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) : Mathematics, 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 \( N \) by sorting them in 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.

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Source : https://www.dcode.fr/kaprekar-algorithm

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