Il Passatempo Fratto

Il forum di Base5, dove è possibile postare problemi, quiz, indovinelli, rompicapo, enigmi e quant'altro riguardi la matematica ricreativa e oltre.

Moderatori: Gianfranco, Bruno

Bruno
Livello 10
Livello 10
Messaggi: 1900
Iscritto il: lun nov 21, 2005 6:07 pm
Località: Bologna

Re: Il Passatempo Fratto

Messaggio da Bruno »

...ha già seminato anche il Passatempo Fratto :D
(Bruno)

...........................
Invisibile un vento
l'ha apena sfioragia
sospension d'un momento;
e la bola iridessente gera 'ndagia.
{Biagio Marin}
................................................................
Meglio soluzioni sbagliate che risposte esatte.
{Rudi Mathematici}

Quelo
Livello 7
Livello 7
Messaggi: 716
Iscritto il: ven giu 16, 2006 3:34 pm

Re: Il Passatempo Fratto

Messaggio da Quelo »

E non avete visto ancora niente... :lol: :lol: :lol:

$403$ quattro quinti

Codice: Seleziona tutto

41/2+47/3+53/4+59/5+71/6+74/7+83/8+86/9+89/10+101/11+107/12+113/13+125/14+122/15+131/16+134/17+137/18+146/19+149/20+158/21+167/22+164/23+173/24+179/25+185/26+188/27+191/28+194/29+197/30+472/31+215/32+212/33+227/34+218/35+233/36+239/38+236/39+251/40+257/42+263/44+254/45+269/46+281/48+436/49+287/50+284/51+293/52+305/54+296/55+535/56+302/57+317/58+329/60+335/62+314/63+859/64+326/65+353/66+359/68+332/69+383/70+365/72+316/75+371/76+344/77+389/78+401/80+356/81+395/84+362/85+386/87+413/88+407/90+398/91+419/92+404/93+422/95+431/96+443/98+410/99+449/100+461/102+467/104+428/105+473/108+479/110+485/112+491/114+446/115+497/116+452/117+458/119+503/120+470/121+509/124+515/126+521/128+557/130+533/132+482/133+488/135+545/136+563/138+569/140+500/143+581/144+512/145+524/147+587/150+593/152+518/153+599/154+536/155+617/156+611/160+530/161+623/162+542/165+629/168+641/170+548/171+635/174+554/175+647/176+653/180+659/182+671/184+677/186+566/187+578/189+683/190+689/192+584/195+311/196+701/198+707/200+590/203+719/204+596/207+731/208+811/209+737/210+743/216+614/217+749/220+626/221+755/224+632/225+761/228+767/230+656/231+773/232+779/234+785/238+791/240+793/242+382/245+668/247+803/248+809/252+674/253+686/255+821/260+692/261+815/264+827/266+839/270+845/272+698/273+716/275+857/276+710/279+851/280+734/285+863/286+869/288+881/290+887/294+722/297+740/299+893/300+875/304+905/306+911/308+917/310+929/312+746/315+752/319+923/320+941/322+758/323+947/324+764/325+953/330+965/336+959/340+776/341+971/342+788/345+977/348+983/350+794/351+989/352+818/357+37/360+824/363+995/364+111/368+43/372+123/374+830/377+61/378+129/380+67/384+842/385+73/390+848/391+909/392+79/396+860/399+159/400+854/403+866/405+177/406+97/408+103/414+183/416+201/418+109/420+872/425+878/429+127/432+213/434+902/435+890/437+219/440+908/441+237/442+243/448+139/450+914/455+151/456+925/459+249/460+157/462+267/464+932/465+163/468+938/475+291/476+169/480+944/483+303/484+309/490+956/493+321/494+962/495+327/496+181/504+333/506+193/510+974/513+339/520+199/522+998/525+222/527+205/528+369/532+246/539+211/540+375/544+223/546+381/550+258/551+229/552+235/558+387/560+82/561+94/567+241/570+393/572+282/575+259/576+411/580+106/585+265/588+318/589+271/594+354/595+417/598+277/600+366/605+423/608+118/609+283/612+447/616+453/620+142/621+289/624+148/627+307/630+402/637+471/638+477/640+489/644+501/646+295/648+519/650+166/651+301/660+172/663+426/665+438/667+313/672+178/675+507/680+531/682+331/684+337/690+202/693+343/696+537/700+349/702+543/704+444/713+355/714+474/715+361/720+486/725+367/726+549/728+214/735+555/736+226/741+373/744+573/748+579/754+379/756+244/759+591/760+256/765+597/770+492/775+397/780+603/782+250/783+615/784+409/792+415/798+633/800+498/805+639/806+421/810+645/812+427/816+262/819+268/825+433/828+657/832+516/833+669/836+274/837+439/840+534/847+681/850+292/855+445/858+451/864+687/868+457/870+699/874+711/880+463/882+705/884+298/891+695/896+652/897+547/899+469/900+723/910+481/912+347/918+729/920+499/924+747/928+511/930+526/931+606/935+505/936+328/945+753/950+771/952+334/957+517/960+523/966+685/968+346/969+964/975+789/980+795/986+801/988+529/990+807/992=1111
[Sergio] / $17$

Pasquale
Livello 12
Livello 12
Messaggi: 2831
Iscritto il: mer mag 25, 2005 2:14 am

Re: Il Passatempo Fratto

Messaggio da Pasquale »

Hallo!
Scusa Sergio, questa sera, quando ancora era visibile la penultima soluzione, mi son messo un po' al lavoro trovando un piccolo allungamento da fare, ma quando sono andato ad inserirlo, ho visto che la precedente soluzione non era più attuale.
Ho trovato anche che le tue nuove aggiunte sono state inserite in modo un po' sparpagliato, per cui controllare se il mio allungamento fosse ancora valido mi è risultato complicato.
Controllare quali siano i numeri già utilizzati e suddividerli in modo opportuno richiede molto tempo.
Dunque un suggerimento: se aggiungi qualcosa ad una soluzione precedente, magari potresti sistemare i nuovi dati in coda, così che si evidenzino meglio le ultime variazioni.
In alternativa, potresti ad esempio dare un ordine crescente ai soli numeratori delle frazioni.
Comunque, riporto la terzina che mi risultava valida rispetto alla precedente soluzione, in modo che possa controllare se è ancora utilizzabile :

769/328 + 853/492 + 907/984 = 5

Se ti riuscisse facilmente di aggiungere le ultime variazioni in coda alla precedente soluzione, mi risulterebbe più facile aggiornare gli appunti sui numeri già utilizzati o meno, suddivisi in numeri primi e non o cos'altro. Grazie
Ultimo quesito: l'ultima soluzione ha comportato soltanto delle aggiunte, oppure hai apportato anche delle variazioni nell'ambito dell'impianto precedente?
Intanto, la medaglietta che avevo proposto è diventata una cosa risibile rispetto al lavoro svolto e dunque lancio un appello per lo studio di qualcosa in aggiunta di più consistente. Occorre qualche :idea: :?:
_________________

\text {   }ciao Immagine ciao
E' la somma che fa il totale (Totò)

Quelo
Livello 7
Livello 7
Messaggi: 716
Iscritto il: ven giu 16, 2006 3:34 pm

Re: Il Passatempo Fratto

Messaggio da Quelo »

Ciao Pasquale,
il mio metodo si basa sullo scegliere un numero con pochi fattori primi e molti divisori, di cui prendo solo quelli fino a 1000
Questo numero sarà l'MCM di tutti i denominatori
Ad ogni denominatore assegno un numeratore con tale che sia MCD (NUM, DEN) = 1
Ogni frazione introduce un contributo che moltiplicato per MCM dà un numero intero, che io chiamo "incrementatore" INC=NUM*MCM/DEN
Calcolo la somma di tutti gli incrementatori e ottengo il numero che diviso per MCM dà la somma di tutte le frazioni
Trovo l'intero più vicino tale per cui la divisione restituisce un numero intero
La differenza DIF è il valore da compensare modulando i numeratori
Scelgo un certo numero di incrementatori tali che il loro MCM sia divisore di DIF e risolvendo un'equazione diofantea trovo come modificare i rispettivi numeratori.
Tutto questo lo faccio fare ad un programma, poi a mano con excel, aggiusto quei pochi numeratori che risultano doppi o multipli dei denominatori
Quindi ad ogni giro non faccio altro che aggiungere un fattore primo a MCM e poi lavoro sulla soluzione proposta.

