Pushforward measure

Template:Infobox number 495 is the integer after 494 and before 496. It is a pentatope number (and so a binomial coefficient ${\tbinom {12}{4}}$).

Kaprekar transformation

The Kaprekar transformation is defined as follows for three-digit numbers:

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 589:

985 − 589 = 396

963 − 369 = 594

954 − 459 = 495

The only three-digit numbers for which this function does not work are repdigits such as 111, which give the answer 0 after a single iteration. All other three-digits numbers work if leading zeros are used to keep the number of digits at 3:

211 – 112 = 099

990 – 099 = 891 (rather than 99 - 99 = 0)

981 – 189 = 792

972 – 279 = 693

963 – 369 = 594

954 − 459 = 495

The number 6174 has the same property for the four-digit numbers.

See also

• Collatz conjecture — sequence of unarranged-digit numbers always ends with the number 1.

References

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