Collatz conjecture kya hai?
Short mein jawab
Koi bhi poorn sankhya lo. Even ho to aadha karo; odd ho to 3 se guna karke 1 jodo; phir dohraao. 1937 mein Lothar Collatz ne anumaan lagaaya ki har shuruaati number aakhir mein 1 tak pahunchta hai. Ab tak aazmaaya har number pahunchta hai, par koi ise prove nahi kar paaya, isliye ye maths ka mashhoor unsolved sawaal hai.
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Niyam
Koi bhi poorn sankhya chuno aur ek hi niyam baar-baar lagaao:
- Agar number even hai, to uska aadha lo.
- Agar number odd hai, to 3 se guna karke 1 jodo.
1 pe ruk jao (1 ke baad niyam phir se 4, 2, 1 hi dega). Is tarah mili numbers ki list Collatz sequence kehlaati hai.
Book ki chaar sequences
| Shuruaat | Sequence | Steps |
|---|---|---|
| 12 | 12, 6, 3, 10, 5, 16, 8, 4, 2, 1 | 9 |
| 17 | 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 | 12 |
| 21 | 21, 64, 32, 16, 8, 4, 2, 1 | 7 |
| 22 | 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 | 15 |
Dhyaan do ki 22 se chalke 17 aata hai aur phir bilkul 17 waala raasta milta hai. Alag shuruaati numbers aksar ek hi raaste pe mil jaate hain, jaise chhoti dhaarayein ek nadi mein milti hain.
7 se ek sequence
Collatz sequence ke numbers 1 tak neeche aane se pehle upar-neeche hote hain, kuch-kuch baadal mein ole jaise.
7 se shuru karke sequence 52 tak chadhti hai aur 16 steps mein 1 pe pahunchti hai. Odd numbers (orange) upar uchhalte hain; even numbers (blue) aadhe hote hain.
2 ki powers aasaan hain
2 ki powers hain 2, 4, 8, 16, 32, 64, …, har ek 2 ko baar-baar khud se guna karke bana. 2 ki har power even hoti hai, aur uska aadha phir 2 ki power hota hai. Isliye sequence bas aadhi hoti jaati hai: 64, 32, 16, 8, 4, 2, 1. Isi wajah se is list ke har shuruaati number ke liye conjecture saaf taur pe sach hai.
Lambe safar
100 se shuru karke sequence hai 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1: ye 25 steps mein 1 pe pahunchti hai. Kuch chhote numbers hairaan karne waali der lete hain. Extra: 27 se shuru karke sequence 111 steps leti hai aur 1 tak aane se pehle 9232 tak chadhti hai.
Kya hum hamesha 1 tak pahunchte hain?
1937 mein German mathematician Lothar Collatz ne anumaan lagaaya ki jawab haan hai, chahe tum koi bhi poorn sankhya lo. Aisa anumaan, jispe bharosa ho par proof na ho, conjecture kehlaata hai. Extra: computers ne bahut bade-bade numbers jaanch liye hain, jo kisi notebook mein likhe bhi na jaa sakein, aur har ek 1 tak pahuncha hai.
Par udaharan jaanchna proof nahi hai. Proof ko dikhaana hoga ki har number, bina kisi apvaad ke, 1 tak pahunchta hai. Ye koi nahi kar paaya, aur na hi kisi ko aisa number mila jo fail ho. Isliye Collatz ka conjecture aaj bhi maths ke sabse mashhoor unsolved sawaalon mein se ek hai.
Poochna aasaan, suljhaana mushkil
Chapter ka summary ek khaas baat kehta hai: numbers ke kai sawaal poochne mein bahut aasaan par suljhaane mein bahut mushkil hote hain. Class 6 ka student Collatz ka niyam ek minute mein samajh leta hai, phir bhi duniya ke bade-bade mathematicians ise suljha nahi paaye. Apni sequences banao: kya sab 1 tak pahunchti hain?
Ek aur mashhoor pattern ke liye padho Kaprekar routine aur 6174. Textbook ke sawaalon ke liye dekho Chapter 3 ke NCERT solutions; poore chapter ke liye Chapter 3 ke notes.
Practice karo
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Collatz sequence mein 13 ke baad kya aata hai?
Jawab dekho
Jawab: (b) 40
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Har sequence 1 tak pahunchti hai, ye anumaan kisne lagaaya?
Jawab dekho
Jawab: (b) Lothar Collatz
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16 se shuru karke 1 tak kitne steps lagte hain?
Jawab dekho
Jawab: (b) 4
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6 se shuru hone waali Collatz sequence likho.
Jawab dekho
Jawab: 6, 3, 10, 5, 16, 8, 4, 2, 1 (8 steps).
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2 ki har power 1 tak kyun pahunchti hai?
Jawab dekho
Jawab: 2 ki power, jaise 32 = 2 × 2 × 2 × 2 × 2, even hoti hai, isliye aadhi karte hain. Nateeja phir 2 ki power hota hai, isliye aadha karte rehte hain: 32, 16, 8, 4, 2, 1. 1 se pehle koi odd number aata hi nahi.
Students ke sawal
Collatz conjecture kya hai?
Ye anumaan ki koi bhi poorn sankhya lekar, even ho to baar-baar aadha karne, aur odd ho to 3 se guna karke 1 jodne se, aakhir mein hamesha 1 aata hai. Lothar Collatz ne 1937 mein ye conjecture diya.
Kya Collatz conjecture sach hai?
Kisi ko nahi pata. Ab tak aazmaaya har shuruaati number 1 tak pahuncha hai, par koi proof nahi hai, isliye ye maths ke sabse mashhoor unsolved sawaalon mein se ek hai.
Kya 100 bhi 1 tak pahunchta hai?
Haan: 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Yaani 25 steps.
Koi sawal atak gaya?
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