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>> [Music kucheza]

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>> DOUG LLOYD: Pengine kufikiri kwamba
kanuni ni tu kutumika kwa kukamilisha kazi.

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Kuandika ni nje.

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Ni haina kitu.

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Hiyo ni pretty kiasi.

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>> Wewe kukusanya yake.

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Wewe kuendesha programu.

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Wewe ni vizuri kwenda.

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>> Lakini amini au, kama
wewe kanuni kwa muda mrefu,

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wewe kweli ili kuja kuona
kanuni kama kitu ambacho ni nzuri.

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Ni kutatua tatizo katika
njia ya kuvutia sana,

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au kuna tu kitu kweli
nadhifu kuhusu njia inaonekana.

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Unaweza kuwa akicheka
saa yangu, lakini ni kweli.

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Na kujirudia ni njia mojawapo
kwa namna ya kupata wazo hili

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ya nzuri, kifahari-kuangalia kanuni.

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Ni kutatua matatizo kwa njia ambazo
ni ya kuvutia, rahisi taswira,

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na kushangaza mfupi.

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>> Kazi njia kujirudia
ni, kazi ya kujirudia

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inaelezwa kama kazi kwamba wito
yenyewe kama sehemu ya utekelezaji wake.

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Hiyo inaweza kuonekana kidogo ajabu,
na tutaweza kuona kidogo

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kuhusu jinsi hii matendo katika wakati huu.

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Lakini tena, hizi
taratibu kujirudia ni

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kwenda kuwa hivyo kifahari
kwa sababu wao wanaenda

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kutatua tatizo hili bila
kuwa kazi hizi zote nyingine

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au hizi mizunguko ya muda mrefu.

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Utaona kwamba hizi kujirudia
taratibu ni kwenda kuangalia hivyo kwa muda mfupi.

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Na kwa kweli ni kwenda kufanya
kanuni yako kuangalia mengi nzuri zaidi.

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>> Mimi nitakupa mfano
ya hii kuona jinsi

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utaratibu kujirudia inaweza kuelezwa.

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Hivyo kama wewe ni ukoo na hii
kutoka hesabu darasani miaka mingi iliyopita,

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kuna kitu kinachoitwa
factorial kazi, ambayo ni kawaida

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ulionyehsa kama mshangao uhakika, ambayo
inaelezwa zaidi ya integers yote mazuri.

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Na kwa njia hiyo n
factorial ni mahesabu

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ni wewe kuzidisha yote ya
nambari chini ya

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au sawa na n together--
integers zote chini ya

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au sawa na n pamoja.

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>> Hivyo 5 factorial ni mara 5
Mara 4 mara 3 2 mara 1.

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Na 4 factorial ni mara 4
Mara 3 2 mara 1 na kadhalika.

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Unaweza kupata wazo.

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>> Kama programmers, hatufanyi
kutumia n, mshangao uhakika.

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Hivyo tutaweza kufafanua factorial
kazi kama ukweli wa n.

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Na tutaweza kutumia factorial kujenga
ufumbuzi kujirudia kwa tatizo hilo.

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Na nadhani unaweza kupata
kuwa ni mengi zaidi kuibua

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rufaa ya iterative
toleo la hii, ambayo

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tutaweza pia tuangalie katika wakati huu.

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>> Hivyo hapa ni michache ya
facts-- pun intended--

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kuhusu factorial--
factorial kazi.

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Factorial ya 1, kama nilivyosema, ni 1.

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Factorial ya 2 ni 2 mara 1.

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Factorial ya 3 ni 3
mara 2 mara 1, na kadhalika.

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Kuongelea 4 na 5 tayari.

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>> Lakini kuangalia hii, ni hii si kweli?

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Si factorial ya 2 tu
2 mara factorial ya 1?

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I mean, factorial ya 1 ni 1.

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Hivyo kwa nini hatuwezi tu kusema kwamba,
tangu factorial ya 2 ni 2 mara 1,

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ni kweli tu mara 2
factorial ya 1?

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>> Na kisha kupanua wazo hilo,
si factorial ya 3

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tu mara 3 factorial ya 2?

