Definition:Kaprekar Mapping

Definition

The Kaprekar mapping is the arithmetic function $K: \Z_{>0} \to \Z_{>0}$ defined on the positive integers as follows:

Let $n \in \Z_{>0}$ be expressed in some number base $b$ (where $b$ is usually $10$).

Let $n'$ be the positive integer created by arranging the digits of $n$ into descending order of size.

Let $n$ be the positive integer created by arranging the digits of $n$ into ascending order of size.

Then:

$K \left({n}\right) = n' - n$

making sure to retain any leading zeroes to ensure that $K \left({n}\right)$ has the same number of digits as $n$.

Also defined as

Some sources do not retain the leading zeroes, and so, for example:

$K \left({1121}\right) = 2111 - 1112 = 999$

and so:

$K \left({999}\right) = 999 - 999 = 0$

instead of:

$K \left({1121}\right) = 2111 - 1112 = 0999$

$K \left({0999}\right) = 9990 - 0999 = 8991$

The mapping as initially specified does retain all leading zeroes.

Also see

• Definition:Kaprekar's Process

Source of Name

This entry was named for Dattathreya Ramchandra Kaprekar.

Sources

• 1949: D.R. Kaprekar: Another Solitaire Game (Scripta Math. Vol. 15: pp. 244 – 245)

• 1955: D.R. Kaprekar: An Interesting Property of the Number $6174$ (Scripta Math. Vol. 21: p. 304)

• Weisstein, Eric W. "Kaprekar Routine." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KaprekarRoutine.html