Generalization and Alternatives of Kaprekar's Routine

TL;DR

This paper generalizes Kaprekar's Routine to broader applications, introduces alternative routines, and discusses open questions about iteration lengths and cycle properties.

Contribution

It presents a novel generalization of Kaprekar's Routine, offering new variants and exploring their properties and open problems.

Findings

Proposed new generalized routines based on Kaprekar's method

Identified specific examples of alternative routines

Raised open questions about iteration lengths and cycle characteristics

Abstract

We extend Kaprekar's Routine for a large class of applications. We also give particular examples of this generalization as alternatives to Kaprekar's Routine and Number. Some open questions about the length of the iterations until reaching either zero or a constant or a cycle, and about the length of the cycles are asked at the end.