Information funnels and multiscale gap-space dynamics in Kaprekar's routine

TL;DR

This paper analyzes the global information-theoretic structure of Kaprekar's routine across different digit lengths, revealing how entropy and dynamics evolve and how gap features influence the process.

Contribution

It provides an exhaustive, multiscale analysis of Kaprekar's routine, introducing entropy funnels and gap-space dynamics to understand its attractors and convergence behavior.

Findings

Average distances remain small despite large state spaces

Entropy decays rapidly before slow tail convergence

Gap features strongly constrain dynamics for D=3, less so for higher D

Abstract

Kaprekar's routine, i.e., sorting the digits of an integer in ascending and descending order and subtracting the two, defines a finite deterministic map on the state space of fixed-length digit strings. While its attractors (such as 495 for D = 3 and 6174 for D = 4) are classical, the global information-theoretic structure of the induced dynamics and its dependence on the digit length D have received little attention. Here an exhaustive analysis is carried out for D in {3,4,5,6}. For each D, all states are enumerated, their attractors and convergence distances are obtained, and the induced distribution over attractors across iterations is used to construct "entropy funnels". Despite the combinatorial growth of the state space, average distances remain small and entropy decays rapidly before entering a slow tail. Permutation symmetry is then exploited by grouping states into digit multisets and, in a further reduction, into low-dimensional digit-gap features. On this gap space, Kaprekar's routine induces a first-order Markov approximation whose transition structure, stationary distribution and drift fields are characterised, showing that simple gap features strongly constrain the dynamics for D=3 but lose predictive power as D increases.