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Kaprekar discovered the Kaprekar constant or $6174$ in $1949$. He showed that $6174$ is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from a set of four digits that are not all identical.

e.g. starting with $1234$, we have $4321 − 1234$ = $3087$, then $8730 − 0378$ = $8352$, and $8532 − 2358$ = $6174$.

But, Why we reach to $6174$ through this process ? I think, subtraction is always divisible by $3$....(not sure)

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    What happens with numbers like $\,1792\,$? Here we get $$9721-1279=8442...$$and the end comes abruptly as I get a number with two equal digits!?2012-08-12
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    @DonAntonio: If you continue your example, you do eventually reach 6174.2012-08-12
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    Sir, if we continue this process,then we will get 1782, which has no digits in common.2012-08-12
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    @ram: Have you studied the cases of 2 and 3 digits first? For 2 digits, one can get cyclic behavior without a fixed point. (E.g. $01 \rightarrow 09 \rightarrow 81 \rightarrow 63 \rightarrow 27 \rightarrow 45 \rightarrow 09 \rightarrow \dots$)2012-08-12
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    @MichaelJoyce, I know: I already did the calculations, but the OP mentions number with 4 *different* digits...2012-08-12
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    Not 4 different, but 4 not all identical, since as you see starting with $nnnn$ immediately goes to $0000$. There are unique Kaprekar constants in 3 and 4 digits, but cyclic "constants" in other lengths-I don't know if it's proven that no larger length has a single fixed point, or results for other bases, but I don't believe there's any general fact that answers the OP's "why."2012-08-12
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    @DonAntonio When the OP says "4 digits which are not all identical", I believe he/she means 4 digits which are not all the same digit.2012-08-12
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    Ah, we know that there are only finitely many Kaprekar's constants in any base (digit lengths that give a unique fixed point under this procedure) and that 495 and 6174 are the only ones in base 10. http://www.emis.ams.org/journals/HOA/IJMMS/2005/182999.pdf2012-08-12
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    With 1112 you go to [0]999 which has a digit sum of 27 (similarly 1113 to 1998 ...)2012-08-12
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    See also http://oeis.org/A099009 and references there.2012-08-12

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