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Na factorial ya 4 ni mara 4
factorial ya 3, na kadhalika?

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Kwa kweli, factorial
ya idadi yoyote unaweza tu

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kuwa walionyesha kama sisi aina
ya kubeba huu milele nje.

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Tunaweza aina ya kujumlisha
Tatizo factorial

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kama ni n mara
factorial ya n bala 1.

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Ni mara n bidhaa za
namba zote chini ya mimi.

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>> Wazo hili, hii
generalization ya tatizo,

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inaruhusu sisi recursively
kufafanua kazi factorial.

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Wakati kufafanua kazi
recursively, kuna

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mambo mawili ambayo yanahitaji kuwa sehemu yake.

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Unahitaji kuwa na kitu kinachoitwa
kesi ya msingi, ambayo, wakati wewe kusababisha yake,

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kuacha mchakato kujirudia.

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>> Vinginevyo, kazi kwamba wito
itself-- kama unaweza imagine--

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inaweza kuendelea milele.

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Kazi wito kazi
wito wito kazi

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kazi wito kazi.

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Kama huna njia
na kuacha ni, mpango wako

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Itakuwa ufanisi kukwama
katika kitanzi usio.

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Itakuwa ajali hatimaye,
kwa sababu kutakuwa na kukimbia nje ya kumbukumbu.

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Lakini hiyo ni kando ya uhakika.

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>> Tunahitaji kuwa na njia nyingine kuacha
mambo badala ya mpango wetu mshindo,

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kwa sababu mpango huo shambulio ni
pengine si nzuri au kifahari.

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Na hivyo sisi wito huu kesi ya msingi.

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Hii ni ufumbuzi rahisi
tatizo ambalo haachi

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mchakato kujirudia kutoka kutokea.

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Hivyo hiyo ni sehemu moja ya
kazi ya kujirudia.

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>> Sehemu ya pili ni kesi ya kujirudia.

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Na hii ni mahali ambapo kujirudia
kwa kweli kuchukua nafasi.

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Hii ni pale ambapo
kazi itakuwa kujiita.

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>> Itakuwa si kujiita katika hasa
njia ile ile ilikuwa inaitwa.

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Ni utakuwa tofauti kidogo
kwamba inafanya tatizo ni

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kujaribu kutatua teeny kidogo kidogo.

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Lakini kwa ujumla hupita mume
ya kutatua wingi wa ufumbuzi

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mwito tofauti chini ya mstari.

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>> Ni yupi kati ya hawa inaonekana
kama kesi ya msingi hapa?

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Ambayo moja ya inaonekana hawa kama
rahisi ufumbuzi wa tatizo?

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Tuna kundi la factorials,
na tunaweza kuendelea

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kwenda on-- 6, 7, 8, 9, 10, na kadhalika.

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>> Lakini moja ya inaonekana hawa kama
kesi nzuri ya kuwa kesi ya msingi.

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Ni ufumbuzi rahisi sana.

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Hatuna kufanya kitu chochote maalum.

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>> Factorial ya 1 ni 1 tu.

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Hatuna kufanya lolote
kuzidisha wakati wote.

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Inaonekana kama kama tunakwenda
kujaribu na kutatua tatizo hili,

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na sisi haja ya kuacha
kujirudia mahali fulani,

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sisi pengine wanataka kuacha
ni wakati sisi kupata 1.

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Hatutaki kuacha kabla ya hapo.

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>> Hivyo kama sisi ni kufafanua
kazi yetu factorial,

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hapa ni mifupa kwa
jinsi sisi anaweza kufanya hivyo.

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Tunahitaji kuziba katika wale wawili things--
kesi ya msingi na kesi ya kujirudia.

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Nini kesi ya msingi?

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Kama n ni sawa na 1, kurudi 1-- hiyo ni
Tatizo kweli rahisi kutatua.

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>> Factorial ya 1 ni 1.

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Siyo 1 mara chochote.

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Ni tu 1.

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Ni ukweli rahisi sana.

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Na hivyo ambayo inaweza kuwa msingi wetu kesi.

