Number patterns aur Collatz Conjecture
Class 6 Maths · Chapter 3: Number Play · NCERT Ganita Prakash section 3.9–3.10
Seekho
Jab numbers pattern mein hon, ek-ek jodne se pehle jode dhoondo: 20 + 20 + 45 + 45 + 30 + 30 = 2 × (20 + 45 + 30) = 190.
2 × 95 = 190
Collatz rule: number even ho to aadha karo; odd ho to 3 se guna karke 1 jodo. Aise karte raho.
Ganitagya Lothar Collatz ne 1937 mein poochha: kya ye rule hamesha 1 tak pahunchta hai, chahe koi bhi number lo? Computers ne bahut bade numbers jaanche aur sab 1 tak pahunche, par koi ise prove nahi kar paaya. Ye ab bhi ek unsolved problem hai!
Apne pasandeeda numbers try karo. Kya har ek 1 tak pahunchta hai?
Yaad rakho
- Collatz: even → aadha; odd → 3 × n + 1.
- Conjecture ek andaaza hai jise abhi tak koi sahi ya galat sabit nahi kar paaya.
Quick check
Collatz rule: 7 ke baad kya aata hai?
- 14
- 22
- 21
- 3
Answer
22. 7 odd hai: 3 × 7 + 1 = 22.
Socho
2 ki powers (1, 2, 4, 8, 16 …) ke liye Collatz rule hamesha 1 tak kyon pahunchta hai?
Ek answer
Ye sab even hain, aur 2 ki power ka aadha pichhli power hai: 16 → 8 → 4 → 2 → 1.
Practice
Teen levels (easy, medium, hard) ke naye sawaalon se practice karo, kuch hands-on, hints aur hal ke saath. Topic poora karne ke liye pichhle 5 mein se 4 sahi karo.