitekt, nakon ˇcegazapoˇcinjeakademsku karijeru kao profesor. Oˇzenio se 1977. godine arhitekticom interijera i 1978. godine dobiva k´cerAnnu. Ern¨oRubik je postao urednik magazina za igre i puzzle, nakon ˇcegaje osnovao vlastiti 'Rubik Studio' koji se bavi dizajnom namjeˇstaja, igrica i mnogih drugih vari- jacija na temu 'Rubikove kocke'. Godine 1990. imenovan je za predsjednika Madarske akademije za inˇzenjerstvo i osnovao je 'International Rubik Foundation' koja pomaˇze i podrˇzava u radu nadarene inˇzenjerei studente industrijskog dizajna. Ern¨oje bio fasciniran geometrijom 3D oblika i prouˇcavaju´ciobjekte u prostoru napravio je prvi uzorak kocke sastavljene od 26 malih kockica razliˇciteboje. Ern¨oje dobio inspiraciju od oblih, glatkih kamenˇci´cas obale Dunava koji su mu dale rjeˇsenje kako napraviti cilindriˇcneoblike i osovine u kocki. Nakon poˇcetnogoduˇsevljenjazbog kombinacija razliˇcitihboja na kocki, Ern¨oRubik se zapitao kako vratiti kocku u 'poˇcetnostanje'. Shvatio je da nasumiˇcnimokretanjem 3 kocke za svog ˇzivota ne´ceuspjeti vidjeti kocku s kojom je zapoˇceo. Tek nakon oz- biljnog promiˇsljanjao logici svakog pokreta kocke, nakon mjesec dana naˇsaoje pravu kombinaciju od mogu´cih43 trilijuna. Sve se to dogodilo u prolje´ce1974. godine, a ve´c 1975. godine Ern¨oRubik je prijavio igraˇcku'Rubikovu kocku' Madarskom zavodu za patente. Izum je najprije nazvan Magiˇcna kocka (mad. B¨uv¨osKocka), a 1980. godine ime joj je promijenjeno u Rubikova kocka u ˇcastizumitelju. Pod nazivom 'Rubikova kocka' je poznata gotovo u svim jezicima osim u njemaˇckom, ˇzidovskom, kineskom, portugalskom i islandskom. U ˇzidovskom jeziku nazivaju je Madarskom kockom, a u ostalima Magiˇcnomkockom. U to vrijeme Madarska je bila izolirana iza "Zeljezneˇ zavjese" i tek se poˇcetkom 1978. ta ˇcudna igraˇcka pojavila u Madarskim trgovinama igraˇcaka. U poˇcetkuje pro- daja tekla slabo sve dok ju nisu otkrili dvojica Madara: poslovni ˇcovjek Tiber Laszi koji ju je odnio 1979. godine na izloˇzbuigraˇcaka u N¨urnberg i Tom Kremer koji je stupio u kontakt s Ideal Toy Company koja je "pokazala" Rubikovu kocku cijelom svi- jetu. U meduvremenu, engleski matematiˇcarDavid Singmaster otkrio je fenomenalne matematiˇcke osobine kocke o kojima piˇseu ˇclankuu ˇcasopisu ’Scientific Americans', 1979. godine, ˇcimesi je Rubikova kocka osigurala joˇsve´cupozornost javnosti. Inter- nacionalni interes za kocku poˇcinje1980. godine, a ve´c1981. potraˇznjaje nadmaˇsila kapacitete proizvodnje. Zanimljivo je da su u to vrijeme dvojica inovatora prijavila svoje izume sliˇcneRu- bikovoj kocki. Terutoshi Ishige je 1976. godine Japanskom zavodu za patente prijavio kocku nalik Rubikovoj kocki, dok je Amerikanac Larry Nichols prijavio svoju kocku i prije Rubika, no sve su ga kompanije za proizvodnju igraˇcaka odbile te je nesretni Larry Nichols cijeli svoj ˇzivot proveo u siromaˇstvu. Vrlo sliˇcneslagalice nastale su vrlo brzo nakon Rubikove kocke (Slika 2.2.). Ern¨oRu- bik patentirao je i 4 × 4 × 4 Kocku osvete (eng. Revenge cube) koja je zahtjevnija zbog svojih 3:7 × 1045 mogu´cihkonfiguracija, a pojavile su se takoder 2 × 2 × 2 Dˇzepna kocka (eng. Pocket