La tua idea non è male, si potrebbe pensare ad un algoritmo che estrae terzine dai numeri rimanenti

L'ultima soluzione disponibile è $403$

Codice: Seleziona tutto

ordine denominatori
41/2+47/3+53/4+59/5+71/6+74/7+83/8+86/9+89/10+101/11+107/12+113/13+125/14+122/15+131/16+134/17+137/18+146/19+149/20+158/21+167/22+164/23+173/24+179/25+185/26+188/27+191/28+194/29+197/30+472/31+215/32+212/33+227/34+218/35+233/36+239/38+236/39+251/40+257/42+263/44+254/45+269/46+281/48+436/49+287/50+284/51+293/52+305/54+296/55+535/56+302/57+317/58+329/60+335/62+314/63+859/64+326/65+353/66+359/68+332/69+383/70+365/72+316/75+371/76+344/77+389/78+401/80+356/81+395/84+362/85+386/87+413/88+407/90+398/91+419/92+404/93+422/95+431/96+443/98+410/99+449/100+461/102+467/104+428/105+473/108+479/110+485/112+491/114+446/115+497/116+452/117+458/119+503/120+470/121+509/124+515/126+521/128+557/130+533/132+482/133+488/135+545/136+563/138+569/140+500/143+581/144+512/145+524/147+587/150+593/152+518/153+599/154+536/155+617/156+611/160+530/161+623/162+542/165+629/168+641/170+548/171+635/174+554/175+647/176+653/180+659/182+671/184+677/186+566/187+578/189+683/190+689/192+584/195+311/196+701/198+707/200+590/203+719/204+596/207+731/208+811/209+737/210+743/216+614/217+749/220+626/221+755/224+632/225+761/228+767/230+656/231+773/232+779/234+785/238+791/240+793/242+382/245+668/247+803/248+809/252+674/253+686/255+821/260+692/261+815/264+827/266+839/270+845/272+698/273+716/275+857/276+710/279+851/280+734/285+863/286+869/288+881/290+887/294+722/297+740/299+893/300+875/304+905/306+911/308+917/310+929/312+746/315+752/319+923/320+941/322+758/323+947/324+764/325+953/330+965/336+959/340+776/341+971/342+788/345+977/348+983/350+794/351+989/352+818/357+37/360+824/363+995/364+111/368+43/372+123/374+830/377+61/378+129/380+67/384+842/385+73/390+848/391+909/392+79/396+860/399+159/400+854/403+866/405+177/406+97/408+103/414+183/416+201/418+109/420+872/425+878/429+127/432+213/434+902/435+890/437+219/440+908/441+237/442+243/448+139/450+914/455+151/456+925/459+249/460+157/462+267/464+932/465+163/468+938/475+291/476+169/480+944/483+303/484+309/490+956/493+321/494+962/495+327/496+181/504+333/506+193/510+974/513+339/520+199/522+998/525+222/527+205/528+369/532+246/539+211/540+375/544+223/546+381/550+258/551+229/552+235/558+387/560+82/561+94/567+241/570+393/572+282/575+259/576+411/580+106/585+265/588+318/589+271/594+354/595+417/598+277/600+366/605+423/608+118/609+283/612+447/616+453/620+142/621+289/624+148/627+307/630+402/637+471/638+477/640+489/644+501/646+295/648+519/650+166/651+301/660+172/663+426/665+438/667+313/672+178/675+507/680+531/682+331/684+337/690+202/693+343/696+537/700+349/702+543/704+444/713+355/714+474/715+361/720+486/725+367/726+549/728+214/735+555/736+226/741+373/744+573/748+579/754+379/756+244/759+591/760+256/765+597/770+492/775+397/780+603/782+250/783+615/784+409/792+415/798+633/800+498/805+639/806+421/810+645/812+427/816+262/819+268/825+433/828+657/832+516/833+669/836+274/837+439/840+534/847+681/850+292/855+445/858+451/864+687/868+457/870+699/874+711/880+463/882+705/884+298/891+695/896+652/897+547/899+469/900+723/910+481/912+347/918+729/920+499/924+747/928+511/930+526/931+606/935+505/936+328/945+753/950+771/952+334/957+517/960+523/966+685/968+346/969+964/975+789/980+795/986+801/988+529/990+807/992=1111

ordine numeratori
37/360+41/2+43/372+47/3+53/4+59/5+61/378+67/384+71/6+73/390+74/7+79/396+82/561+83/8+86/9+89/10+94/567+97/408+101/11+103/414+106/585+107/12+109/420+111/368+113/13+118/609+122/15+123/374+125/14+127/432+129/380+131/16+134/17+137/18+139/450+142/621+146/19+148/627+149/20+151/456+157/462+158/21+159/400+163/468+164/23+166/651+167/22+169/480+172/663+173/24+177/406+178/675+179/25+181/504+183/416+185/26+188/27+191/28+193/510+194/29+197/30+199/522+201/418+202/693+205/528+211/540+212/33+213/434+214/735+215/32+218/35+219/440+222/527+223/546+226/741+227/34+229/552+233/36+235/558+236/39+237/442+239/38+241/570+243/448+244/759+246/539+249/460+250/783+251/40+254/45+256/765+257/42+258/551+259/576+262/819+263/44+265/588+267/464+268/825+269/46+271/594+274/837+277/600+281/48+282/575+283/612+284/51+287/50+289/624+291/476+292/855+293/52+295/648+296/55+298/891+301/660+302/57+303/484+305/54+307/630+309/490+311/196+313/672+314/63+316/75+317/58+318/589+321/494+326/65+327/496+328/945+329/60+331/684+332/69+333/506+334/957+335/62+337/690+339/520+343/696+344/77+346/969+347/918+349/702+353/66+354/595+355/714+356/81+359/68+361/720+362/85+365/72+366/605+367/726+369/532+371/76+373/744+375/544+379/756+381/550+382/245+383/70+386/87+387/560+389/78+393/572+395/84+397/780+398/91+401/80+402/637+404/93+407/90+409/792+410/99+411/580+413/88+415/798+417/598+419/92+421/810+422/95+423/608+426/665+427/816+428/105+431/96+433/828+436/49+438/667+439/840+443/98+444/713+445/858+446/115+447/616+449/100+451/864+452/117+453/620+457/870+458/119+461/102+463/882+467/104+469/900+470/121+471/638+472/31+473/108+474/715+477/640+479/110+481/912+482/133+485/112+486/725+488/135+489/644+491/114+492/775+497/116+498/805+499/924+500/143+501/646+503/120+505/936+507/680+509/124+511/930+512/145+515/126+516/833+517/960+518/153+519/650+521/128+523/966+524/147+526/931+529/990+530/161+531/682+533/132+534/847+535/56+536/155+537/700+542/165+543/704+545/136+547/899+548/171+549/728+554/175+555/736+557/130+563/138+566/187+569/140+573/748+578/189+579/754+581/144+584/195+587/150+590/203+591/760+593/152+596/207+597/770+599/154+603/782+606/935+611/160+614/217+615/784+617/156+623/162+626/221+629/168+632/225+633/800+635/174+639/806+641/170+645/812+647/176+652/897+653/180+656/231+657/832+659/182+668/247+669/836+671/184+674/253+677/186+681/850+683/190+685/968+686/255+687/868+689/192+692/261+695/896+698/273+699/874+701/198+705/884+707/200+710/279+711/880+716/275+719/204+722/297+723/910+729/920+731/208+734/285+737/210+740/299+743/216+746/315+747/928+749/220+752/319+753/950+755/224+758/323+761/228+764/325+767/230+771/952+773/232+776/341+779/234+785/238+788/345+789/980+791/240+793/242+794/351+795/986+801/988+803/248+807/992+809/252+811/209+815/264+818/357+821/260+824/363+827/266+830/377+839/270+842/385+845/272+848/391+851/280+854/403+857/276+859/64+860/399+863/286+866/405+869/288+872/425+875/304+878/429+881/290+887/294+890/437+893/300+902/435+905/306+908/441+909/392+911/308+914/455+917/310+923/320+925/459+929/312+932/465+938/475+941/322+944/483+947/324+953/330+956/493+959/340+962/495+964/975+965/336+971/342+974/513+977/348+983/350+989/352+995/364+998/525=1111

