**6174** is known as **Kaprekar’s constant**^{} after the Indian mathematician D. R. Kaprekar. This number is notable for the following rule:

1- Take any four-digit number, using at least two different digits. (Leading zeros are allowed.)

2- Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary.

3- Subtract the smaller number from the bigger number.

4- Go back to step 2 and repeat.

The above process, known as Kaprekar’s routine, will always reach its fixed point, 6174, in at most 7 iterations. Once 6174 is reached, the process will continue yielding 7641 – 1467 = 6174. For example, choose 3524:

- 5432 – 2345 = 3087
- 8730 – 0378 = 8352
- 8532 – 2358 = 6174
- 7641 – 1467 =
**6174**

The only four-digit numbers for which Kaprekar’s routine does not reach 6174 are repdigits such as 1111, which give the result 0000 after a single iteration. All other four-digit numbers eventually reach 6174 if leading zeros are used to keep the number of digits at 4.

Therefore the number 6174 is the only number unchanged by Kaprekar’s operation — our mysterious number is unique. The number 495 is the unique kernel for the operation on three-digit numbers, and all three-digit numbers reach 495 using the operation.

## 1 comment

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