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?
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.