
samyuktha spodanam

ganithashaasthrathil, prashnatthinte samyojanathe prashnatthinte inputt, parimithikal, athirukal enniva engane badhikkunnu ennathinaalulla oru prashnatthinte sankeernathayude druthagathiyilulla valarchayaan kompinettoriyal spodanam. chila prashnangalude pariharikkaanaavaathathine nyaayeekarikkan chilappol kompinettoriyal spodanam upayogikkunnu. chila ganithashaasthra pravartthanangal, chila pasilukaludeyum geyimukaludeyum vishakalanam, ackerman functionaayi maathrkayaakkaan kazhiyunna chila paatholajikkal udaaharanagal enniva atharam prashnangalude udaaharanangalil ulppedunnu.
udaaharanagal.
laattin chathurangal.
oro variyilum oro nirayilum oro gadakavum krithyamaayi orikkal sambhavikkunna savisheshathayulla oru koottam gadakangalil ninnulla entrikal ulla oru sreniyaanu laattin square oof oordar. moonnaamathe kramatthilulla oru laattin squarinte oru udaaharanam nalkunnu,
oru laattin chathuratthinte oru saadhaarana udaaharanam poortthiyaaya sudoku pasil aayirikkum. oru laattin square oru samyojitha vasthuvaan (oru beejaganita vasthuvin vipareethamaayi) kaaranam entrikalude krameekaranam maathramaan pradhanam, allathe entrikal yadhaarthathil enthaanennalla. kramatthinte oru pravartthanamenna nilayil laattin chathurangalude ennam (entrikal varaykkunna settil ninnu swathanthramaayi) inipparayunna pattikayil chithreekarichirikkunnathu kompinettoriyal spodanathinte oru udaaharanam nalkunnu.
sudoku.
sudoku polulla griddle kalikkunna chila pasilukalilum oru kompinettoriyal spodanam sambhavikkam. oro gadakavum valippathinte upavibhagangalil ("boxukal" ennu vilikkunnu) krithyamaayi orikkal sambhavikkunna adhika savisheshathayulla oru tharam laattin squareanu sudoku. koottupalisha vardhikkumbol sambhavikkunna kompinettoriyal spodanam, thaazhepparayunna pattikayil chithreekarichirikkunnathu, nirmmikkanum vishakalanam cheyyaanum pariharikkanum kazhiyunna sudocasinte gunangalkk paridhikal srishtikkunnu.
kalikal.
kompinettoriyal sankeernnatha oru parihara paridhiyilekku nayikkunna oru geyiminte oru udaaharanam chess (64 chathurangalum 32 kashanangalumulla oru geyim) pariharikkunnathilaanu. chess oru pariharikkapetta kaliyalla. 2005ll aarr kashanangalo athil kuravo ulla ellah chess geyim endingukalum pariharichu, ithu oro sthaanathinteyum phalam kaanikkunnu. oru chess kashanam koodi cherthu tablebase poortthiyaakkaan patthu varsham koodi vendivannu, angane 7 kashanagalulla tablebase poortthiyaayi. oru chess endingle oru kashanam koodi cherkkunnath (angane 8-kashanam tablebase undaakkunnu) adhika kompinettoriyal sankeernnatha kaaranam asaadhyamaayi kanakkaakkappedunnu.
koodaathe, valia chess vakabhedangalilum ananthamaaya chessilum board valuppam vardhikkunnathinaal valia chess polulla geyimukal pariharikkanulla saadhyatha kooduthal budhimuttaayitheerunnu.
computing.
ashayavinimayathinum bahumukha idathinum samanamaaya reethiyil computing parithasthithikalil kompinettoriyal spodanam sambhavikkam. a" ennu vilikkunna oru booliyan variable mathramulla oru lalithamaaya samvidaanathe sangalppikkuka. eee samvidaanathinu rund saadhyamaaya avasthakalundu, "a" = sharee allengil "a" = thettu. mattoru booliyan variable "bii" cherkkunnath systethinu saadhyamaaya naalu avasthakal nalkum, "eiii" = sharee, "bii" = sharee, "eiii" = sharee, "bii" = thettu, eiii "= thettu, bii = sharee, eiii" = thettu, eiii "= thettu, bii" = thettu. n" booliyan ulla oru systethinu 2 "n" saadhyamaaya avasthakalundu, athesamayam "z" anuvadaneeyamaaya moolyangalulla (booliyante 2 (sharee, thettu) ennathinekkaal) "n" veriyabilukalude oru systethinu "z" n "saadhyamaaya avasthakalundaakum.
