Generalized Kaprekar Routine

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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 .


Contributed by: Noel Patson (March 2011)
Open content licensed under CC BY-NC-SA




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