[lotharbot]

Another fixed point of the operation is zero, which is reached in a single step if you start with a number with all the same digits (1111, for example.) Zero is stable and attracting, it just has a very small basin of attraction.

[waterhouse]

If you want to see precisely which numbers lead to which fixed points, and in how many steps, I've posted it here for all 4-digit numbers:

http://pastebin.com/3iXT777N

[sesqu]

Here's a plot: http://imageshack.us/f/192/stepsd.png/

It might be interesting to plot the mapping instead.

edit: one has already been made: http://news.ycombinator.com/item?id=2626879

[ableal]

Thanks, but there's a bug in your code/results - for example, 1011 does not converge to 0, it does hit 6174 in 5 steps. Only 1111, 2222, etc. yield 0.

I suspect the handling of 3-digit values. I also slipped there at first ;-)

[waterhouse]

For 1011, the next step is 1110 - 0111 = 999, and then... apparently you're supposed to handle that as 9990 - 0999, rather than 999 - 999. Huh. I see.

But I'm inclined to consider that a bug in the definition, rather than in the code. :-} The four-digit-ness came from applying the operation to four-digit numbers, and saying "oh yeah and we'll zero-extend the results to four digits" introduces an ugly element of redundancy.

However, we can make it not ugly anymore by making "zero-extend everything to make it be 4 digits" the primary condition. Instead of applying it to 4-digit numbers, we'll make every number have 4 digits and then apply the Kaprekar procedure. So we'll now be working with the numbers 0000-9999, rather than 1000-9999. (There's no way to zero-extend numbers bigger than 9999 to be 4 digits, so that's all.) Here are the results for 0000-9999:

http://pastebin.com/TQ3cMshV

I think the frequencies of the number of steps to equilibrium are somewhat nicer, as well, when you count 0000-0999. As a simple metric, they have more factors of small primes, especially 2.

  1000-9999:
  0 steps:    1 =   1
  1 steps:  365 =   5 *  73
  2 steps:  519 =   3 * 173
  3 steps: 2124 = 2^2 * 3^2 *  59
  4 steps: 1124 = 2^2 * 281
  5 steps: 1379 =   7 * 197
  6 steps: 1508 = 2^2 *  13 *  29
  7 steps: 1980 = 2^2 * 3^2 *   5 *  11

  0000-9999:
  0 steps:    2 =   2
  1 steps:  392 = 2^3 * 7^2
  2 steps:  576 = 2^6 * 3^2
  3 steps: 2400 = 2^5 *   3 * 5^2
  4 steps: 1272 = 2^3 *   3 *  53
  5 steps: 1518 =   2 *   3 *  11 *  23
  6 steps: 1656 = 2^3 * 3^2 *  23
  7 steps: 2184 = 2^3 *   3 *   7 *  13