numeri rimanenti
[141, 206, 278, 338, 358, 370, 376, 388, 394, 412, 424, 430, 454, 466, 478, 487, 502, 508, 514, 538, 541, 553, 556, 559, 562, 564, 565, 568, 571, 574, 577, 582, 583, 586, 592, 601, 602, 604, 607, 610, 613, 618, 619, 622, 625, 628, 631, 634, 636, 642, 643, 649, 654, 655, 658, 661, 662, 664, 666, 670, 673, 676, 678, 679, 688, 691, 694, 697, 703, 706, 708, 709, 712, 717, 718, 721, 724, 727, 730, 732, 733, 738, 739, 742, 745, 750, 751, 757, 762, 763, 766, 768, 769, 772, 774, 777, 778, 781, 786, 787, 790, 796, 797, 799, 802, 804, 808, 813, 814, 817, 820, 822, 823, 826, 829, 831, 834, 835, 838, 841, 843, 844, 846, 849, 852, 853, 856, 861, 862, 865, 867, 871, 873, 876, 877, 879, 883, 885, 886, 888, 889, 892, 894, 895, 898, 901, 903, 904, 906, 907, 913, 915, 916, 919, 921, 922, 926, 927, 933, 934, 937, 939, 940, 942, 943, 946, 948, 949, 951, 954, 955, 958, 961, 963, 967, 970, 972, 973, 976, 978, 979, 981, 982, 984, 985, 987, 991, 993, 994, 996, 997, 999]

numeri primi rimanenti
[487, 541, 571, 577, 601, 607, 613, 619, 631, 643, 661, 673, 691, 709, 727, 733, 739, 751, 757, 769, 787, 797, 823, 829, 853, 877, 883, 907, 919, 937, 967, 991, 997]
[Sergio] / $17$

Quelo
Livello 7
Livello 7
Messaggi: 716
Iscritto il: ven giu 16, 2006 3:34 pm

Re: Il Passatempo Fratto

Messaggio da Quelo »

Ho rettificato il precedente messaggio, nella 432 c'è un errore, l'ultima valida è la 403

...

C'era un errore, in quanto il linguaggio di programmazione utilizzato non è preciso sulle divisioni di numeri grandi
Nel compensare la cosa sono passato direttamente a $456$, pari a nove decimi

Comincia a diventare sempre più laboriosa la sistemazione finale, penso che mi fermerò per un po'

Codice: Seleziona tutto

ordine denominatori
47/2+53/3+59/4+71/5+83/6+86/7+89/8+101/9+107/10+113/11+125/12+122/13+733/14+134/15+137/16+146/17+149/18+158/19+167/20+173/21+179/22+131/23+191/24+194/25+197/26+562/27+215/28+212/29+227/30+218/31+233/32+236/33+239/34+251/35+257/36+254/37+263/38+269/39+281/40+278/41+293/42+305/44+284/45+311/46+317/48+793/49+329/50+314/51+335/52+347/54+326/55+353/56+332/57+359/58+371/60+799/62+338/63+365/64+344/65+389/66+395/68+356/69+401/70+413/72+419/74+362/75+431/76+386/77+443/78+449/80+398/81+461/82+467/84+404/85+422/87+383/88+473/90+428/91+485/92+446/93+452/95+491/96+503/98+458/99+497/100+509/102+515/104+482/105+521/108+557/110+470/111+545/112+563/114+778/115+569/116+500/117+512/119+581/120+524/121+530/123+587/124+593/126+599/128+617/130+611/132+536/133+542/135+895/136+635/138+781/140+548/143+647/144+554/145+566/147+653/148+659/150+671/152+584/153+985/154+578/155+683/156+689/160+590/161+695/162+701/164+596/165+719/168+707/170+602/171+731/174+614/175+917/176+737/180+755/182+749/184+626/185+761/186+632/187+662/189+767/190+773/192+668/195+785/196+791/198+797/200+674/203+803/204+686/205+692/207+809/208+698/209+821/210+815/216+710/217+827/220+716/221+839/222+845/224+722/225+857/228+863/230+734/231+869/232+875/234+881/238+887/240+893/242+746/245+905/246+752/247+911/248+923/252+758/253+764/255+955/256+776/259+929/260+788/261+941/264+947/266+953/270+959/272+794/273+818/275+965/276+824/279+971/280+842/285+977/286+830/287+983/288+989/290+995/294+129/296+848/297+854/299+43/300+141/304+61/306+159/308+177/310+67/312+866/315+860/319+183/320+201/322+872/323+73/324+878/325+213/328+79/330+890/333+97/336+219/340+908/341+103/342+914/345+109/348+237/350+926/351+243/352+932/357+127/360+938/363+249/364+267/368+944/369+291/370+139/372+303/374+956/377+151/378+309/380+157/384+974/385+163/390+998/391+321/392+169/396+94/399+327/400+258/403+106/405+339/406+282/407+181/408+381/410+193/414+375/416+387/418+199/420+318/425+118/429+211/432+393/434+142/435+354/437+411/440+166/441+417/442+223/444+423/448+229/450+366/451+402/455+235/456+172/459+447/460+241/462+453/464+178/465+265/468+426/475+471/476+271/480+438/481+202/483+477/484+489/490+277/492+474/493+501/494+214/495+507/496+283/504+519/506+301/510+226/513+531/518+537/520+289/522+244/525+486/527+295/528+543/532+498/533+516/539+307/540+549/544+313/546+573/550+534/551+331/552+262/555+337/558+579/560+250/561+268/567+343/570+591/572+597/574+564/575+349/576+603/580+274/585+355/588+582/589+633/592+361/594+606/595+639/598+367/600+618/605+645/608+292/609+373/612+298/615+657/616+669/620+316/621+379/624+334/627+636/629+397/630+642/637+681/638+687/640+699/644+705/646+409/648+711/650+346/651+717/656+421/660+358/663+654/665+415/666+678/667+433/672+376/675+723/680+729/682+427/684+439/690+382/693+445/696+708/697+747/700+457/702+732/703+753/704+750/713+463/714+762/715+469/720+771/725+487/726+789/728+388/735+795/736+499/738+801/740+394/741+505/744+807/748+813/754+517/756+412/759+831/760+424/765+511/768+843/770+774/775+430/777+786/779+523/780+849/782+436/783+867/784+529/792+535/798+873/800+804/805+879/806+541/810+885/812+903/814+547/816+454/819+909/820+466/825+553/828+915/832+822/833+921/836+472/837+559/840+834/847+927/850+846/851+478/855+565/858+502/861+571/864+933/868+577/870+939/874+951/880+583/882+963/884+601/888+508/891+641/896+514/897+852/899+607/900+987/902+993/910+613/912+619/918+631/920+625/924+876/925+643/928+649/930+916/931+906/935+655/936+958/943+526/945+661/950+673/952+538/957+679/960+685/962+691/966+769/968+556/969+976/975+727/980+721/984+677/986+739/988+751/990+623/992+694/999=1355

