Tool using the Kaprekar algorithm. It uses an integer N, and arranges its digits in ascending and descending order, before subtracting them.
Kaprekar Algorithm - dCode
Tag(s) : Arithmetics, Algorithm
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Kaprekar Algorithm
Calculus through Kaprekar Algorithm
Answers to Questions (FAQ)
How to calculate Kaprekar sequence?
The Kaprekar routine from a number $ N $ consists of creating 2 other numbers $ N_1 $ and $ N_2 $ by arranging the digits of $ N $ through sorting by ascending order to $ N_1 $ and decreasing for $ N_2 $. Kaprekar then forms a new number $ N $ such that $ N_2-N_1 = N $ and repeats the process until arriving at a previously found number.
Example: $ N = 7533 $, $ N_1 = 3357 $, $ N_2 = 7533 $, replace $ N $ with $ 7533 - 3357 = 4176 $
$ N = 4176 $, $ N_1 = 1467 $, $ N_2 = 7641 $ then replace $ N $ with $ 7641 - 1467 = 6174 $
$ N = 6174 $, $ N_1 = 1467 $, $ N_2 = 7641 $ replace $ N $ with $ 7641 - 1467 = 6174 $, which creates an infinite loop on the constant 6174, which is the Kaprekar constant for 4 digits.
What are Kaprekar constants and Kaprekar loops?
Loops are repetition of values or constants that appears in the algorithm depending on the size of the number $ N $.
| Number of digits | Constant/Loop |
|---|---|
| 3 | 495 |
| 4 | 6174 |
| 5 | 53955, 59994 or 62964, 71973, 83952, 74943 or 61974, 82962, 75933, 63954 |
| 6 | 420876, 851742, 750843, 840852, 860832, 862632, 642654 or 631764 or 549945 |
| 7 | 7509843, 9529641, 8719722, 8649432, 7519743, 8429652, 7619733, 8439552 |
| 8 | 43208766, 85317642, 75308643, 84308652, 86308632, 86326632, 64326654 or 64308654, 83208762, 86526432 or 97508421 or 63317664 |
When one of these numbers is reached, either it remains constant or it follows the cycle by looping to infinity.
How to prove that 6174 is the only 4-digit Kaprekar constant?
The mathematical proof of the existence of 6174 as the only fixed point number of the algorithm is a bit long and consists in enumerating a few possible cases and proving that only one case has no contradiction. The proof: here
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Kaprekar Algorithm on dCode.fr [online website], retrieved on 2024-12-21,
