This Demonstration shows fractal patterns in the number of steps required to reach a fixed point or cyclic behavior in an application of Kaprekar's routine applied to the natural numbers for different bases.
To apply the Kaprekar routine for a positive integer
greater than 1, arrange the digits of
in base
in descending (
) and ascending (
) order, compute
(discarding any initial 0s). Repeating this procedure eventually leads to a cycle or a fixed point. The number of steps required,
, plotted as a colored square at the position
, produces beautiful fractal patterns. This Demonstration explores a subset of these patterns by allowing the setting of starting values and ranges for
and
.
Browse all topics
