Kaprekar Routine
The Kaprekar routine is an algorithm 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 
(discarding any initial
0s) and iterate, where 
is sometimes called the Kaprekar
function. The algorithm reaches 0 (a degenerate case), a constant, or a cycle, depending
on the number of digits in 
and the value of
. The list of values is sometimes called a Kaprekar
sequence, and the result 
is sometimes
called a Kaprekar number (Deutsch and Goldman 2004), though this nomenclature should
be deprecated because of confusing with the distinct sort of Kaprekar
number.
In base-10, the numbers 
for which 
are given by 495, 6174, 549945, 631764, ...
(OEIS A099009). Similarly, the numbers 
for which iterating 
gives a cycle
of length 
are given by 53955, 59994, 61974,
62964, 63954, 71973, ... (OEIS A099010).
Iterating the Kaprekar map in base-10, all 1- and 2-digit numbers give 0. Exactly 60 3-digit numbers, namely 100, 101, 110, 111, 112, 121, 122, 211, 212, 221, ... (OEIS A090429), reach 0, while the rest give 495 in at most 6 iterations. Exactly 77 4-digit numbers, namely 1000, 1011, 1101, 1110, 1111, 1112, 1121, 1211, ... (OEIS A069746), reach 0, while the remainder give 6174 in at most 8 iterations. The value 6174 is sometimes known as Kaprekar's constant (Deutsch and Goldman 2004). This pattern breaks down for 5-digit numbers, which may converge to 0 or one of the 10 constants 53955, 59994, 61974, 62964, 63954, 71973, 74943, 75933, 82962, 83952.
The following table summarizes the possible cycles in various bases 
and the first few
numbers of digits.
 | possible
cycles for  , 2, ... base- digits |
| 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) , ... |
The figure above (similar to that appearing on the cover of the above issue of The Mathematics Teacher) shows the number of steps required for the Kaprekar routine
to reach a fixed point for values of 
to 9999, partitioned
into rows of length 100 (Deutsch and Goldman 2004). In this plot, numbers having
fewer than 4 digits are padded with leading 0s, thus resulting in all values converging
to 6174.
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