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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Tool using the Kaprekar algorithm. It uses an integer N, and arranges its digits in ascending and descending order, before subtracting them.

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.

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.

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 (link)

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