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Kama sisi kupata kupita 1 katika hii
kazi, tutaweza tu kurudi 1.

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>> Nini kujirudia
kesi pengine kuangalia kama?

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Kwa kila idadi vingine
badala 1, nini mfano?

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Naam, kama sisi ni kuchukua
factorial ya n,

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ni mara n factorial ya n bala 1.

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>> Kama sisi ni kuchukua factorial ya 3,
ni mara 3 factorial ya 3 bala 1,

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au 2.

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Na hivyo kama sisi siyo
kuangalia saa 1, vinginevyo

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Mara kurudi n
factorial ya n bala 1.

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Ni pretty moja kwa moja.

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>> Na kwa ajili ya kuwa na kidogo
safi na zaidi ya kifahari kificho,

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kujua kwamba kama tuna single-mstari loops
au moja-line matawi masharti,

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tunaweza kujikwamua yote ya
braces curly inayowazunguka.

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Ili tuweze kuimarisha huu kwa hili.

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Hii ina sawa
utendaji kama hii.

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>> Mimi tu kuchukua mbali curly
inakabiliwa na, kwa sababu kuna mstari mmoja tu

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ndani ya matawi hayo masharti.

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Basi hizi kuishi identically.

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Kama n ni sawa na 1, kurudi 1.

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Vinginevyo kurudi mara n
factorial ya n bala 1.

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Hivyo sisi ni kufanya tatizo ndogo.

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Kama n kuanza nje kama 5, tunakwenda
kurudi mara 5 factorial ya 4.

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Na tutaweza kuona katika dakika tunapozungumza
kuhusu wito stack-- katika video nyingine

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ambapo sisi majadiliano juu
piga stack-- tutaweza kujifunza

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kuhusu nini hasa mchakato huu kazi.

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>> Lakini wakati factorial ya 5 inasema
kurudi mara 5 factorial ya 4, na 4

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ni kwenda kusema, OK, vizuri, kurudi
Mara 4 factorial ya 3.

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Na kama unaweza kuona, tuko
aina ya inakaribia 1.

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Sisi ni kupata karibu na
karibu na kwamba kesi ya msingi.

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>> Na mara moja sisi hit kesi ya msingi,
yote ya kazi ya awali

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jibu walikuwa wanatafuta.

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Factorial ya 2 alikuwa akisema kurudi
2 mara factorial ya 1.

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Naam, factorial ya 1 anarudi 1.

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Hivyo wito kwa factorial
ya 2 wanaweza kurudi mara 2 kwa 1,

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na kutoa kwamba nyuma ya factorial ya
3, ambayo ni kusubiri kwa kuwa matokeo.

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>> Na kisha unaweza mahesabu
matokeo yake, mara 3 2 ni 6,

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na kuwapa nyuma kwa factorial ya 4.

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Na tena, tuna
video juu ya wito mkusanyiko

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ambapo hii ni mfano kidogo
zaidi ya nini mimi kusema hivi sasa.

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Lakini hii ni yake.

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Hii peke yake ni ufumbuzi wa
kuhesabu factorial ya idadi.

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>> Ni mistari minne tu ya kificho.

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Hiyo ni pretty baridi, sawa?

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Ni aina ya sexy.

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>> Hivyo kwa ujumla, lakini si
siku zote, kazi ya kujirudia

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anaweza kuchukua nafasi kitanzi katika
yasiyo ya kujirudia kazi.

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Hivyo hapa, bega kwa bega, ni iterative
toleo la kazi factorial.

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Wote hawa mahesabu
hasa kitu kimoja.

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>> Wote wawili mahesabu factorial ya n.

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Toleo la upande wa kushoto
anatumia kujirudia kwa kufanya hivyo.

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Toleo la juu ya haki
anatumia iteration ya kufanya hivyo.

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>> Na taarifa, tuna kutangaza
kutofautiana integer bidhaa.

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Na kisha sisi kitanzi.

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Hivyo muda mrefu kama n ni kubwa kuliko 0, sisi
kuweka kuzidisha kuwa bidhaa hiyo na N

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na decrementing n mpaka
sisi mahesabu ya bidhaa.

