K-cycles

The number 6174 has the well-known property that it equals the difference between the numbers formed by rearranging its digits in descending and ascending order: 7641−1467=6174.The general process of finding this difference is called the Kaprekar process. Repeating this process indefinitely on n-digit numbers leads to one or more cycles, called Kaprekar cycles.An examination of Kaprekar cycles with even n=2m indicates that the great majority of such cycles consist of numbers with digit sum 9m. In this article, these cycles are called K-cycles, and a method using differences of digits is employed to obtain a complete classification.K-cycles have periods of 1, 3, 5 or 7. The number of such cycles is determined in the final section of this paper.