What question did this study set out to answer?

The aim is to classify K-cycles generated by the Kaprekar process and understand their properties.

July 4, 2026

K-cycles

Key Points

The aim is to classify K-cycles generated by the Kaprekar process and understand their properties.
Utilized a method based on differences of digits to classify K-cycles.
Focused on cycles with even digit counts, n=2m.
Analyzed the period of these cycles, identifying specific lengths.
K-cycles were found to have periods of 1, 3, 5, or 7.
The majority of K-cycles consist of numbers whose digit sum is 9m.
A complete classification of these cycles was obtained.

Abstract

The number 6174 has the well-known property that it equals the difference between the numbers formed by rearranging its digits in descending and ascending order: 7641−1467=6174.The general process of finding this difference is called the Kaprekar process. Repeating this process indefinitely on n-digit numbers leads to one or more cycles, called Kaprekar cycles.An examination of Kaprekar cycles with even n=2m indicates that the great majority of such cycles consist of numbers with digit sum 9m. In this article, these cycles are called K-cycles, and a method using differences of digits is employed to obtain a complete classification.K-cycles have periods of 1, 3, 5 or 7. The number of such cycles is determined in the final section of this paper.

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