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Hivyo kazi hizi mbili, tena,
kufanya hasa kitu kimoja.

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Lakini hawana kufanya hivyo katika
hasa kwa njia hiyo.

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>> Sasa, inawezekana kwa
kuwa na msingi zaidi ya moja

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kesi au zaidi ya moja
kujirudia kesi, kulingana

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juu ya nini kazi yako ni kujaribu kufanya.

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Wewe ni si lazima tu mdogo kwa
kesi moja ya msingi au kujirudia moja

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kesi.

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Hivyo mfano wa kitu
na kesi nyingi msingi

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Haya inaweza kuwa
Fibonacci idadi mlolongo.

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>> Unaweza kukumbuka kutoka
msingi siku shule

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kwamba mlolongo Fibonacci inaelezwa
kama hii kitu cha kwanza ni 0.

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Kipengele cha pili ni 1.

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Wote wale ni tu kwa ufafanuzi.

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>> Kisha kila kipengele nyingine inaelezwa
kama jumla ya n bala 1 na n bala 2.

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Hivyo kipengele cha tatu
itakuwa 0 pamoja na 1 ni 1.

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Na kisha kipengele cha nne
itakuwa kipengele cha pili, 1,

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pamoja na kipengele cha tatu, 1.

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Na kwamba itakuwa 2.

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Na kadhalika na kadhalika.

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>> Hivyo katika kesi hii, tuna kesi mbili msingi.

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Kama n ni sawa na 1, kurudi 0.

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Kama n ni sawa na 2, kurudi 1.

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Vinginevyo, kurudi Fibonacci wa n
bala 1 pamoja Fibonacci wa n bala 2.

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>> Hivyo hiyo ni kesi nyingi msingi.

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Je kuhusu kesi nyingi kujirudia?

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Naam, kuna kitu
aitwaye dhana Collatz.

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Mimi si kwenda kusema,
unajua nini, yaani,

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kwa sababu ni kweli mwisho wetu
Tatizo kwa video hii tu.

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Na ni zoezi wetu
kufanya kazi ya pamoja.

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>> Hivyo hapa ni nini
Collatz dhana is--

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inatumika kwa kila sifuri.

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Na speculates kwamba ni
kila mara inawezekana kupata nyuma

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kwa 1 kama wewe kufuata hatua hizi.

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Kama n ni 1, kuacha.

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Sisi tumepewa nyuma 1 ikiwa n ni 1.

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>> Vinginevyo, kwenda kwa njia hii
mchakato tena juu ya n kugawanywa na 2.

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Na kuona kama unaweza kupata nyuma 1.

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Vinginevyo, kama n ni isiyo ya kawaida, kupitia
mchakato huu tena juu ya 3n pamoja na 1,

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au mara 3 n pamoja na 1.

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Hivyo hapa tuna moja kesi ya msingi.

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Kama n ni sawa na 1, kuacha.

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Sisi siyo kufanya kujirudia tena.

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>> Lakini tuna kesi mbili kujirudia.

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Kama n ni hata, tunafanya kujirudia moja
kesi, wito n kugawanywa na 2.

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Kama n ni isiyo ya kawaida, sisi kufanya mbalimbali
kujirudia kesi juu mara 3 n pamoja na 1.

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>> Na hivyo lengo kwa video hii ni
kuchukua pili, pause video,

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na kujaribu na kuandika hii
kazi ya kujirudia Collatz

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ambapo kupita katika thamani n, na
ni mahesabu ya hatua wangapi ni

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inachukua kupata 1 kama wewe kuanza kutoka n
na wewe kufuata hatua hizo hadi hapo juu.

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Kama n ni 1, inachukua hatua 0.

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Vinginevyo, ni kwenda
kuchukua hatua moja pamoja na hata hivyo

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hatua nyingi inachukua juu ya n ama
kugawanywa na 2 kama n ni hata, au 3n pamoja na 1

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kama n ni isiyo ya kawaida.

