Maximum distances in the four-digit Kaprekar process

2020/10/22 by Devlin, Pat, Zeng, Tony · 2 citations
#00A08 #05A15 #11B37 #11B75 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2010.11756

Abstract

For natural numbers x and b, the classical Kaprekar function is defined as K_b (x) = D-A, where D is the rearrangement of the base-b digits of x in descending order and A is ascending. The bases b for which K_b has a 4-digit non-zero fixed point were classified by Hasse and Prichett, and for each base this fixed point is known to be unique. In this article, we determine the maximum number of iterations required to reach this fixed point among all four-digit base-b inputs, thus answering a question of Yamagami. Moreover, we also explore---as a function of b---the fraction of four-digit inputs for which iterating K_b converges to this fixed point.

Cited by

• Information funnels and multiscale gap-space dynamics in Kaprekar's routine
• Four-digit Kaprekar dynamics in odd bases

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