Kaprekar's routine and the number 6174

Who was Kaprekar?

D. R. Kaprekar was a mathematics teacher in a government school in Devlali, Maharashtra. He loved playing with numbers and found many beautiful patterns in them. In 1949 he discovered a surprising property of 4-digit numbers, now known as Kaprekar’s routine.

The routine, step by step

1. Take a 4-digit number with at least two different digits.
2. Arrange its digits to make the largest number. Call it A.
3. Arrange the same digits to make the smallest number. Call it B.
4. Subtract: C = A − B.
5. Repeat the steps with C.

The book’s example starts with 6382: 8632 − 2368 = 6264, then 6642 − 2466 = 4176, then 7641 − 1467 = 6174. Here is the year 2026:

Why does it stop at 6174?

Once you reach 6174, the largest number from its digits is 7641 and the smallest is 1467, and 7641 − 1467 = 6174 again. So the routine keeps giving 6174 for ever. This number is called the Kaprekar constant. Every 4-digit number whose digits are not all the same reaches it, in at most 7 rounds.

Numbers with a zero

When a 0 is one of the digits, keep it as a leading digit in the smallest number, as Kaprekar did. From 2, 0, 2 and 6, the smallest arrangement is 0226, and from 5, 0, 8 and 5 it is 0558. In the same way, if a difference comes out with only three digits, such as 999, treat it as 0999 and continue: 9990 − 0999 = 8991.

This matters. If you insist on the smallest 4-digit arrangement (5058 instead of 0558), you usually still reach 6174, but the number of rounds can change, and numbers such as 1000 or 1011 break down: 1011 gives 1110 − 1011 = 99 and gets stuck. NCERT’s answer key uses the 4-digit way, so for 5683 it counts 8 rounds instead of 7.

How many rounds?

Start	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

Try your own year of birth: most numbers need between 3 and 7 rounds.

3-digit numbers: 495

The same steps with 3-digit numbers lead to 495, which then repeats: 954 − 459 = 495. For example, starting with 908: 980 − 089 = 891, 981 − 189 = 792, 972 − 279 = 693, 963 − 369 = 594, 954 − 459 = 495.

Playing with the largest and smallest

Pratibha makes the largest and smallest 4-digit numbers from 4, 7, 3 and 2: 7432 and 2347. Their difference is 5085 and their sum is 9779. Other digits give other results: from 9, 8, 1 and 0, the largest and smallest 4-digit numbers are 9810 and 1089, with difference 8721 (more than 5085), while 1, 2, 3 and 4 give 4321 + 1234 = 5555 (less than 9779).

For another number pattern, read palindromic numbers, or see a famous unsolved one in the Collatz conjecture. For the textbook’s questions, see the Chapter 3 NCERT solutions; for the whole chapter, see the Chapter 3 notes.