ordine numeratori
43/300+47/2+53/3+59/4+61/306+67/312+71/5+73/324+79/330+83/6+86/7+89/8+94/399+97/336+101/9+103/342+106/405+107/10+109/348+113/11+118/429+122/13+125/12+127/360+129/296+131/23+134/15+137/16+139/372+141/304+142/435+146/17+149/18+151/378+157/384+158/19+159/308+163/390+166/441+167/20+169/396+172/459+173/21+177/310+178/465+179/22+181/408+183/320+191/24+193/414+194/25+197/26+199/420+201/322+202/483+211/432+212/29+213/328+214/495+215/28+218/31+219/340+223/444+226/513+227/30+229/450+233/32+235/456+236/33+237/350+239/34+241/462+243/352+244/525+249/364+250/561+251/35+254/37+257/36+258/403+262/555+263/38+265/468+267/368+268/567+269/39+271/480+274/585+277/492+278/41+281/40+282/407+283/504+284/45+289/522+291/370+292/609+293/42+295/528+298/615+301/510+303/374+305/44+307/540+309/380+311/46+313/546+314/51+316/621+317/48+318/425+321/392+326/55+327/400+329/50+331/552+332/57+334/627+335/52+337/558+338/63+339/406+343/570+344/65+346/651+347/54+349/576+353/56+354/437+355/588+356/69+358/663+359/58+361/594+362/75+365/64+366/451+367/600+371/60+373/612+375/416+376/675+379/624+381/410+382/693+383/88+386/77+387/418+388/735+389/66+393/434+394/741+395/68+397/630+398/81+401/70+402/455+404/85+409/648+411/440+412/759+413/72+415/666+417/442+419/74+421/660+422/87+423/448+424/765+426/475+427/684+428/91+430/777+431/76+433/672+436/783+438/481+439/690+443/78+445/696+446/93+447/460+449/80+452/95+453/464+454/819+457/702+458/99+461/82+463/714+466/825+467/84+469/720+470/111+471/476+472/837+473/90+474/493+477/484+478/855+482/105+485/92+486/527+487/726+489/490+491/96+497/100+498/533+499/738+500/117+501/494+502/861+503/98+505/744+507/496+508/891+509/102+511/768+512/119+514/897+515/104+516/539+517/756+519/506+521/108+523/780+524/121+526/945+529/792+530/123+531/518+534/551+535/798+536/133+537/520+538/957+541/810+542/135+543/532+545/112+547/816+548/143+549/544+553/828+554/145+556/969+557/110+559/840+562/27+563/114+564/575+565/858+566/147+569/116+571/864+573/550+577/870+578/155+579/560+581/120+582/589+583/882+584/153+587/124+590/161+591/572+593/126+596/165+597/574+599/128+601/888+602/171+603/580+606/595+607/900+611/132+613/912+614/175+617/130+618/605+619/918+623/992+625/924+626/185+631/920+632/187+633/592+635/138+636/629+639/598+641/896+642/637+643/928+645/608+647/144+649/930+653/148+654/665+655/936+657/616+659/150+661/950+662/189+668/195+669/620+671/152+673/952+674/203+677/986+678/667+679/960+681/638+683/156+685/962+686/205+687/640+689/160+691/966+692/207+694/999+695/162+698/209+699/644+701/164+705/646+707/170+708/697+710/217+711/650+716/221+717/656+719/168+721/984+722/225+723/680+727/980+729/682+731/174+732/703+733/14+734/231+737/180+739/988+746/245+747/700+749/184+750/713+751/990+752/247+753/704+755/182+758/253+761/186+762/715+764/255+767/190+769/968+771/725+773/192+774/775+776/259+778/115+781/140+785/196+786/779+788/261+789/728+791/198+793/49+794/273+795/736+797/200+799/62+801/740+803/204+804/805+807/748+809/208+813/754+815/216+818/275+821/210+822/833+824/279+827/220+830/287+831/760+834/847+839/222+842/285+843/770+845/224+846/851+848/297+849/782+852/899+854/299+857/228+860/319+863/230+866/315+867/784+869/232+872/323+873/800+875/234+876/925+878/325+879/806+881/238+885/812+887/240+890/333+893/242+895/136+903/814+905/246+906/935+908/341+909/820+911/248+914/345+915/832+916/931+917/176+921/836+923/252+926/351+927/850+929/260+932/357+933/868+938/363+939/874+941/264+944/369+947/266+951/880+953/270+955/256+956/377+958/943+959/272+963/884+965/276+971/280+974/385+976/975+977/286+983/288+985/154+987/902+989/290+993/910+995/294+998/391=1355

numeri rimanenti
[188, 206, 302, 479, 488, 568, 586, 604, 610, 622, 628, 634, 652, 658, 664, 670, 676, 688, 706, 709, 712, 718, 724, 730, 742, 743, 745, 757, 763, 766, 772, 787, 790, 796, 802, 808, 811, 817, 823, 826, 829, 835, 838, 841, 844, 853, 856, 859, 862, 865, 871, 877, 883, 886, 889, 892, 894, 898, 901, 904, 907, 913, 919, 922, 934, 937, 940, 942, 946, 948, 949, 954, 961, 964, 967, 970, 972, 973, 978, 979, 981, 982, 991, 994, 996, 997]

numeri primi rimanenti
[479, 709, 743, 757, 787, 811, 823, 829, 853, 859, 877, 883, 907, 919, 937, 967, 991, 997]
[Sergio] / $17$

Quelo
Livello 7
Livello 7
Messaggi: 716
Iscritto il: ven giu 16, 2006 3:34 pm

Re: Il Passatempo Fratto

Messaggio da Quelo »

