Maximum distances in the four-digit Kaprekar process

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.