oro nodinum "z" kuttikal ulla "n" uyaramulla oru vrukshathinte ila nodukalaayi saadhyamaaya avasthakale kanakkakkaam. ila nodukalude eee drutha vardhana thirayal polulla mekhalakalil upayogapradamaakum, kaaranam valare dooram erangaathe thanne niravadhi phalangal axus cheyyaan kazhiyum. atharam gadanakal kaikaaryam cheyyumbol ithu oru thadassamakaam.
oru object ooriyantad bhaashayile oru klaas sreniye avarude maathaapithaakkalil ninnu paarambaryamaayi labhikkuna vividha tharatthilulla vasthukkalulla oru vrukshamaayi kanakkakkaam. thaarathamyathil ("a" <"b" pole) vyathyastha klaasukal samyojippikkendathunde, sambhavikkaavunna kombineshanukalude ennam pottitherikkum. oro tharatthilulla thaarathamyavum prograam cheyyendathundenkil, ithu taamasiyaathe cheria ennam classukalkku polum asaadyamaakum. onniladhikam anantharaavakaashangalkk ithu pariharikkan kazhiyum, upavarggangalkk onniladhikam maathaapithaakkal undakaan anuvadikkunnathiloode, athinaal nilavilulla oru sreniyeyum thadassappeduthaathe, oro kuttiyekkaalum kurachu rakshakarthaakkal klaasukal pariganikkaam.
vyathyastha pachakkarikal avayude poorvvika enangalil ninnu anantharaavakaashamaayi labhikkuna oru vargeekaranamaanu oru udaaharanam. oro pachakariyudeyum ruchiye mattullavayumayi thaarathamyam cheyyaan sramikkunnathu budhimuttaayitheerunnu, kaaranam shreniyil janithakashaasthrathekkuristhurulla vivarangal maathrame adangiyittulloo, ruchiyekkurichu paraamarsikkunnilla. ennirunnaalum, kaaratt/kaaratt, kaaratt/urulakkizhangu, kaaratt/mulappicha urulakkizhangu, urulakkizhangu/urulakkizhangu, urulakkizhangu/mulappicha mulappicha mulappicha/mulappicha ivayellaam avarude nilavile poorvvikarude adisthaanathilulla shreni nilanirthikkondu oru prathyeka classil ninnu anantharaavakaashamaayi gunikkaan kazhiyum, appol mukalil paranjavayellaam oru ruchikaramaaya/ruchikaramaaya thaarathamyathiloode maathrame nadappilakkaan kazhiyoo.
ganithashaasthram.
nammal "n" nte factorial edukkunnuvennu karuthuka
pinne 1! = 1,2! = 2,3! = 6,4! = 24. ennirunnaalum, thaarathamyena cheria "n" ennathinu polum nammal valare valia samkhyakalilekku vegathil athunnu. udaaharanathinu, 100! ±, mikka calculaterukalilum pradarshippikkaan kazhiyaathathra valuthum nireekshikkaavunna prapanchatthile adisthaana kanangalude ennathekkaal valare valuthumaaya oru samkhya.
ashayavinimayam.
administration, computing ennivayil, oru prakriyayil oorganaiseshanukal cherkkappedumbol ashayavinimaya lainukalil athivegam vardhikkunna vardhanavaanu kompinettoriyal spodanam. (eee valarchaye palappozhum "exponential" ennu visheshippikkaarundengil yadhaarthathil ithu polynomialaanu.)
oru prathyeka vishayathekkurichu rund samghadanakalkku ashayavinimayam nadathendathundengil, oru adhock reethiyil nerittu ashayavinimayam nadathunnathu eluppamaayirikkum-aashayavinimayathinte oru chaanal maathrame aavashyamulloo. ennirunnaalum, oru moonnaamathe samghadana cherkkukayaanenkil, moonnu vyathyastha chaanalukal aavasyamaanu. naalaamathe samghadanaye cherkkunnathinu aarr chaanalukal aavasyamaanu; anchu, patthu; aarr, pathinanju; muthalaayava.
pothuve, athu edukkum
soothravaakyam _ 2
n" oorganaiseshanukalkkulla ashayavinimaya lineukal, ithu "n" gadakangalude 2-samyojanangalude ennam maathramaan (binomial coffificientum kaanuka).
eee ashayavinimayam eppol ottathavana aavasyamallennu thirichariyukayum vivarangal kaimarunnathinulla pothuvaaya allengil idatharam maargam srishtikkukayum cheyyuka ennathaanu badal sameepanam. aadya jodic kooduthal jolie aavasyamaanu ennathaanu poraayma, kaaranam oronnum mattonnine manassilakkunnathinulla upariplavamaaya eluppa sameepanathekkaal athinte aantharika sameepanam pothuvilekku parivartthanam cheyyanam.