Non so se un giorno si può considerare un po', comunque $479$

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ordine denominatori
47/2+53/3+59/4+71/5+83/6+89/7+101/8+107/9+113/10+122/11+125/12+131/13+137/14+134/15+149/16+146/17+167/18+158/19+173/20+179/21+191/22+188/23+197/24+194/25+227/26+955/27+233/28+862/29+239/30+218/31+251/32+236/33+257/34+254/35+263/36+898/37+281/38+278/39+973/40+284/41+767/42+997/43+311/44+314/45+317/46+329/48+326/49+347/50+332/51+305/52+293/54+488/55+359/56+356/57+365/58+371/60+383/62+362/63+389/64+386/65+395/66+401/68+398/69+419/70+413/72+431/74+404/75+443/76+422/77+449/78+461/80+428/81+467/82+479/84+446/85+485/86+452/87+491/88+497/90+458/91+503/92+470/93+482/95+509/96+515/98+983/99+521/100+545/102+623/104+524/105+563/108+269/110+500/111+587/112+581/114+536/115+593/116+530/117+542/119+599/120+573/121+554/123+611/124+617/126+937/128+566/129+641/130+635/132+970/133+584/135+647/136+653/138+659/140+590/143+671/144+596/145+614/147+677/148+683/150+689/152+626/153+695/154+632/155+701/156+707/160+662/161+719/162+737/164+668/165+743/168+749/170+674/171+755/172+353/174+692/175+761/176+773/180+785/182+791/184+686/185+797/186+698/187+710/189+803/190+809/192+716/195+815/196+821/198+827/200+722/203+839/204+734/205+746/207+857/208+752/209+863/210+758/215+845/216+764/217+881/220+776/221+869/222+887/224+788/225+875/228+893/230+794/231+905/232+911/234+923/238+917/240+985/242+818/245+941/246+824/247+947/248+953/252+830/253+842/255+889/256+965/258+848/259+971/260+854/261+977/264+177/266+61/270+995/272+866/273+872/275+67/276+878/279+141/280+908/285+159/286+890/287+73/288+757/290+79/294+183/296+914/297+926/299+97/300+932/301+201/304+103/306+213/308+219/310+109/312+944/315+938/319+237/320+243/322+956/323+127/324+974/325+249/328+139/330+998/333+151/336+267/340+282/341+157/342+291/344+94/345+979/348+303/350+106/351+309/352+118/357+169/360+142/363+321/364+327/368+166/369+339/370+181/372+375/374+318/377+193/378+381/380+199/384+354/385+178/387+211/390+366/391+393/392+223/396+202/399+411/400+402/403+214/405+417/406+423/407+229/408+447/410+235/414+453/416+471/418+241/420+426/425+226/429+477/430+265/432+489/434+244/435+438/437+501/440+250/441+519/442+271/444+507/448+277/450+474/451+486/455+283/456+262/459+531/460+289/462+537/464+268/465+295/468+498/473+534/475+543/476+307/480+549/481+274/483+361/484+579/490+313/492+564/493+591/494+292/495+597/496+331/504+603/506+337/510+298/513+343/516+639/518+657/520+349/522+316/525+582/527+355/528+669/532+606/533+618/539+961/540+681/544+367/546+687/550+636/551+373/552+334/555+379/558+642/559+699/560+346/561+358/567+397/570+705/572+711/574+654/575+409/576+717/580+376/585+415/588+678/589+723/592+421/594+708/595+729/598+427/600+747/602+732/605+753/608+382/609+433/612+388/615+771/616+789/620+394/621+439/624+412/627+750/629+457/630+762/637+795/638+801/640+807/644+424/645+813/646+445/648+831/650+436/651+843/656+463/660+454/663+786/665+469/666+804/667+487/672+466/675+849/680+867/682+499/684+873/688+511/690+472/693+505/696+822/697+879/700+517/702+834/703+885/704+846/713+523/714+852/715+529/720+876/725+535/726+909/728+894/731+478/735+915/736+541/738+921/740+502/741+547/744+927/748+933/754+565/756+508/759+939/760+514/765+553/768+951/770+571/774+906/775+526/777+942/779+577/780+963/782+538/783+981/784+601/792+583/798+987/800+948/805+993/806+607/810+613/812+619/814+625/816+954/817+556/819+631/820+562/825+643/828+649/832+972/833+655/836+568/837+661/840+978/847+673/850+996/851+586/855+679/858+691/860+604/861+685/864+709/868+721/870+727/874+733/880+739/882+883/884+751/888+610/891+633/896+622/897+512/899+763/900+769/902+634/903+781/910+787/912+793/918+799/920+811/924+652/925+823/928+829/930+718/931+658/935+835/936+670/943+676/945+841/946+853/950+859/952+694/957+871/960+865/962+877/966+745/968+706/969+766/975+901/980+929/984+895/986+907/988+790/989+919/990+913/992+724/999=1484

ordine numeratori
47/2+53/3+59/4+61/270+67/276+71/5+73/288+79/294+83/6+89/7+94/345+97/300+101/8+103/306+106/351+107/9+109/312+113/10+118/357+122/11+125/12+127/324+131/13+134/15+137/14+139/330+141/280+142/363+146/17+149/16+151/336+157/342+158/19+159/286+166/369+167/18+169/360+173/20+177/266+178/387+179/21+181/372+183/296+188/23+191/22+193/378+194/25+197/24+199/384+201/304+202/399+211/390+213/308+214/405+218/31+219/310+223/396+226/429+227/26+229/408+233/28+235/414+236/33+237/320+239/30+241/420+243/322+244/435+249/328+250/441+251/32+254/35+257/34+262/459+263/36+265/432+267/340+268/465+269/110+271/444+274/483+277/450+278/39+281/38+282/341+283/456+284/41+289/462+291/344+292/495+293/54+295/468+298/513+303/350+305/52+307/480+309/352+311/44+313/492+314/45+316/525+317/46+318/377+321/364+326/49+327/368+329/48+331/504+332/51+334/555+337/510+339/370+343/516+346/561+347/50+349/522+353/174+354/385+355/528+356/57+358/567+359/56+361/484+362/63+365/58+366/391+367/546+371/60+373/552+375/374+376/585+379/558+381/380+382/609+383/62+386/65+388/615+389/64+393/392+394/621+395/66+397/570+398/69+401/68+402/403+404/75+409/576+411/400+412/627+413/72+415/588+417/406+419/70+421/594+422/77+423/407+424/645+426/425+427/600+428/81+431/74+433/612+436/651+438/437+439/624+443/76+445/648+446/85+447/410+449/78+452/87+453/416+454/663+457/630+458/91+461/80+463/660+466/675+467/82+469/666+470/93+471/418+472/693+474/451+477/430+478/735+479/84+482/95+485/86+486/455+487/672+488/55+489/434+491/88+497/90+498/473+499/684+500/111+501/440+502/741+503/92+505/696+507/448+508/759+509/96+511/690+512/899+514/765+515/98+517/702+519/442+521/100+523/714+524/105+526/777+529/720+530/117+531/460+534/475+535/726+536/115+537/464+538/783+541/738+542/119+543/476+545/102+547/744+549/481+553/768+554/123+556/819+562/825+563/108+564/493+565/756+566/129+568/837+571/774+573/121+577/780+579/490+581/114+582/527+583/798+584/135+586/855+587/112+590/143+591/494+593/116+596/145+597/496+599/120+601/792+603/506+604/861+606/533+607/810+610/891+611/124+613/812+614/147+617/126+618/539+619/814+622/897+623/104+625/816+626/153+631/820+632/155+633/896+634/903+635/132+636/551+639/518+641/130+642/559+643/828+647/136+649/832+652/925+653/138+654/575+655/836+657/520+658/935+659/140+661/840+662/161+668/165+669/532+670/943+671/144+673/850+674/171+676/945+677/148+678/589+679/858+681/544+683/150+685/864+686/185+687/550+689/152+691/860+692/175+694/957+695/154+698/187+699/560+701/156+705/572+706/969+707/160+708/595+709/868+710/189+711/574+716/195+717/580+718/931+719/162+721/870+722/203+723/592+724/999+727/874+729/598+732/605+733/880+734/205+737/164+739/882+743/168+745/968+746/207+747/602+749/170+750/629+751/888+752/209+753/608+755/172+757/290+758/215+761/176+762/637+763/900+764/217+766/975+767/42+769/902+771/616+773/180+776/221+781/910+785/182+786/665+787/912+788/225+789/620+790/989+791/184+793/918+794/231+795/638+797/186+799/920+801/640+803/190+804/667+807/644+809/192+811/924+813/646+815/196+818/245+821/198+822/697+823/928+824/247+827/200+829/930+830/253+831/650+834/703+835/936+839/204+841/946+842/255+843/656+845/216+846/713+848/259+849/680+852/715+853/950+854/261+857/208+859/952+862/29+863/210+865/962+866/273+867/682+869/222+871/960+872/275+873/688+875/228+876/725+877/966+878/279+879/700+881/220+883/884+885/704+887/224+889/256+890/287+893/230+894/731+895/986+898/37+901/980+905/232+906/775+907/988+908/285+909/728+911/234+913/992+914/297+915/736+917/240+919/990+921/740+923/238+926/299+927/748+929/984+932/301+933/754+937/128+938/319+939/760+941/246+942/779+944/315+947/248+948/805+951/770+953/252+954/817+955/27+956/323+961/540+963/782+965/258+970/133+971/260+972/833+973/40+974/325+977/264+978/847+979/348+981/784+983/99+985/242+987/800+993/806+995/272+996/851+997/43+998/333=1484

