Sort the digits to 'wxyz', where each letter is a 0-9 digit.

'wxyz' - 'zyxw' = 999(w - z) + 90(x - y)

w - z is between 1 and 9, since w > z (we have ruled out numbers like 1111). x - y is between 0 and 9. So there are at most 90 such numbers. In fact there are fewer because x-y <= w-z.

This is why there are many collisions in the first step.

Numberphile video on the topic: https://youtu.be/d8TRcZklX_Q?si=t9x2HLWYOpPiTbn4

Previous discussion: https://news.ycombinator.com/item?id=39018769

Interestingly, there is a common pattern for fixed-points and cycles for different number lengths: numbers made of 4, 5 and 9. For example

length 3: fixed-point 495 length 5: 2-cycle containing 59994 length 6: fixed-point 59994

Similarly for digits 6, 1, 4, 7:

length 4: 6174 length 5: 4-cycle containing 61974 length 6: fixed-point 631764

No attempts at generalization to larger numbers of digits?

From https://en.wikipedia.org/wiki/D._R._Kaprekar

A similar constant for 3 digits is 495.[7] However, in base 10 a single such constant only exists for numbers of 3 or 4 digits; for other digit lengths or bases other than 10, the Kaprekar's routine algorithm described above may in general terminate in multiple different constants or repeated cycles, depending on the starting value