Can anyone share the proof for the following?

Today when I was surfing the internet I stumbled across this video.This was fasinating.

Take any 3 digit number, rearrange the numbers to form the largest number and the smallest number possible.

Subtract the two, repeat the process and eventually you will end at the number 495.

If you do this with any 4 digit number you get 6174.

I really wanted to know the proof of this. Can anybody please share the proof?Please

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`2 \times 3`	$2 \times 3$
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Comments

The algebraic proof is pretty intricate, and I'm too tired for $\LaTeX$ , so here's a table of values.

Ved Pradhan - 11 months, 4 weeks ago

If $a-c=0$ or $a-c=1$ , the values leave 3 digit numbers and approach zero.

For four digit numbers, a similar proof can be used, but it might take more time.

Ved Pradhan - 11 months, 4 weeks ago

Kaprekar's constant

Pi Han Goh - 11 months, 4 weeks ago

I think it is a result by iteration.

Zakir Husain - 11 months, 3 weeks ago

Some number which don't get to $495$ : $111,222,333,444,555,666,777,888,999$

Zakir Husain - 11 months, 3 weeks ago

@Zakir Husain : As you can see if you look in the reply to my comment, the numbers that don't converge to 495 are numbers where the difference between the biggest and smallest numbers are 1 or 0. For the numbers where this is 1 or 0, the sequence converges to 0.

Ved Pradhan - 11 months, 3 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$