numeri rimanenti
[163, 206, 212, 302, 335, 338, 548, 557, 569, 578, 628, 664, 712, 730, 742, 772, 778, 796, 802, 808, 826, 838, 844, 856, 886, 892, 904, 916, 922, 934, 940, 949, 958, 959, 964, 967, 976, 982, 991, 994]

numeri primi rimanenti
[163, 557, 569, 967, 991]
[Sergio] / $17$

Quelo
Livello 7
Livello 7
Messaggi: 716
Iscritto il: ven giu 16, 2006 3:34 pm

Re: Il Passatempo Fratto

Messaggio da Quelo »

In attesa che i tempi siano maturi per una nuova soluzione, sono andato ad indagare insiemi diversi da quello proposto da Pasquale ed ho scoperto che alcuni ammettono soluzioni complete e altri no.
Il caso più semplice è l'insieme {2,3,4,5,6,7} che ammette una sola soluzione:

{2..7}

$\displaystyle \frac{5}{2}+\frac{4}{3}+\frac{7}{6}=5 \quad \quad$ 5/2+4/3+7/6=5

Gli insiemi {2..9} e {2..11} non ammettono soluzioni mentre i successivi fino a {2..19} presentano diverse soluzioni (per 17 e 19 ne riporto solo alclune)

Da qui la congettura
Dato un insieme di numeri naturali consecutivi da 2 a 2n+1 con n > 5 è possibile, usando tutti i numeri, comporre n frazioni irriducibili la cui somma sia un numero intero
Chiaramente è una congettura ludica che non ha alcun fondamento matematico :wink:

{2..13}

$\displaystyle \frac{9}{2}+\frac{8}{3}+\frac{11}{4}+\frac{6}{5}+\frac{13}{10}+\frac{7}{12}=13 \quad \quad$ 9/2+8/3+11/4+6/5+13/10+7/12=13

$\displaystyle \frac{3}{2}+\frac{11}{4}+\frac{8}{5}+\frac{7}{6}+\frac{9}{10}+\frac{13}{12}=9 \quad \quad$ 3/2+11/4+8/5+7/6+9/10+13/12=9

$\displaystyle \frac{11}{2}+\frac{3}{4}+\frac{8}{5}+\frac{7}{6}+\frac{9}{10}+\frac{13}{12}=11 \quad \quad$ 11/2+3/4+8/5+7/6+9/10+13/12=11

$\displaystyle \frac{2}{3}+\frac{7}{4}+\frac{8}{5}+\frac{13}{6}+\frac{9}{10}+\frac{11}{12}=8 \quad \quad$ 2/3+7/4+8/5+13/6+9/10+11/12=8

Codice: Seleziona tutto

{2..15}
15/2+8/3+9/4+14/5+13/6+7/10+11/12=19
15/2+14/3+13/4+8/5+7/6+9/10+11/12=20
9/2+10/3+7/4+14/5+13/6+11/12+8/15=16
5/2+8/3+9/4+13/6+10/7+11/12+15/14=13
9/2+8/3+5/4+13/6+10/7+11/12+15/14=14
9/2+8/3+11/4+13/6+10/7+5/12+15/14=15
9/2+10/3+15/4+11/6+8/7+13/12+5/14=16
5/2+8/3+11/4+7/6+9/10+13/12+14/15=12
7/2+8/3+13/4+5/6+9/10+11/12+14/15=13
7/2+14/3+9/4+5/6+13/10+11/12+8/15=14
9/2+14/3+7/4+11/6+13/10+5/12+8/15=15
11/2+8/3+5/4+13/6+9/10+7/12+14/15=14
13/2+8/3+7/4+11/6+9/10+5/12+14/15=15
15/2+4/3+8/5+13/6+12/7+9/10+11/14=16
5/2+8/3+13/6+12/7+9/10+11/14+4/15=11
9/2+4/3+5/6+12/7+13/10+11/14+8/15=11
3/2+9/4+8/5+11/6+13/10+7/12+14/15=10
9/2+7/4+8/5+11/6+3/10+13/12+14/15=12
9/2+13/4+8/5+11/6+3/10+7/12+14/15=13
13/2+9/4+8/5+11/6+3/10+7/12+14/15=14
3/2+5/4+6/7+13/10+11/12+9/14+8/15=7
3/2+11/4+6/7+13/10+5/12+9/14+8/15=8
5/2+13/4+6/7+3/10+11/12+9/14+8/15=9
11/2+3/4+6/7+13/10+5/12+9/14+8/15=10
13/2+5/4+6/7+3/10+11/12+9/14+8/15=11
13/2+11/4+6/7+3/10+5/12+9/14+8/15=12
8/3+11/4+14/5+13/6+9/10+7/12+2/15=12
14/3+9/4+8/5+7/6+11/10+13/12+2/15=12
14/3+11/4+2/5+13/6+9/10+7/12+8/15=12

Codice: Seleziona tutto

{2..17}
9/2+8/3+11/4+16/5+7/6+17/10+13/12+14/15=18
9/2+8/3+13/4+16/5+17/6+7/10+11/12+14/15=19
9/2+8/3+17/4+14/5+11/6+13/10+7/12+16/15=19
9/2+14/3+17/4+16/5+13/6+11/10+7/12+8/15=21
9/2+16/3+11/4+8/5+7/6+13/10+17/12+14/15=19
9/2+16/3+17/4+8/5+7/6+13/10+11/12+14/15=20
11/2+8/3+9/4+14/5+17/6+13/10+7/12+16/15=19
11/2+8/3+9/4+16/5+13/6+17/10+7/12+14/15=19
13/2+8/3+9/4+16/5+17/6+7/10+11/12+14/15=20
17/2+8/3+9/4+14/5+11/6+13/10+7/12+16/15=21
17/2+14/3+9/4+16/5+13/6+11/10+7/12+8/15=23
17/2+16/3+9/4+8/5+7/6+13/10+11/12+14/15=22
9/2+8/3+16/5+13/6+12/7+17/10+11/14+4/15=17
11/2+16/3+12/5+17/6+4/7+9/10+13/14+8/15=19
13/2+4/3+12/5+11/6+16/7+9/10+17/14+8/15=17
17/2+16/3+12/5+11/6+4/7+9/10+13/14+8/15=21

