"6174" and "495"

"6174" and "495" are two interesting numbers in number theory.

"6174" is Kaprekar's constant and "495" is pentatope number.

There is a common property of these two numbers.

Case 1:

1. Take any four-digit number with at least two digits different. (Leading zeros are allowed.)
2. Arrange the digits in ascending and then in descending 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.

The above operation, known as Kaprekar's operation, will always reach 6174 in at most 7 steps and it stops there. Once 6174 is reached, the process will keep yielding 7641 – 1467 = 6174. For example, choose 3524:

5432 – 2345 = 3087
8730 – 0378 = 8352
8532 – 2358 = 6174


Case 2:

1. Start with a three-digit number with at least two digits different.
2. Arrange the digits in ascending and then in descending order to get two three-digit numbers, adding leading zeros if necessary.
3. Subtract the smaller number from the bigger number.
4. Go back to step 2.

Repeating this process will always reach 495 in a few steps. Once 495 is reached, the process stops because 954 – 459 = 495.

Example

For example, choose 598:

985 − 589 = 396
963 − 369 = 594
954 − 459 = 495

FYI: Dattaraya Ramchandra Kaprekar was an Indian mathematician who discovered many interesting properties in number theory. Having never received any formal postgraduate training, for his entire career (1930-1962) he was a schoolteacher at Nashik in Maharashtra, India.