Kaprekar's Routine

I first encountered the algorithm that Wolfram MathWorld calls Kaprekar's Routine last year and actually made a post about it titled Birth Year Magic. I've created this post because I was reminded of the similarity between Kaprekar's Routine and the Odd-Even Algorithm that I've written about in a number of previous posts:

• Odds and Evens
• Attractors, Vortices and Captives
• Binary Odds and Evens
• Odds and Evens: Statistics

Dattaraya Ramchandra Kaprekar is something of an inspiration for me because our situations are similar academic-wise.

In Kaprekar's Routine, numbers get mapped to what are called fixed points (attractors in my terminology) or end up in loops (vortices in my terminology). Here is what Wolfram MathWorld has to say about the algorithm:

The Kaprekar routine is an algorithm discovered in 1949 by D. R. Kaprekar for 4-digit numbers, but which can be generalised to \(k\)-digit numbers. To apply the Kaprekar routine to a number \(n\), arrange the digits in descending \( n^{'} \) and ascending \( n^{''} \) order. Now compute \( K(n)=n^{'}-n^{''} \) (discarding any initial 0s) and iterate, where \(K(n)\) 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 \(k\) and the value of \(n\). The list of values is sometimes called a Kaprekar sequence, and the result K(n) 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 n for which \(K(n)=n\) are given by 495, 6174, 549945, 631764, ... (OEIS A099009). Similarly, the numbers \(n\) for which iterating \(K(n)\) gives a cycle of length \(k \geq 2\) 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.

I developed an SageMath algorithm (permalink) to determine the behaviour of numbers of any length over a given range. In my earlier post, my algorithms only applied to three and four digit numbers. Here is the behaviour for numbers in the range from 26400 to 26425:

26400 --> [26400, 63954, 61974, 82962, 75933, 63954]

26401 --> [26401, 62964, 71973, 83952, 74943, 62964]

26402 --> [26402, 61974, 82962, 75933, 63954, 61974]

26403 --> [26403, 61974, 82962, 75933, 63954, 61974]

26404 --> [26404, 61974, 82962, 75933, 63954, 61974]

26405 --> [26405, 62964, 71973, 83952, 74943, 62964]

26406 --> [26406, 63954, 61974, 82962, 75933, 63954]

26407 --> [26407, 73953, 63954, 61974, 82962, 75933, 63954]

26408 --> [26408, 83952, 74943, 62964, 71973, 83952]

26409 --> [26409, 93951, 85932, 74943, 62964, 71973, 83952, 74943]

26410 --> [26410, 62964, 71973, 83952, 74943, 62964]

26411 --> [26411, 52965, 70983, 94941, 84942, 73953, 63954, 61974, 82962, 75933, 63954]

26412 --> [26412, 51975, 81972, 85932, 74943, 62964, 71973, 83952, 74943]

26413 --> [26413, 51975, 81972, 85932, 74943, 62964, 71973, 83952, 74943]

26414 --> [26414, 51975, 81972, 85932, 74943, 62964, 71973, 83952, 74943]

26415 --> [26415, 52965, 70983, 94941, 84942, 73953, 63954, 61974, 82962, 75933, 63954]

26416 --> [26416, 53955, 59994, 53955]

26417 --> [26417, 63954, 61974, 82962, 75933, 63954]

26418 --> [26418, 73953, 63954, 61974, 82962, 75933, 63954]

26419 --> [26419, 83952, 74943, 62964, 71973, 83952]

26420 --> [26420, 61974, 82962, 75933, 63954, 61974]

26421 --> [26421, 51975, 81972, 85932, 74943, 62964, 71973, 83952, 74943]

26422 --> [26422, 41976, 82962, 75933, 63954, 61974, 82962]

26423 --> [26423, 41976, 82962, 75933, 63954, 61974, 82962]

26424 --> [26424, 41976, 82962, 75933, 63954, 61974, 82962]

26425 --> [26425, 42966, 71973, 83952, 74943, 62964, 71973]

All the numbers end in a loop of one type or another. Let's compare what happens to today's number (I'm 26402 days old) under the two different algorithms:

• Kaprekar's Routine:
  
  26402 --> [26402, 61974, 82962, 75933, 63954, 61974]
  
  
• Odd Even Algorithm: 
  
  26402 -->[26402, 26388, 26367, 26363, 26355, 26360, 26349]

Kaprekar's Routine causes the number to end up in a loop [61974, 82962, 75933, 63954, 61974] while the Odd Even Algorithm leads to an attractor, or fixed point, 26349. Go back to my post Birth Year Magic to find out more about Kaprekar.