Codice: Seleziona tutto

{2..19}
9/2+8/3+15/4+18/5+13/6+16/7+19/10+11/12+17/14=23
11/2+16/3+15/4+8/5+17/6+18/7+9/10+19/12+13/14=25
13/2+8/3+15/4+18/5+19/6+16/7+9/10+11/12+17/14=25
15/2+8/3+19/4+18/5+13/6+16/7+9/10+11/12+17/14=26
15/2+16/3+11/4+8/5+17/6+18/7+9/10+19/12+13/14=26
15/2+16/3+17/4+18/5+11/6+8/7+9/10+13/12+19/14=27
17/2+16/3+15/4+8/5+11/6+18/7+9/10+19/12+13/14=27
19/2+8/3+15/4+18/5+13/6+16/7+9/10+11/12+17/14=27
9/2+10/3+13/4+18/5+17/6+8/7+11/12+19/14+16/15=22
9/2+10/3+19/4+8/5+11/6+18/7+17/12+13/14+16/15=22
13/2+10/3+9/4+18/5+17/6+8/7+11/12+19/14+16/15=23
19/2+10/3+9/4+8/5+17/6+18/7+11/12+13/14+16/15=25
[Sergio] / $17$

Pasquale
Livello 12
Livello 12
Messaggi: 2831
Iscritto il: mer mag 25, 2005 2:14 am

Re: Il Passatempo Fratto

Messaggio da Pasquale »

Incredibile! Oltre 110 e lode, baci e abbracci, cosa altro si potrebbe offrire per questi approfonditi studi ?

Intanto, restando nel campo delle frazioni e sempre a titolo di breve "passatempo periodico", butto lì un nuovo quesito a più basso livello.

Si tratterebbe di individuare quante più frazioni diverse possibili, generatrici di parti periodiche, tali che la somma delle frazioni stesse, generi un numero con parte periodica costituita da quante più cifre possibili fra loro diverse, poste in ordine crescente e/o decrescente. Dunque al massimo un periodo di 20 cifre, ottenuto dalla somma di quante più frazioni possibili.

Es: 12,2434342...... +
.... 23,2143412......
.... 41,22122122......
.....76,6789876
_________________

\text {   }ciao Immagine ciao
E' la somma che fa il totale (Totò)

Quelo
Livello 7
Livello 7
Messaggi: 716
Iscritto il: ven giu 16, 2006 3:34 pm

Re: Il Passatempo Fratto

Messaggio da Quelo »

Dopo lunghi studi ed elaborazioni possiamo dare una risposta definitiva al quesito di Pasquale:

Il numero massimo di frazioni è $499$

Questa di seguito è solo una delle possibili soluzioni, che potrebbero essere migliaia

Codice: Seleziona tutto

ordine denominatori
303/2+508/3+793/4+942/5+497/6+426/7+623/8+832/9+113/10+458/11+959/12+632/13+241/14+914/15+929/16+834/17+785/18+786/19+809/20+554/21+61/22+796/23+845/24+686/25+471/26+670/27+543/28+822/29+857/30+534/31+599/32+298/33+655/34+382/35+139/36+972/37+723/38+896/39+751/40+384/41+569/42+922/43+409/44+448/45+393/46+926/47+737/48+968/49+981/50+446/51+509/52+718/53+545/54+438/55+813/56+404/57+349/58+664/59+811/60+727/62+628/63+838/65+257/66+417/68+436/69+307/70+683/72+815/74+844/75+251/76+514/77+293/78+977/80+849/82+283/84+486/85+681/86+284/87+711/88+937/90+894/91+867/92+386/93+549/94+648/95+227/96+993/98+956/99+699/100+743/102+557/104+758/105+859/106+359/108+487/110+146/111+985/112+917/114+326/115+659/116+746/117+603/118+178/119+869/120+542/123+695/124+571/126+548/129+707/130+263/132+192/133+64/135+83/136+125/138+233/140+764/141+274/143+395/144+366/145+778/147+211/148+223/150+107/152+452/153+361/154+934/155+661/156+226/159+327/160+536/161+463/164+724/165+103/168+277/170+122/171+757/172+965/174+654/175+669/176+788/177+587/180+191/182+843/184+614/185+505/186+262/187+927/188+332/189+729/190+662/195+865/196+895/198+881/200+346/203+905/204+768/205+730/207+883/208+994/209+989/210+515/212+606/215+553/216+794/217+173/220+766/221+983/222+639/224+674/225+763/228+181/230+830/231+383/232+419/234+804/235+365/236+401/238+269/240+422/245+871/246+898/247+81/248+887/252+776/253+892/255+673/258+810/259+891/260+866/261+955/264+324/265+753/266+877/270+305/272+790/273+356/275+415/276+932/279+183/280+625/282+716/285+267/286+454/287+791/288+653/290+317/294+712/295+541/296+808/297+904/299+433/300+960/301+573/304+601/306+853/308+179/310+755/312+842/315+821/318+500/319+289/322+398/323+706/325+875/328+856/329+889/330+466/333+907/336+367/340+320/341+337/342+321/344+802/345+593/348+331/350+412/351+375/352+721/354+668/357+499/360+199/364+717/368+244/369+519/370+906/371+127/372+311/374+597/376+762/377+431/378+229/380+566/385+640/387+839/390+484/391+685/392+79/396+394/399+657/400+576/403+919/406+334/407+677/408+823/410+990/413+841/414+921/416+353/418+709/420+268/423+469/424+556/425+502/429+941/430+739/432+807/434+142/435+964/437+373/440+890/441+745/442+619/444+913/450+818/451+886/455+605/456+676/459+507/460+535/462+747/464+568/465+149/468+879/470+939/472+402/473+872/475+249/476+722/477+521/480+586/481+634/483+243/490+911/492+582/493+801/494+584/495+863/496+491/504+987/506+991/510+358/513+485/516+642/517+831/518+581/520+565/522+772/525+618/527+563/528+281/530+512/531+313/532+852/533+734/539+631/540+447/544+635/546+489/550+710/551+787/552+976/555+971/558+428/559+453/560+166/561+679/564+97/570+537/572+933/574+158/575+789/580+256/583+908/585+71/588+732/589+169/590+995/592+529/594+678/595+121/598+479/600+219/602+411/608+652/609+474/611+671/612+194/615+501/616+641/620+974/621+847/624+626/627+978/629+389/630+769/636+726/637+567/638+363/644+704/645+157/646+970/649+443/650+362/651+961/656+579/658+953/660+604/663+996/665+343/666+388/667+923/672+478/675+973/680+271/682+979/684+827/688+578/689+421/690+988/693+427/696+824/697+771/700+803/702+562/703+622/705+947/708+948/713+167/714+218/715+733/720+214/725+915/728+610/731+202/735+201/736+511/738+73/740+982/741+101/742+67/744+213/748+335/752+137/754+457/756+488/759+239/760+278/765+162/767+109/770+89/774+316/775+292/777+338/779+997/780+873/782+854/783+633/784+577/792+692/795+797/798+750/799+151/800+916/805+523/806+131/812+687/814+829/816+878/817+998/819+381/820+862/825+237/826+761/828+206/833+591/836+698/837+607/840+193/846+405/848+339/850+128/851+242/855+449/858+909/860+958/861+647/864+163/868+701/870+503/874+951/880+197/882+445/884+596/885+355/888+524/893+938/897+134/899+719/900+250/901+749/902+254/903+291/910+643/912+379/918+309/920+467/924+302/925+999/928+547/930+314/931+876/935+461/936+613/940+498/943+691/944+526/945+347/946+781/950+773/952+439/954+694/957+963/962+835/966+482/969+538/975+967/980+617/984+397/986+949/992=3086

