Definition:Kaprekar's Process

Definition

Kaprekar's process is the repeated application of the Kaprekar mapping to a given positive integer.

Examples

Kaprekar's Process on $4527$

From $4527$:

\(\ds 7542 - 2457\)

\(=\)

\(\ds 5085\)

\(\ds 8550 - 0558\)

\(=\)

\(\ds 7992\)

\(\ds 9972 - 2799\)

\(=\)

\(\ds 7173\)

\(\ds 7731 - 1377\)

\(=\)

\(\ds 6354\)

\(\ds 6543 - 3456\)

\(=\)

\(\ds 3087\)

\(\ds 8730 - 0378\)

\(=\)

\(\ds 8352\)

\(\ds 8532 - 2358\)

\(=\)

\(\ds 6174\)

\(\ds 7641 - 1467\)

\(=\)

\(\ds 6174\)

$\blacksquare$

Also known as

Kaprekar's process is also known as the Kaprekar routine or the Kaprekar sequence.

Sometimes rendered as the Kaprekar process.

Also defined as

Some sources define the Kaprekar mapping so as not to retain the leading zeroes, and so, for example:

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

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

instead of:

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

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

The process as initially specified does retain all leading zeroes.

Also see

• Definition:Modified Kaprekar Process
• Results about Kaprekar's process can be found here.

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)

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $495$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $495$

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