Kaprekar routine aur 6174

Kaprekar kaun the?

D. R. Kaprekar Devlali, Maharashtra ke ek sarkaari school mein maths teacher the. Unhe numbers se khelna bahut pasand tha aur unhone numbers mein kai sundar patterns khoje. 1949 mein unhone 4-digit numbers ki ek hairaan karne waali khoobi khoji, jise ab Kaprekar routine kehte hain.

Routine, step by step

1. Kam se kam do alag digits waala ek 4-digit number lo.
2. Uske digits se sabse bada number banao. Ise A kaho.
3. Unhi digits se sabse chhota number banao. Ise B kaho.
4. Ghataao: C = A − B.
5. C ke saath steps dohraao.

Book ka udaharan 6382 se shuru hota hai: 8632 − 2368 = 6264, phir 6642 − 2466 = 4176, phir 7641 − 1467 = 6174. Ye raha saal 2026:

6174 pe kyun rukta hai?

6174 pe pahunchne ke baad uske digits se sabse bada number 7641 aur sabse chhota 1467 banta hai, aur 7641 − 1467 = 6174 phir se. Isliye routine hamesha 6174 hi deta rehta hai. Is number ko Kaprekar constant kehte hain. Jis 4-digit number ke saare digits same nahi, wo zyada se zyada 7 rounds mein yahan pahunchta hai.

Zero waale numbers

Jab digits mein 0 ho, to sabse chhote number mein use aage rakho, jaisa Kaprekar karte the. 2, 0, 2 aur 6 se sabse chhota 0226 hai, aur 5, 0, 8 aur 5 se 0558. Isi tarah agar fark sirf teen digits ka aaye, jaise 999, to use 0999 maano aur aage badho: 9990 − 0999 = 8991.

Ye zaroori hai. Agar tum sabse chhota 4-digit number hi lo (0558 ki jagah 5058), to zyaadatar phir bhi 6174 milta hai, par rounds badal sakte hain, aur 1000 ya 1011 jaise numbers atak jaate hain: 1011 se 1110 − 1011 = 99 aata hai aur routine ruk jaata hai. NCERT ki answer key 4-digit waala tareeka use karti hai, isliye 5683 ke liye 7 ki jagah 8 rounds ginti hai.

Kitne rounds?

Shuruaat	Rounds	Chain
6382	3	6264, 4176, 6174
2026	6	5994, 5355, 1998, 8082, 8532, 6174
5683	7	5085, 7992, 7173, 6354, 3087, 8352, 6174
2014	7	4086, 8172, 7443, 3996, 6264, 4176, 6174

Apna janam-saal aazmaao: zyaadatar numbers ko 3 se 7 rounds lagte hain.

3-digit numbers: 495

Yahi steps 3-digit numbers ke saath 495 tak le jaate hain, jo phir dohraata hai: 954 − 459 = 495. Jaise 908 se: 980 − 089 = 891, 981 − 189 = 792, 972 − 279 = 693, 963 − 369 = 594, 954 − 459 = 495.

Sabse bade aur sabse chhote se khel

Pratibha 4, 7, 3 aur 2 se sabse bada aur sabse chhota 4-digit number banaati hai: 7432 aur 2347. Inka fark 5085 aur jod 9779 hai. Doosre digits doosre nateeje dete hain: 9, 8, 1 aur 0 se sabse bada aur sabse chhota 4-digit number 9810 aur 1089 hain, fark 8721 (5085 se zyada), jabki 1, 2, 3 aur 4 se 4321 + 1234 = 5555 (9779 se kam).

Ek aur number pattern ke liye padho palindrome numbers kya hain, ya ek mashhoor unsolved pattern Collatz conjecture kya hai mein dekho. Textbook ke sawaalon ke liye dekho Chapter 3 ke NCERT solutions; poore chapter ke liye Chapter 3 ke notes.