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>> Sasa, nimekuwa kuweka juu ya screen hapa
michache ya mtihani mambo kwa ajili yenu,

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michache ya kesi ya vipimo kwa ajili yenu, ili kuona
nini hawa mbalimbali idadi Collatz ni,

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na pia mfano
ya hatua ambazo

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haja ya kuwa wamekwenda kupitia hivyo unaweza
aina ya kuona mchakato huu katika hatua.

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Hivyo kama n ni sawa na
1, Collatz ya n ni 0.

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Huna kufanya
chochote kupata nyuma 1.

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Wewe ni tayari pale.

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>> Kama n ni 2, inachukua
hatua moja ya kupata 1.

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Unaweza kuanza kwa 2.

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Naam, 2 si sawa na 1.

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Hivyo ni kwenda kuwa hatua moja
pamoja na hata hivyo wengi hatua hiyo

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inachukua n kugawanywa na 2.

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>> 2 kugawanywa na 2 ni 1.

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Hivyo inachukua hatua moja pamoja na hata hivyo
hatua nyingi inachukua kwa 1.

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1 inachukua hatua sifuri.

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>> Kwa 3, kama unaweza kuona, kuna
hatua chache kabisa kushiriki.

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Wewe kwenda kutoka 3.

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Na kisha kwenda kwa
10, 5, 16, 8, 4, 2, 1.

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Inachukua hatua saba kupata nyuma 1.

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Na kama unaweza kuona, kuna
wanandoa wengine kesi mtihani hapa

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mtihani nje mpango wako.

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Hivyo tena, pause video.

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Na nitakwenda kuruka nyuma sasa kwa
nini mchakato halisi ni hapa,

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nini dhana hii ni.

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>> Kuona kama unaweza kufikiri
jinsi ya kufafanua Collatz ya n

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hivyo kwamba mahesabu ya jinsi wengi
hatua inachukua kupata kwa 1.

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Hivyo hopefully, una paused video
na wewe si tu kusubiri kwa ajili yangu

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kukupa jibu hapa.

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Lakini kama wewe ni, vizuri,
hapa ni jibu hata hivyo.

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>> Hivyo hapa ni ufafanuzi iwezekanavyo
ya kazi Collatz.

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Msingi wetu case-- ikiwa n ni
sawa na 1, sisi kurudi 0.

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Haina kuchukua yoyote
hatua ya kupata nyuma 1.

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>> Vinginevyo, tuna mbili kujirudia cases--
moja kwa idadi hata na kimoja cha isiyo ya kawaida.

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Njia ya mimi mtihani kwa idadi hata
ni kuangalia kama n Mod 2 sawa na 0.

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Hii ni kimsingi, tena,
kuuliza swali,

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kama unakumbuka kile mod is-- kama mimi
mgawanyiko n na 2 ni hakuna salio?

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Hiyo itakuwa hata idadi.

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>> Na hivyo kama n Mod 2 sawa na 0 ni
kupima ni hii hata idadi.

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Kama ni hivyo, nataka kurudi 1,
kwa sababu hii ni dhahiri

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kuchukua hatua moja pamoja na Collatz ya
chochote idadi ni nusu ya mimi.

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Vinginevyo, nataka kurudi 1
pamoja na Collatz ya mara 3 n pamoja na 1.

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Hiyo ilikuwa ni mwingine
hatua kujirudia kwamba sisi

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inaweza kuchukua kwa mahesabu
Collatz-- idadi ya hatua

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inachukua kupata nyuma
kwa 1 kutokana na idadi.

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Hivyo hopefully, mfano huu
aliwapa kidogo

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ya ladha ya taratibu kujirudia.

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Hopefully, unafikiri kificho ni
kidogo nzuri zaidi kama zinatekelezwa

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katika kifahari, kujirudia njia.

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Lakini hata kama si, kujirudia ni
kweli zana yenye nguvu hata hivyo.

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Na hivyo ni dhahiri kitu
kupata kichwa yako karibu,

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kwa sababu wewe utakuwa na uwezo wa kujenga
mipango pretty baridi kwa kutumia recursion

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kwamba ili vinginevyo kuwa ngumu kuandika
kama unatumia mizunguko na iteration.

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Mimi nina Doug Lloyd.

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Hii ni CS50.

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