Kaprekar Routine

A routine discovered in 1949 by D. R. Kaprekar for 4-digit numbers, but which can be generalized to -digit numbers. Toapply 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 numberof 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 , thereis a unique number to which converges in at most iterations Iff is Even. Forany 4-digit number in base-10, the routine terminates on the number 6174 after seven or fewer steps (where it enters the1-cycle ).

2. 0, 0, 9, 21, (45), (49), ...,

3. 0, 0, (32, 52), 184, (320, 580, 484), ...,

4. 0, 30, 201, (126, 138), (570, 765), (2550), (3369), (3873), ...,

5. 8, (48, 72), 392, (1992, 2616, 2856, 2232), (7488, 10712, 9992, 13736, 11432), ...,

6. 0, 105, (430, 890, 920, 675, 860, 705), 5600, (4305, 5180), (27195), (33860), (42925), (16840, 42745, 35510), ...,

7. 0, (144, 192), (1068, 1752, 1836), (9936, 15072, 13680, 13008, 10608), (55500, 89112, 91800, 72012, 91212, 77388), ...,

8. 21, 252, (1589, 3178, 2723), (1022, 3122, 3290, 2044, 2212), (17892, 20475), (21483, 25578, 26586, 21987), ...,

9. (16, 48), (320, 400), (2256, 5312, 3856), (3712, 5168, 5456), 41520, (34960, 40080, 55360, 49520, 42240), ...,

10. 0, 495, 6174, (53955, 59994), (61974, 82962, 75933, 63954), (62964, 71973, 83952, 74943), ...,

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.

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