A routine discovered in 1949 by D. R. Kaprekar for 4-digit numbers, but which can be generalized to -digit numbers. To apply the Kaprekar routine to a number , arrange the digits in descending () and ascending () order. Now compute and iterate. The algorithm reaches 0 (a degenerate case), a constant, or a cycle, depending on the number of digits in and the value of .
For a 3-digit number in base 10, the Kaprekar routine reaches the number 495 in at most six iterations. In base , there is a unique number to which converges in at most iterations Iff is Even. For any 4-digit number in base-10, the routine terminates on the number 6174 after seven or fewer steps (where it enters the 1-cycle ).
See also 196-Algorithm, Kaprekar Number, RATS Sequence
References
Eldridge, K. E. and Sagong, S. ``The Determination of Kaprekar Convergence and Loop Convergence of All 3-Digit Numbers.''
Amer. Math. Monthly 95, 105-112, 1988.
Kaprekar, D. R. ``An Interesting Property of the Number 6174.'' Scripta Math. 15, 244-245, 1955.
Trigg, C. W. ``All Three-Digit Integers Lead to...'' The Math. Teacher, 67, 41-45, 1974.
Young, A. L. ``A Variation on the 2-digit Kaprekar Routine.'' Fibonacci Quart. 31, 138-145, 1993.
© 1996-9 Eric W. Weisstein