ordine numeratori
61/22+64/135+67/744+71/588+73/740+79/396+81/248+83/136+89/774+97/570+101/742+103/168+107/152+109/770+113/10+121/598+122/171+125/138+127/372+128/851+131/812+134/899+137/754+139/36+142/435+146/111+149/468+151/800+157/646+158/575+162/767+163/868+166/561+167/714+169/590+173/220+178/119+179/310+181/230+183/280+191/182+192/133+193/846+194/615+197/882+199/364+201/736+202/735+206/833+211/148+213/748+214/725+218/715+219/602+223/150+226/159+227/96+229/380+233/140+237/826+239/760+241/14+242/855+243/490+244/369+249/476+250/901+251/76+254/903+256/583+257/66+262/187+263/132+267/286+268/423+269/240+271/682+274/143+277/170+278/765+281/530+283/84+284/87+289/322+291/910+292/777+293/78+298/33+302/925+303/2+305/272+307/70+309/920+311/374+313/532+314/931+316/775+317/294+320/341+321/344+324/265+326/115+327/160+331/350+332/189+334/407+335/752+337/342+338/779+339/850+343/666+346/203+347/946+349/58+353/418+355/888+356/275+358/513+359/108+361/154+362/651+363/644+365/236+366/145+367/340+373/440+375/352+379/918+381/820+382/35+383/232+384/41+386/93+388/667+389/630+393/46+394/399+395/144+397/986+398/323+401/238+402/473+404/57+405/848+409/44+411/608+412/351+415/276+417/68+419/234+421/690+422/245+426/7+427/696+428/559+431/378+433/300+436/69+438/55+439/954+443/650+445/884+446/51+447/544+448/45+449/858+452/153+453/560+454/287+457/756+458/11+461/936+463/164+466/333+467/924+469/424+471/26+474/611+478/675+479/600+482/969+484/391+485/516+486/85+487/110+488/759+489/550+491/504+497/6+498/943+499/360+500/319+501/616+502/429+503/874+505/186+507/460+508/3+509/52+511/738+512/531+514/77+515/212+519/370+521/480+523/806+524/893+526/945+529/594+534/31+535/462+536/161+537/572+538/975+541/296+542/123+543/28+545/54+547/930+548/129+549/94+553/216+554/21+556/425+557/104+562/703+563/528+565/522+566/385+567/638+568/465+569/42+571/126+573/304+576/403+577/792+578/689+579/658+581/520+582/493+584/495+586/481+587/180+591/836+593/348+596/885+597/376+599/32+601/306+603/118+604/663+605/456+606/215+607/840+610/731+613/940+614/185+617/984+618/527+619/444+622/705+623/8+625/282+626/627+628/63+631/540+632/13+633/784+634/483+635/546+639/224+640/387+641/620+642/517+643/912+647/864+648/95+652/609+653/290+654/175+655/34+657/400+659/116+661/156+662/195+664/59+668/357+669/176+670/27+671/612+673/258+674/225+676/459+677/408+678/595+679/564+681/86+683/72+685/392+686/25+687/814+691/944+692/795+694/957+695/124+698/837+699/100+701/870+704/645+706/325+707/130+709/420+710/551+711/88+712/295+716/285+717/368+718/53+719/900+721/354+722/477+723/38+724/165+726/637+727/62+729/190+730/207+732/589+733/720+734/539+737/48+739/432+743/102+745/442+746/117+747/464+749/902+750/799+751/40+753/266+755/312+757/172+758/105+761/828+762/377+763/228+764/141+766/221+768/205+769/636+771/700+772/525+773/952+776/253+778/147+781/950+785/18+786/19+787/552+788/177+789/580+790/273+791/288+793/4+794/217+796/23+797/798+801/494+802/345+803/702+804/235+807/434+808/297+809/20+810/259+811/60+813/56+815/74+818/451+821/318+822/29+823/410+824/697+827/688+829/816+830/231+831/518+832/9+834/17+835/966+838/65+839/390+841/414+842/315+843/184+844/75+845/24+847/624+849/82+852/533+853/308+854/783+856/329+857/30+859/106+862/825+863/496+865/196+866/261+867/92+869/120+871/246+872/475+873/782+875/328+876/935+877/270+878/817+879/470+881/200+883/208+886/455+887/252+889/330+890/441+891/260+892/255+894/91+895/198+896/39+898/247+904/299+905/204+906/371+907/336+908/585+909/860+911/492+913/450+914/15+915/728+916/805+917/114+919/406+921/416+922/43+923/672+926/47+927/188+929/16+932/279+933/574+934/155+937/90+938/897+939/472+941/430+942/5+947/708+948/713+949/992+951/880+953/660+955/264+956/99+958/861+959/12+960/301+961/656+963/962+964/437+965/174+967/980+968/49+970/649+971/558+972/37+973/680+974/621+976/555+977/80+978/629+979/684+981/50+982/741+983/222+985/112+987/506+988/693+989/210+990/413+991/510+993/98+994/209+995/592+996/665+997/780+998/819+999/928=3086
Tonando alla mia congettura, al momento è stata dimostrata per 5 < n < 20 e n = 499
In onore di Pasquale, che ha proposto il questito, potremmo battezzarla:
Congettura di Pas-Quelo sulle frazioni irriducibili a somma intera

Dato un insieme di 2n numeri naturali consecutivi da 2 a 2n+1 con n > 5 è possibile, usando tutti i numeri, comporre n frazioni irriducibili la cui somma sia un numero intero
[Sergio] / $17$

Gianfranco
Supervisore del sito
Supervisore del sito
Messaggi: 1459
Iscritto il: ven mag 20, 2005 9:51 pm
Località: Sestri Levante
Contatta:

Re: Il Passatempo Fratto

Messaggio da Gianfranco »

Assolutamente fenomenale!
Complimenti a Pasquale per aver proposto il problema!
Complimenti a Quelo per i risultati raggiunti!
Sono pieno di ammirazione per voi, grazie!

E buon 25 aprile!
Pace e bene a tutti.
Gianfranco

Pasquale
Livello 12
Livello 12
Messaggi: 2831
Iscritto il: mer mag 25, 2005 2:14 am

Re: Il Passatempo Fratto

Messaggio da Pasquale »

Ringrazio per la citazione della mia "ispirazione", ma la corona d'alloro appartiene giustamente a Quelo ed all'impegno ed ingegno profusi nella ricerca di un risultato, inizialmente inimmaginabile. :shock:
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:D
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b5.jpg
b5.jpg (4.47 KiB) Visto 509 volte
_________________

\text {   }ciao Immagine ciao
E' la somma che fa il totale (Totò)

Quelo
Livello 7
Livello 7
Messaggi: 716
Iscritto il: ven giu 16, 2006 3:34 pm

Re: Il Passatempo Fratto

Messaggio da Quelo »

Grazie, in questo caso, come previsto dal regolamento, mi fregerò del numero 17


Aggiungo anche una risposta all'altro quesito, quello delle frazioni con parte periodica in sequenza:

Il risultato che vogliamo ottenere ha una forma come questa: $\displaystyle 0,\overline{98765432101234567890}$

Che equivale alla frazione $\displaystyle \frac{98765432101234567890}{99999999999999999999}$

Una soluzione immediata è $\displaystyle 98765432101234567890\frac{1}{10^{20}-1}$

Che conta appunto 98765432101234567890 frazioni tutte pari a $0,\overline{00000000000000000001}$

Se però vogliamo che le frazioni siano tutte diverse possiamo usare tutti i numeri da 1 a 14054567377 più 16890954137

$\displaystyle \frac{16890954137+\sum_{n=1}^{14054567377}n}{10^{20}-1}$

Che conta 14054567378 frazioni tutte diverse
[Sergio] / $17$

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