cube) koja nema srediˇsnjegdijela i lakˇsaje za slaganje jer ima 3; 674; 160 razliˇcitihkonfiguracija te 5×5×5 Profesorska kocka (eng. Professor's cube) teˇzaverzija Rubikove kocke jer ima 2:83 × 1074 mogu´cihkonfiguracija. Slika 2.2. Varijacije Rubikove kocke Godine 2005. grˇckiinovator Panagiotis Verde konstruirao je 6 × 6 × 6 Rubikovu kocku koja ima 1:57 × 10116 mogu´cihkonfiguracija, a 2007. uspjeˇsnoje konstruirao i 7 × 7 × 7 kocku koja ima 1:95 × 10160 mogu´cihkonfiguracija. 4 Te kocke su izradene V-cube tehnologijom. Odlikuju se odliˇcnomkvalitetom, vrlo lako se okre´cui njihov savrˇsenimehanizam im omogu´cava da gotovo nikad ne zap- inju. No, vrlo su zahtjevne jer sadrˇzemnogo ve´cibroj razliˇcitih kombinacija nego standardna kocka. Nemaju fiksirane centre (ˇstopove´cava broj mogu´cihkonfiguracija i do 6:29 × 10116 za 6 × 6 × 6 kocku, i 7:80 × 10160 za 7 × 7 × 7 kocku) pa je zbog toga izazov joˇsve´ci, a i pogreˇsnaorijentacija moˇzedovesti do nemogu´cnostislaganja kocke. Godine 2008. predstavljen je prvi primjerak elektroniˇcke 3 × 3 × 3 Rubikove kocke, a 2009. godine predstavljena je i druga elektroniˇcka kocka nazvana 'Rubik's TouchCube' koja radi na senzore koji su osjetljivi na dodir (Slika 2.3.). Stranice se okre´cupovlaˇcenjemprsta po povrˇsinikocke, a kocka ima ugraden mehanizam koji se brine da se registriraju samo dodiri na onoj strani kocke koja je okrenuta prema gore. Slika 2.3. Elektroniˇckiprimjerak Rubikove kocke 2.2. Varijacije Rubikove kocke Od same pojave Rubikove 3 × 3 × 3 kocke do danas pojavile su se brojne varijacije koje su u osnovi sliˇcneRubikovoj kocki: 2.2.1. Rubik 360 Rubik 360 predstavljen je u veljaˇci2009. godine. Sas- toji se od 3 prozirne sfere u kojima se nalazi 6 ˇsarenih kuglica. Cilj je da se ˇsarenekuglice iz unutarnje sfere kroz srediˇsnjusferu, koje ima samo dvije rupice, pre- mjeste u utore iste boje na vanjskoj sferi, pritom da su utor i kuglica iste boje. Rubik 360 nije mogu´ceras- taviti pa sloˇziti u ˇzeljenipoloˇzaj te ima samo jedno Slika 2.4. Rubik 360 rjeˇsenje. 2.2.2. Sudokocka Sudokocka je varijacija Rubikove 3 × 3 × 3 kocke. Sve strane kocke su iste boje i sadrˇzebrojeve od 1 do 9. Slaganje ove kocke teˇzeje od Rubikove kocke jer svaki broj mora biti na toˇcnoodgovaraju´coj poziciji, a centralni broj mora takoder biti u odgovaraju´coj orijentaciji. Kod Sudokocke postoji viˇseod jednog Slika 2.5. Sudokocka rjeˇsenja. 5 2.2.3. Square one Square One je varijacija originalne Rubikove kocke koja zakretanjem daje tijelo koje nema oblik kocke. Sastoji se od triju slojeva. Gornji i donji sloj podi- jeljeni su kao pita u 8 dijelova: 4 kutna koja oblikom podsje´caju na zmaja i 4 rubna dijela u obliku trokuta. Srednji sloj podijeljen je u 2 dijela duˇzneke linije jednog od preostala dva sloja. Svaki se sloj moˇzeslo- Slika 2.6. Square One bodno zakretati. 2.2.4. Pyraminx Pyraminx je slagalica u obliku tetraedra. Svaka je strana obojena razliˇcitombojom, te je podijeljena na 9 trokuta koji su razdijeljeni u 3 sloja: prvi sloj sadrˇzi jedan trokut, drugi sloj sadrˇzitri trokuta, a tre´cisloj sadrˇzi5 trokuta.
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