Algebraic Kaprekar routine architecture II

TL;DR

This paper explores algebraic relations in the Kaprekar routine, identifying group structures and analyzing cycles and transformation trees for numbers with up to five digits.

Contribution

It introduces algebraic relations and group structures related to the Kaprekar routine, extending analysis to multi-digit numbers and their cycles.

Findings

Identifies algebraic relations for Kaprekar transformations

Establishes group structures like the Klein group in this context

Analyzes cycles and transformation trees for 2- to 5-digit numbers

Abstract

In general terms, we establish algebraic relations that numbers must satisfy in order for their images to match after one or several transformations. Some groups associated with these relationships are identified, such as the Klein group. Such equivalences are applied to numbers of 2, 3, 4 or 5 digits. The relationship between cycles and the transformation trees structure are analyzed.