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Kaprekar Algorithm

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

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Kaprekar Algorithm -

Tag(s) : Arithmetics, Algorithm

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# Kaprekar Algorithm

## Calculus through Kaprekar Algorithm

### 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 digitsConstant/Loop
3495
46174
553955, 59994 or 62964, 71973, 83952, 74943 or 61974, 82962, 75933, 63954
6420876, 851742, 750843, 840852, 860832, 862632, 642654 or 631764 or 549945
77509843, 9529641, 8719722, 8649432, 7519743, 8429652, 7619733, 8439552
843208766, 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 (link)

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